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+{"text":"\\section{Introduction}\n\nPressure-balanced magnetic structures in the form of strong magnetic\nenhancements (humps) and depressions (holes) that are quasi-stationary in\nthe plasma frame, with no or little change in the magnetic field direction,\nare commonly observed in regions of the solar wind and of planetary\nmagnetosheaths with relatively large $\\beta $ and a dominant (generally\nion) temperature in the transverse direction ( see, for instance, \\cite{S11, Genot09b} and\nreferences therein). The origin of these structures is still not fully\nunderstood, but they are usually viewed as nonlinearly saturated states of the\nmirror instability (MI) discovered by Vedenov and Sagdeev \n\\cite{VedenovSagdeev}. It is a kinetic instability whose growth rate\nwas first obtained under the assumption of cold electrons, a regime where\nthe contributions of the parallel electric field $E_{\\Vert }$ can be\nneglected. However, in realistic space plasmas, the electron temperature can\nhardly be ignored \\cite{Stverak08}. The linear theory retaining the electron\ntemperature and its possible anisotropy, in the quasi-hydrodynamic limit\n(which neglects finite Larmor radius corrections), was developed in the case\nof bi-Maxwellian distribution functions by several authors \n\\cite{Stix62}--\\cite{Hell07}. A general estimate of the growth rate \nunder the sole condition that it is small compared with the ion\ngyrofrequency (a condition reflecting  close vicinity to threshold) is\npresented in \\cite{KPS12a}. The instability then develops in\nquasi-perpendicular directions, making the parallel magnetic perturbation\ndominant. This analysis includes in particular regimes with a significant\nelectron temperature anisotropy for which the instability extends beyond the\nion Larmor radius. In the limit where the instability is limited to scales\nlarge compared with the ion Larmor radius, only the leading order\ncontribution in terms of the small parameter $\\gamma \/(|k|_{z}v_{\\|i})$ is\nto be retained in estimating Landau damping, and the growth rate is given by \n\\begin{eqnarray}\n&&\\gamma =\\frac{2}{\\sqrt{\\pi }}\\frac{T_{\\Vert i}}{T_{\\perp i}}\\frac\n|k_{z}|v_{\\Vert i}}{E}\\Big \\{\\Gamma  -\\frac{1}{\\beta _{\\perp }}\\Big (1+\\frac{\\beta _{\\perp }-\\beta\n_{\\Vert }}{2}\\Big )\\frac{k_{z}^{2}}{k_{\\perp }^{2}}  \\nonumber \\\\\n&&\\qquad -\\frac{3}{4(1+\\theta _{\\perp })}\\Big (\\frac{T_{\\perp i}}{T_{\\Vert i\n}-1\\Big )(1+F)k_{\\perp }^{2}r_{L}^{2}\\Big \\},  \\label{growth_rate}\n\\end{eqnarray\nwhere \n\\begin{equation}\n\\Gamma =\\frac{T_{\\perp i}}{T_{\\parallel i}}\\frac{(\\theta _{\\parallel\n}+\\theta _{\\perp })^{2}+2\\theta _{\\parallel }(\\theta _{\\perp }^{2}+1)}\n2\\theta _{\\parallel }(1+\\theta _{\\perp })(\\theta _{\\parallel }+1)}-1-\\frac{\n}{\\beta _{\\perp }}  \\label{newthreshold}\n\\end{equation\nmeasures the distance to threshold and \n\\begin{eqnarray}\nE &=&\\frac{1+\\theta _{\\perp }}{(1+\\theta _{\\parallel })^{2}}\\left[ 2+\\theta _{\\perp\n}(4+\\theta _{\\perp })+\\theta _{\\parallel }^{2}\\right]  \\nonumber \\\\\nF &=&\\frac{T_{\\Vert e}}{T_{\\Vert e}+T_{\\Vert i}}\\Big \\{-1+\\frac{\\theta\n_{\\perp }}{\\theta _{\\Vert }}  \\nonumber \\\\\n&&-\\frac{2}{3}\\frac{T_{\\Vert i}}{T_{\\perp i}}\\Big [\\Big (\\frac{T_{\\Vert i}}\nT_{\\perp i}}-1\\Big )\\frac{1}{\\beta _{\\perp i}}-\\theta _{\\perp }\\Big (\\frac\nT_{\\perp e}}{T_{\\Vert e}}-1\\Big )\\Big ]\\Big \\}.  \\nonumber\n\\end{eqnarray\nHere, ${T_{\\perp \\alpha }}$ and ${T_{\\Vert \\alpha }}$ are the perpendicular\nand parallel (relative to the ambient magnetic field $\\mathbf{B}_{0}$ \ntaken in the $z$ direction)\ntemperatures of the species $\\alpha $ ($\\alpha =i$ for ions and $\\alpha =e$\nfor electrons ), $\\theta _{\\perp }={T_{\\perp e}}\/{T_{\\perp i}}$, $\\theta\n_{\\Vert }={T_{\\Vert e}}\/{T_{\\Vert i}}$ and $\\beta _{\\perp }=\\beta _{\\perp\ni}+\\beta _{\\perp e}$ with $\\beta _{\\perp \\alpha }=8\\pi p{_{\\perp \\alpha }\/B\n_{0}^{2}$ where $p{_{\\perp \\alpha }}$ is the perpendicular thermal pressure\n(similar definition for $\\beta _{\\Vert }$). Furthermore, the parallel\nthermal velocity is defined as $v_{\\Vert \\alpha}=\\sqrt{{2T_{\\Vert \\alpha}}\/{m_{\\alpha}}}$,\nand $r_{L}=({2{T_{\\perp i}\/m_{p})^{1\/2}\/\\Omega _{i}}}\n$ denotes the ion Larmor radius ($\\Omega _{i}=eB_{0}\/m_{i}c$ is the ion\ngyrofrequency).\n\nThe growth rate given by Eq. (\\ref{growth_rate}) has the same structure \nas in the cold electron regime considered  \nin \\cite{Hall79} in the case of bi-Maxwellian ions \nand generalized in \\cite{Pokho05} and \\cite\n{Hell07} to an arbitrary distribution function. The first term within the\ncurly brackets  provides the threshold condition which \ncoincides with that \ngiven in\n\\cite{Stix62}--\\cite{Hall79}. The second one reflects the magnetic field\nline elasticity and the third one (where $F$ depends on the electron\ntemperatures due to  the coupling between the species induced by the\nparallel electric field which is relevant for hot electrons)\nprovides the arrest of the instability at small scales by finite Larmor\nradius (FLR) effects.\n\nAn aim of this letter is to extend to hot electrons the weakly nonlinear\nanalysis  previously developed for cold electrons \\cite{KPS07a,KPS07b}. \nSince in this asymptotics, FLR contributions appear only at the linear\nlevel, the idea is to use the drift kinetic formalism\nto calculate the nonlinear terms. We show that the equation governing\nthe evolution of  weakly nonlinear mirror modes has\nthe same form as in the case of cold electrons. In particular, the sign of the\nnonlinear coupling coefficient that prescribes the shape of\nmirror structures, is not changed. This equation \nis of gradient type equation with a  free\nenergy (or a Lyapunov functional) which is unbounded  from below.\nThis leads to  finite-time blowing-up solutions \\cite{ZK12}, associated with \nthe existence of a subcritical bifurcation \n\\cite{KPS07a,KPS07b}. To describe subcritical stationary mirror\nstructures in the strongly nonlinear regime, we present an anisotropic MHD model \nwhere the perpendicular and parallel pressures are determined\nfrom the drift kinetic equations in the adiabatic\napproximation, in the form of prescribed functions of the magnetic field amplitude.\n\n\\section{Basic equations}\n\nA main condition characterizing mirror modes,  at least near threshold,  is\nprovided by the force balance equation \n\\begin{eqnarray}\n&&-\\nabla \\Big(p_{\\perp }+\\frac{B^{2}}{8\\pi }\\Big)+\\Big[1+ \\frac{4\\pi }{B^{2\n}(p_{\\perp }-p_{\\Vert })\\Big]\\frac{(\\mathbf{B}\\cdot \\nabla) \\mathbf{B}}{4\\pi \n}  \\nonumber \\\\\n&&+ \\mathbf{B}(\\mathbf{B}\\cdot\\nabla) \\Big (\\frac{p_\\perp -p_\\|}{B^2} \\Big )\n-\\nabla \\cdot \\mathbf{\\Pi}=0,  \\label{balance-nl}\n\\end{eqnarray}\nwhere the pressure tensor, viewed as the  the sum of the contributions of the various species,\nhas been written as the sum of a gyrotropic part\ncharacterized by the parallel ($p_{\\Vert }=\\sum_{\\alpha}p_{\\Vert \\alpha }$) and perpendicular \n($p_{\\perp }=\\sum_{\\alpha }p_{\\perp \\alpha }$) pressures, and \nof a gyroviscous contribution $\\mbox{\\boldmath $\\Pi$}$\noriginating from the sole ion FLR effects when concentrating on scales large compared with the\nelectron Larmor radius.  As mentioned above,\nFLR effects  arising only at the linear level with\nrespect to the amplitude of the perturbations, the other linear and nonlinear \ncontributions can be evaluated from the drift kinetic equation for each particle species\n\\begin{equation}\n\\frac{\\partial f_{\\alpha }}{\\partial t}+v_{\\Vert }\\mathbf{b}\\cdot \\nabla\nf_{\\alpha }+\\Big[-\\mu \\mathbf{b}\\cdot \\nabla B+\\frac{e_{\\alpha }}{m_{\\alpha\n} }E_{\\Vert }\\Big]\\frac{\\partial f_{\\alpha }}{\\partial v_{\\Vert }}=0\n\\label{mainkin}\n\\end{equation\n\n\nWe ignore the transverse electric drift which is subdominant for mirror\nmodes. In this approximation, both ions and electrons move in the direction\nof the magnetic field (defined by the unit vector $\\mathbf{b}=\\mathbf{B}\/B$)\nunder the effect of the magnetic force $\\mu \\mathbf{\\ b}\\cdot\\nabla B$ and\nthe parallel electric field $E_{\\Vert }=-\\mathbf{b}\\cdot \\nabla\\phi$ where\nthe magnetic moment $\\mu=v_{\\perp }^{2}\/(2B)$ is an adiabatic invariant\nwhich plays the role of a parameter in Eq. (\\ref{mainkin}). Here $\\phi$ is\nthe electric potential. The quasi-neutrality condition $n_{e}=n_{i}\\equiv n\n, where $n_{\\alpha }=B\\int f_{\\alpha }d\\mu dv_{\\Vert }d\\varphi \\equiv \\int\nf_{\\alpha}d^{3}v$, is used to close the system and eliminate $E_\\|$.\n\nIn this framework where FLR effects are neglected, the gyrotropic pressures \n are given  by \n$p_{\\alpha \\Vert }\\equiv m_{\\alpha }\\int v_{\\Vert }^{2}f_{\\alpha }d^{3}v\n=m_{\\alpha }B\\int v_{\\Vert }^{2}f_{\\alpha }d\\mu dv_{\\Vert}d\\varphi $,  \nand $p_{\\alpha \\perp } \\equiv \\frac{1}{2}m_{\\alpha }\\int v_{\\perp }^{2}f_{\\alpha }d^{3}v\n=m_{\\alpha }B^{2}\\int \\mu f_{\\alpha }d\\mu dv_{\\Vert}d\\varphi$.\n\nThe asymptotic equation governing the mirror dynamics near threshold is obtained by\nexpanding Eqs. (\\ref{balance-nl}), (\\ref{mainkin}) and the quasi-neutrality\ncondition, with  the pressure tensor elements for each species\ncomputed near a bi-Maxwellian equilibrium state characterized by the\ntemperatures $T_{\\perp\\alpha }$ and $T_{\\Vert\\alpha }$.\n\n\\section{Linear instability}\n\nBefore turning to the nonlinear regime, we briefly review the derivation of\nthe MI linear growth rate in the simplified framework provided by the drift\nkinetic approximation which is only valid at scales large enough for FLR\neffects to be subdominant.\n\nLinearizing  Eq. (\\ref{balance-nl}) about the background field $\\mathbf\nB_{0}}$ and equilibrium pressures $p_\\perp^{(0)}$ and  $p_\\|^{(0)}$, \nand considering  perturbations $\\widetilde{\\mathbf{B}}$ and \n$p_\\perp^{(1)}\\propto  e^{-i\\omega t+i\\mathbf{k\\cdot r}}$, we get \n\\begin{equation}\np_{\\perp }^{(1)}+\\frac{B_{0}\\widetilde{B}_{z}}{4\\pi }=-\\frac{k_{z}^{2}}\nk_{\\perp }^{2}}\\Big(1+\\frac{\\beta _{\\perp }-\\beta _{\\Vert }}{2}\\Big)\\frac\nB_{0}\\widetilde{B}_{z}}{4\\pi }.  \\label{pressures-1}\n\\end{equation\nHere, $p_{\\perp }^{(1)}$ has to be calculated from the linearized drift kinetic\nequation\n\\begin{equation}\n\\frac{\\partial f_{\\alpha }^{(1)}}{\\partial t}+v_{\\Vert }\\frac{\\partial\nf_{\\alpha }^{(1)}}{\\partial z}+\\Big [-\\mu \\frac{\\partial \\widetilde{B}_{z}}\n\\partial z}+\\frac{e_{\\alpha }}{m_{\\alpha }}E_{\\Vert }\\Big]\\frac{\\partial\nf_{\\alpha }^{(0)}}{\\partial v_{\\Vert }}=0,  \\label{kin-1}\n\\end{equation\nwhere we assume each $f_{\\alpha }^{(0)}$ to be a bi-Maxwellian distribution\nfunction \n\\begin{equation}\nf_{\\alpha }^{(0)}=A_{\\alpha }\\exp \\Big [-\\frac{v_{\\parallel }^{2}}\nv_{\\parallel \\alpha }^{2}}-\\frac{\\mu B_{0}m_{\\alpha }}{T_{\\perp \\alpha }\n\\Big ],  \\label{bi-maxwell}\n\\end{equation\nwith $A_{\\alpha }~=~n_{0}m_{\\alpha }\/(2\\pi \\sqrt{\\pi }v_{\\parallel \\alpha\n}T_{\\perp \\alpha })$. \n\n\nEquation (\\ref{kin-1}) is solved in Fourier representation, as \n\\begin{equation}\nf_{\\alpha }^{(1)}=-\\frac{\\mu \\widetilde{B}_{z}+\\frac{e_{\\alpha }} {m_{\\alpha\n}}\\phi }{\\omega -k_{z}v_{\\Vert }}k_{z}\\frac{\\partial f_{\\alpha }^{(0)}}{\n\\partial v_{\\Vert }}.  \\label{kin-gen-1}\n\\end{equation}\nThe neutrality condition \nallows one to express the potential $\\phi$ in terms of $\\widetilde{B}\n_{z}$. Indeed, assuming   $\\displaystyle\n\\zeta ={\\sqrt{\\pi }\\omega }\/(|k_{z}|v_{\\parallel i})\\ll 1}$\n(so that the contribution from the Landau pole is small),\n\\begin{equation}\n\\int f_{i}^{(1)}dv_{z}d\\mu d\\varphi =-\\frac{n_{0}}{B_{0}T_{\\parallel i}}\n\\Big[ T_{\\perp i}\\frac{\\widetilde{B}_{z}}{B_{0}}+e\\phi \\Big ] \\Big [ 1+ \ni\\zeta \\Big ].  \\nonumber\n\\end{equation}\nSimilarly, neglecting the electron\nLandau resonance contribution because of the small mass ratio, \n\\begin{equation}\n\\int f_{e}^{(1)}dv_{\\|}d\\mu d\\varphi =-\\frac{n_{0}}{B_{0}T_{\\Vert e}}\\Big[\nT_{\\perp e}\\frac{\\widetilde{B}_{z}}{B_{0}}-e\\phi \\Big].  \\nonumber\n\\end{equation}\nConsequently,\n\\begin{equation}\ne\\phi \\approx \\frac{T_{\\perp i}}{1+\\theta _{\\parallel }}\\Big[(\\theta _{\\perp\n}-\\theta _{\\parallel })-\\frac{\\theta _{\\parallel }(1+\\theta _{\\perp })}{\n1+\\theta _{\\parallel }}i\\zeta \\Big]\\frac{\\widetilde{B}_{z}}{B_{0}}.\n\\label{phi-1}\n\\end{equation\nWe thus recover that for mirror modes, the parallel electric field\nvanishes when the electrons are cold ($\\theta_\\perp = \\theta_\\| = 0$).\nInterestingly, when $\\theta _{\\perp }=\\theta _{\\parallel }$, only the Landau\npole contributes to $\\phi$.\n\nIt is now necessary to evaluate \n\\begin{equation}\np_{\\perp }^{(1)}=2\\frac{\\widetilde{B}_{z}}{B_{0}}p_{\\perp\n}^{(0)}+B_{0}^{2}\\sum_{\\alpha }m_{\\alpha }\\int \\mu f_{\\alpha }^{(1)}d\\mu\ndv_{\\Vert }d\\varphi .  \\nonumber\n\\end{equation\nUsing \n\\begin{eqnarray}\n\\int \\frac{k_{z}v_{\\Vert }}{\\omega -k_{z}v_{\\Vert }}f_{i}^{(0)}d\\mu\ndv_{\\Vert }d\\varphi  &=&-\\frac{n_{0}}{B_{0}}(1+i\\zeta )  \\nonumber \\\\\n\\int \\frac{k_{z}v_{\\Vert }}{\\omega -k_{z}v_{\\Vert }}f_{e}^{(0)}d\\mu\ndv_{\\Vert }d\\varphi  &=&-\\frac{n_{0}}{B_{0}},  \\nonumber\n\\end{eqnarray\nwe get \n\\begin{equation}\np_{\\perp }^{(1)}=-\\beta _{\\perp }\\frac{B_{0}^{2}}{4\\pi }\\Big[\\frac{1}{\\beta\n_{\\perp }}+\\Gamma +\\frac{T_{\\perp i}}{T_{\\Vert i}}\\frac{i\\zeta D}{2(1+\\theta\n_{\\perp })}\\Big]\\frac{\\widetilde{B}_{z}}{B_{0}}.  \\nonumber\n\\end{equation\nSubstituting this expression into the linearized force balance equation\nyields the linear instability growth rate given by Eq. (\\ref{growth_rate}),\nup to the FLR term which is not captured by the drift kinetic approximation.\nNote that the growth rate given by Eq. (\\ref{growth_rate}) is consistent\nwith the applicability condition $\\gamma \/|k_{z}|\\ll v_{{\\Vert }i}$ near\nthreshold ($\\Gamma \\ll 1$), as \n$k_{z}$ and   $(k_{z}\/k_{\\perp })^2$ scale like  $\\Gamma $,  while \n$\\gamma$ like  $\\Gamma ^{2}$. \n\n\\section{General pressure estimates}\n\nAs demonstrated in \\cite{KPS07a,KPS07b}, the scalings resulting from the\nlinear theory near threshold imply an adiabaticity condition to leading order.\nIt is thus enough to consider the stationary kinetic equation \n\\begin{equation}\nv_{\\Vert }\\mathbf{b}\\cdot \\nabla f_{\\alpha }-(\\mathbf{b}\\cdot \\nabla )\\left[\n\\mu B+\\frac{e_{\\alpha }}{m_{\\alpha }}\\phi \\right] \\frac{\\partial f_{\\alpha } \n}{\\partial v_{\\Vert }}=0.  \\label{Vlasov_stat}\n\\end{equation}\nIt turns out that Eq. (\\ref{Vlasov_stat}) is exactly solvable, the\ngeneral solution being an arbitrary function $f_{\\alpha}=g_{\\alpha }(\\mu\n,W_{\\alpha })$ of the particle energy $\\displaystyle{\\ W_{\\alpha }=\n{v_{\\Vert }^{2}}\/{2}+\\mu B+\\frac{e_{\\alpha }}{m_{\\alpha }} \\phi}$, and of $\\mu$.\nTo find the function $g_{\\alpha }(\\mu ,W_{\\alpha })$, we use the\nadiabaticity argument which means that, to leading order, $g_{\\alpha }$ as a\nfunction of $\\mu$ and $W_{\\alpha }$ retains its form during\nthe evolution. Therefore, the function $g_{\\alpha }(\\mu ,W_{\\alpha })$ is\nfound by matching with the initial distribution function $f_{\\alpha }^{(0)}$\ngiven by Eq. (\\ref{bi-maxwell}) which corresponds to $\\phi =0$ and \nW_{\\alpha }=\\frac{v_{\\Vert }^{2}}{2}+\\mu B_{0}$. We get \n\\begin{eqnarray}\n&&g_{\\alpha }(\\mu ,W_{\\alpha })=A_{\\alpha }\\exp \\Big[ -\\frac{v_{\\parallel\n}^{2}}{v_{\\parallel \\alpha }^{2}}-\\frac{\\mu B_{0}m_{\\alpha }}{T_{\\perp\n\\alpha }}\\Big]  \\nonumber \\\\\n&&\\quad=A_{\\alpha }\\exp \\Big [ -\\frac{2W_{\\alpha }} {v_{\\parallel \\alpha}^{2}}+\n\\mu B_{0}m_{\\alpha }\\Big( \\frac{1}{T_{\\parallel \\alpha }}-\\frac{1}{\nT_{\\perp \\alpha }}\\Big) \\Big] .  \\label{g-alpha}\n\\end{eqnarray}\nThus, $g_{\\alpha }(\\mu ,W_{\\alpha })$ is a Boltzmann distribution function\nwith respect to $W_{\\alpha }$ but, at fixed $W_{\\alpha }$, it displays an\nexponential growth relatively to $\\mu $ if $T_{\\perp \\alpha }>$ \nT_{\\parallel \\alpha }$. This effect can however be compensated by the\ndependence of $W_{\\alpha }$ in $\\mu$. This means that only a fraction of the\nphase space $(\\mu ,W_{\\alpha })$ is accessible, a property possibly related\nwith the existence of trapped and untrapped particles.\n\nNote that expanding Eq. (\\ref{g-alpha}) relatively to $\\widetilde{B}_{z}\/B_0$\nand $e\\phi ^{(1)}\/T_{\\perp i}$ reproduces the first order contribution to the\ndistribution function given by Eq. (\\ref{kin-gen-1}) with $\\omega=0$, and also\nthe second order correction found in \\cite{KPS07a,KPS07b} \nin the case of cold electrons. It should be\nemphasized that Eq. (\\ref{g-alpha}) only assumes adiabaticity and remains\nvalid for finite perturbations.\n\nThe function $g_{\\alpha }$ can also be rewritten in terms of $v_{\\Vert }$, \nv_{\\perp }$ and $\\phi $ as \n\\begin{eqnarray}\n&&g_{\\alpha }=A_{\\alpha }\\exp \\Big[-\\frac{m_{\\alpha }v_{\\Vert }^{2}}{\n2T_{\\parallel \\alpha }}-\\frac{e_{\\alpha }\\phi }{T_{\\parallel \\alpha }}\\Big]\n\\times  \\nonumber \\\\\n&&\\qquad \\exp \\left\\{ -\\frac{m_{\\alpha }v_{\\perp }^{2}}{2T_{\\perp \\alpha }}\n\\Big (\\frac{T_{\\perp \\alpha }}{T_{\\parallel \\alpha }}-\\frac{B_{0}}{B}\\Big [ \n\\frac{T_{\\perp \\alpha }}{T_{\\parallel \\alpha }}-1\\Big ]\\Big )\\right\\} , \n\\nonumber\n\\end{eqnarray\nwhich can be viewed as the bi-Maxwellian distribution function with the\nrenormalized transverse temperature \n\\begin{equation}\nT_{\\perp \\alpha }^{(eff)}=T_{\\perp \\alpha }\\left[ \\frac{T_{\\perp \\alpha }}{\nT_{\\parallel \\alpha }}-\\frac{B_{0}}{B}\\Big(\\frac{T_{\\perp \\alpha }}{\nT_{\\parallel \\alpha }}-1\\Big)\\right] ^{-1}.  \\nonumber\n\\end{equation\nNote the Boltzmann factor $\\exp {-[e_{\\alpha }\\phi \/T_{\\parallel \\alpha }]}$\nin the expression of $g_{\\alpha }$. For cold electrons, the ion distribution\nfunction was obtained in \\cite{Const02} by assuming that it \nremains bi-Maxwellian, and owing to the invariance of the kinetic energy and\nof the magnetic moment. This estimate, obtained by neglecting both time\ndependency (and consequently  Landau resonance) and finite Larmor radius\ncorrections, reproduces the closure condition given in \\cite{PRS06}.\n\nAfter rewriting Eq. (\\ref{g-alpha}) in the form \n\\begin{equation}\ng_{\\alpha }=A_{\\alpha }\\exp \\Big[-\\frac{e_{\\alpha }\\phi }{T_{\\parallel\n\\alpha }}-\\frac{v_{\\Vert }^{2}}{v_{\\parallel \\alpha }^{2}}-\\frac{\\mu\nB_{0}m_{\\alpha }}{T_{\\perp \\alpha }}\\Big(1+\\frac{T_{\\perp \\alpha }}{\nT_{\\parallel \\alpha }}\\frac{B-B_{0}}{B_{0}}\\Big)\\Big],  \\nonumber\n\\end{equation}\nthe quasi-neutrality condition gives \n\\begin{eqnarray}\n&&\\left( 1+\\frac{T_{\\perp i}}{T_{\\parallel i}}\\frac{B-B_{0}}{B_{0}}\\right)\n^{-1}\\exp \\left( -\\frac{e\\phi }{T_{\\parallel i}}\\right) =  \\nonumber \\\\\n&&\\left( 1+\\frac{T_{\\perp e}}{T_{\\parallel e}}\\frac{B-B_{0}}{B_{0}}\\right)\n^{-1}\\exp \\left( \\frac{e\\phi }{T_{\\parallel e}}\\right)  \\nonumber\n\\end{eqnarray}\nor \n\\begin{eqnarray}\n&&e\\phi =(T_{\\parallel i}^{-1}+T_{\\parallel e}^{-1})^{-1}\\times  \\nonumber \\\\\n&&\\log \\left[ \\left( 1+\\frac{T_{\\perp e}}{T_{\\parallel e}}\\frac{B-B_{0}}{\nB_{0}}\\right) \\left( 1+\\frac{T_{\\perp i}}{T_{\\parallel i}}\\frac{B-B_{0}}{\nB_{0}}\\right) ^{-1}\\right] .  \\label{potential}\n\\end{eqnarray\nInterestingly, the electron density (and thus also that of the ions)\n\\begin{equation}\nn_{e}=n_{0}\\frac{B}{B_{0}}\\left( 1+\\frac{T_{\\perp e}}{T_{\\parallel e}}\\frac{\nB-B_{0}}{B_{0}}\\right) ^{-1}\\exp \\left[ \\frac{e\\phi }{T_{\\parallel e}}\\right]\n\\nonumber\n\\end{equation\nhas the usual Boltzmann factor $\\exp \\left[ e\\phi \/T_{\\parallel e}\\right] $\nand also an algebraic prefactor depending on the magnetic field $B$. In the\ncase of isotropic electron temperature ($T_{\\perp e}=T_{\\parallel e}\\equiv\nT_{e}$), the electron density has the usual Boltzmann form $n_{e}=n_{0}\\exp\n\\left[ e\\phi \/T_{e}\\right] $.\n\nEquation (\\ref{potential}) shows that the potential vanishes in two\ncases: for cold electrons and also when electron and ion temperature anisotropies \n$a_e$ and $a_i$ (with $a_\\alpha = T_{\\perp \\alpha} \/T_{\\|\\alpha}$) \nare equal, a case considered in the linear theory of the\nmirror instability \\cite{Stix62,hasegawa,Hall79}.\n\nIn order  to evaluate explicitly the perpendicular pressure for each species \n\\begin{eqnarray}\n&&p_{\\perp \\alpha }=m_{\\alpha }B^{2}\\int \\mu g_{\\alpha }d\\mu dv_{\\Vert\n}d\\varphi   \\nonumber \\\\\n&&\\ \\ =n_{0}T_{\\perp \\alpha }\\frac{B^{2}}{B_{0}^{2}}\\Big(1+\\frac{T_{\\perp\n\\alpha }}{T_{\\parallel \\alpha }}\\frac{B-B_{0}}{B_{0}}\\Big)^{-2}\\exp \\Big(\n\\frac{e_{\\alpha }\\phi }{T_{\\parallel \\alpha }}\\Big),  \\nonumber\n\\end{eqnarray\nwhere $e\\phi $ is given by Eq. (\\ref{potential}), it is convenient to\nintroduce the functions \n\\begin{eqnarray}\nS_{\\perp i}(u) &=&\\left( \\frac{1+u}{1+a_{i}u}\\right) ^{2}\\left( \\frac\n1+a_{i}u}{1+a_{e}u}\\right) ^{c_{i}}  \\label{Sperpi} \\\\\nS_{\\perp e}(u) &=&\\left( \\frac{1+u}{1+a_{e}u}\\right) ^{2}\\left( \\frac\n1+a_{e}u}{1+a_{i}u}\\right) ^{c_{e}},  \\label{Sperpe}\n\\end{eqnarray\nwith the notations $u=(B-B_{0})\/B_{0}$ and  $c_{\\alpha }=T_{\\parallel \\alpha\n}^{-1}\/(T_{\\parallel i}^{-1}+T_{\\parallel e}^{-1})$. The two latter\nfunctions transform one into the other by exchanging the subscripts $i$ and \ne$. The ion and electron perpendicular pressures are then written as \np_{\\perp \\alpha }=n_{0}T_{\\perp \\alpha }S_{\\perp \\alpha }(u)$. In the\nspecial case of cold electrons, \n\\begin{equation}\np_{\\perp }=n_{0}T_{\\perp i}\\frac{B^{2}}{B_{0}^{2}}\\Big(1+\\frac{T_{\\perp i}}\nT_{\\parallel i}}\\frac{B-B_{0}}{B_{0}}\\Big)^{-2},  \\label{p_perb-B}\n\\end{equation\nwhich is algebraic relatively to $B$. From  this expression \nas well as from the general formula for  $p_{\\perp }=p_{\\perp i}+p_{\\perp e}$\ngiven by Eqs. (\\ref{Sperpi}) and (\\ref{Sperpe}) it follows that the perpendicular\nand  magnetic pressures are anticorrelated. When $B$  increases (decreases), \nthe ratio of the perpendicular to the magnetic \npressure, i.e. the local $\\beta _{\\perp }$,\ndecreases (increases), which  corresponds to a reduction (an increase)\nof the distance to threshold. This implies that the instability \ncannot  saturate at  small amplitudes. \n\nSimilarly, for the parallel pressure, we have \n\\begin{equation}\np_{_{\\parallel \\alpha }}=n_{0}T_{_{\\parallel \\alpha }}\\frac{B}{B_{0}}\\Big(1\n\\frac{T_{\\perp \\alpha }}{T_{\\parallel \\alpha }}\\frac{B-B_{0}}{B_{0}}\\Big\n^{-1}\\exp \\Big(-\\frac{e_{\\alpha }\\phi }{T_{\\parallel \\alpha }}\\Big). \n\\nonumber\n\\end{equation\nthat rewrites $p_{\\Vert \\alpha} =n_{0}T_{\\Vert \\alpha}S_{\\Vert \\alpha}(u)$ with \n\\begin{eqnarray}\nS_{\\| i}(u) &=&\\left (\\frac{1+u}{1+a _{i}u}\\right ) \\left (\\frac{1+a _{i}u}\n1+a _{e}u}\\right)^{c_i}  \\label{Sparali} \\\\\nS_{\\| e}(u) &=&\\left(\\frac{1+u}{1+a _{e}u}\\right) \\left (\\frac{1+a _{e}u}{\n1+a _{i}u}\\right)^{c_e}.  \\label{Sparale}\n\\end{eqnarray\n\n\\section{ The weakly nonlinear regime}\n\nAs it follows from Eq. (\\ref{pressures-1}), in the linear regime near \nthreshold, the fluctuations of perpendicular and magnetic\npressures  almost compensate each other. \nIn the weakly nonlinear regime, the second order\ncorrection to the total (perpendicular plus magnetic) pressure is thus relevant\nand leads to a local shift of $\\Gamma$. To find this correction, we consider  the\nexpansions of  the perpendicular pressures of the ions and electrons in the \n$u$ variable. Because of the symmetry between the functions $S_{\\perp\ni}(u)$ and $S_{\\perp e}(u)$, it is enough to consider the expansion \n\\begin{eqnarray}\nS_{\\perp i}(u) &=&1+u\\Big (2-2a _{i}-c_{i}(a _{e}-a _{i})\\Big ) \\nonumber \\\\\n&&+u^{2}\\Big [c_{i}\\Big (a _{e}a _{i}-a _{i}^{2}+\\frac{1}{2}(a_{e}-a _{i})^{2}\\Big)  \n\\nonumber \\\\\n&&-4a _{i}+3a _{i}^{2}+\\frac{1}{2}c_{i}^{2}(a _{e}-a\n_{i})^{2}-2c_{i}(a _{e}-a _{i})  \\nonumber \\\\\n&&+2\\alpha _{i}c_{i}(a _{e}-a _{i})+1\\Big ]+O\\left( u^{3}\\right)  \\nonumber\n\\end{eqnarray\nAs a result, the second order contributions to the  perpendicular ion  pressure\nis  given by \n\\begin{eqnarray}\n&&p_{i\\perp }^{(2)}=n_{0}T_{\\perp i}\\Big \n3a _{i}^{2}-4a _{i}+1+c_{i}(a _{e}-a _{i})  \\nonumber \\\\\n&&\\qquad \\times \\Big(\\frac{1}{2}(c_{i}+1)(a _{e}-a _{i})-2+3a _{i\n\\Big)\\Big ]u^2,  \\nonumber\n\\end{eqnarray\nwith an analogous formula for the perpendicular electron pressure, \nobtained by exchanging the $i$ and $e$ indices.\n   Furthermore,  the threshold condition rewrites\n\\begin{eqnarray}\n&&\\frac{B_{0}^{2}}{4\\pi }+n_{0}\\Big \\{T_{\\perp i}\\left[ 2-2a\n_{i}-c_{i}\\left( a _{e}-a _{i}\\right) \\right]   \\nonumber \\\\\n&&\\qquad +T_{\\perp e}\\left[ 2-2a _{e}+c_{e}\\left( a _{e}-a\n_{i}\\right) \\right] \\Big \\}=0.  \\label{new-threshold}\n\\end{eqnarray\nThe quadratic contributions to the pressure balance (\\ref{balance-nl}), \noriginating from $p_{i\\perp }^{(2)}+p_{e\\perp\n}^{(2)}+\\left( B-B_{0}\\right) ^{2}\/(8\\pi )$, are collected in a  term \n $\\Lambda \\left( \\frac{B-B_{0}}{B_{0}}\\right) ^{2}$ with \n\\begin{eqnarray}\n\\Lambda  &=&n_{0}\\Big \\{T_{\\perp i}\\Big (3a _{i}^{2}-4a _{i}+1 \n\\nonumber \\\\\n&&+c_{i}(a _{e}-a _{i})\\Big [\\frac{1}{2}(1+c_{i})(a _{e}-a\n_{i})-2+3a _{i}\\Big ]\\Big )  \\nonumber \\\\\n&&+T_{\\perp e}\\Big (3a _{e}^{2}-4a _{e}+1+c_{e}(a _{e}-a _{i}) \n\\nonumber \\\\\n&&\\times \\Big [\\frac{1}{2}(1+c_{e})(a _{e}-a _{i})+2-3a _{e}\\Big \n\\Big )\\Big \\}+\\frac{B_{0}^{2}}{8\\pi }.  \\label{eqlambda}\n\\end{eqnarray\nThe value $\\Lambda _{c}$ of  $\\Lambda $ at\nthreshold is obtained  by expressing ${B_{0}^{2}}\/{8\\pi }$ by\nmeans of Eq.  (\\ref{new-threshold}), which gives \n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}\\Big \\{T_{\\perp i}\\Big [3a _{i}^{2}-4a _{i}+1 \n\\nonumber \\\\\n&&+c_{i}(a _{e}-a _{i})\\Big (\\frac{1}{2}(1+c_{i})(a _{e}-a\n_{i})-2+3a _{i}\\Big )  \\nonumber \\\\\n&&-\\frac{1}{2}\\Big (2-2a _{i}-c_{i}(a _{e}-a _{i})\\Big )\\Big ] \n\\nonumber \\\\\n&&+T_{\\perp e}\\Big [3a _{e}^{2}-4a _{e}+1+c_{e}(a _{e}-a _{i}) \n\\nonumber \\\\\n&&\\times \\Big (\\frac{1}{2}(1+c_{e})(a _{e}-a _{i})+2-3a _{e}] \n\\nonumber \\\\\n&&-\\frac{1}{2}\\Big (2-2a _{e}+c_{e}(a _{e}-a _{i})\\Big )\\Big ]\\Big \\}. \\nonumber\n\\end{eqnarray\nAfter some algebra, defining $\\lambda _{c}=\\Lambda _{c}\/(n_{0}T_{\\perp i})$, one gets \n\\begin{eqnarray}\n&&\\frac{\\lambda _{c}}{\\alpha _{i}}=\\frac{T_{\\perp i}}{T_{\\parallel i}}\\Big \n3+3\\frac{\\theta _{\\perp }^{3}}{\\theta _{\\parallel }^{2}}-\\frac{1}{2}\\frac\n\\left( \\theta _{\\perp }-\\theta _{\\parallel }\\right) ^{2}}{\\theta _{\\parallel\n}^{2}\\left( 1+\\theta _{\\parallel }\\right) ^{2}}  \\nonumber \\\\\n&&\\qquad \\times \\left( 4\\theta _{\\perp }+4\\theta _{\\parallel\n}^{2}+\\allowbreak 5\\left( \\theta _{\\perp }+1\\right) \\theta _{\\parallel\n}\\right) \\Big]  \\nonumber \\\\\n&&\\qquad -\\frac{3}{2\\theta _{\\parallel }\\left( 1+\\theta _{\\parallel }\\right) \n}\\left[ \\left( \\theta _{\\perp }+\\theta _{\\parallel }\\right) ^{2}+2\\theta\n_{\\parallel }(1+\\theta _{\\perp }^{2})\\right]. \\label{eqlambda_c}\n\\end{eqnarray}\n\nProceeding as in \\cite{KPS07a}, retaining the contribution of the \nabove  quadratic terms to the pressure balance, leads one to supplement \na nonlinear contribution to  Eq. (\\ref{growth_rate}) that becomes \n\\begin{eqnarray}\n&&\\frac{\\partial u}{\\partial t}=\\frac{2}{\\sqrt{\\pi }}\\frac{T_{\\Vert i}}\nT_{\\perp i}}\\frac{v_{\\Vert i}}{D}{\\widehat{{\\cal K}_z }}\n\\Big \\{ \\Gamma u -\\frac{\\chi}{\\beta _{\\perp }}(\\Delta_\\perp )^{-1}\\partial _{zz}u \\nonumber \\\\\n&& +\\frac{3}{4}\\Big (\\frac{T_{\\perp i}}{T_{\\Vert i}}-1\\Big \n\\frac{1+F}{1+\\theta _{\\perp }}r_{L}^{2}\\Delta _{\\perp }u\n-\\frac{\\lambda _{c}}{2(1+\\theta _{\\perp })}u^2 \\Big \\}\\label{NL}\n\\end{eqnarray\nHere the integral operator ${\\widehat{{\\cal K}}_z}$ reduces in Fourier representation\nto $|k_z|$ and  $\\chi = 1+\\frac{\\beta_{\\perp }-\\beta _{\\Vert }}{2}$. Furthermore, within the present\napproximation, $u$ coincides with ${\\widetilde B}_z\/B_0$.\n\n\nEquation (\\ref{NL}) extends the result of \\cite{KPS07a, Calif08} valid for \ncold electrons. As in the latter case,  this equation is a gradient type equation,\n\\[\n\\frac{\\partial u}{\\partial t}=-{\\widehat{{\\cal K}_z }}\\frac{\\delta F}{\\delta u},\n\\]\nfor which the free\nenergy (written in dimensionless variables)\n\\[\nF=\\int \\left \\{  \\frac 12 \\left [ -\\Gamma u^2 +(\\partial_z u)^2 +u\\Delta_\\perp ^{-1}\\partial _{zz}u\\right ] +\\frac 13\\lambda _{c}u^3\\right \\} d{\\bf r}\n\\] \nis  unbounded  from below due to the integral $\\int \\lambda _{c}u^3 d{\\bf r}$.\nThis leads to a blow-up behavior, associated with a subcritical bifurcation \\cit\n{KPS07a}, \\cite{KPS07b}. Saturation at large values of the amplitude \nand formation of stationary structures requires additional nonlinear\neffects such as the influence of resonant particles on the nonlinear coupling \\cite{PSKH09}.\nEquation (\\ref{NL}) that does not include saturation processes is not suitable to address the \nquestion of the reduction of the temperature anisotropy by the development of the mirror\ninstability mentioned in \\cite{VedenovSagdeev}. This effect is reproduced by the \nquasi-linear theory \\cite{SS64}, and was also studied in the context of the \nso-called FLR-Landau fluid model\nthat, like the present asymptotics, retains a linear description of the Landau resonance and \nof FLR effects, but includes all the hydrodynamic nonlinearities and does not a priori\nprescribe a pressure balance condition. It was  observed in this case \nthat during the saturation phase, the mean temperatures rapidly evolve in a way as to   \nreduce the distance to threshold \\cite{Borgogno07}.\n\nAs demonstrated in \\cite{KPS07a,KPS07b}, \nthe sign of the nonlinear coupling $\\lambda _{c}$ \ndefines the type of the mirror structures, namely holes ($\\lambda _{c}>0$) or \nhumps ($\\lambda _{c}<0$), near threshold. This sign is strongly dependent on the\nequilibium distribution function \\cite{HKP09} . It is nevertheless of interest to consider the case\nwhere both  ions and electrons have a bi-Maxwellian distribution function.\nIt turns out that the sign of $\\lambda _{c}$\ncan then be determined analytically in a few special cases.\n\n\\noindent\n{ \\it (i) Limit $\\protect\\theta_\\| \\ll \\protect\\theta_\\perp$:}\n\\begin{equation}\n\\frac{\\Lambda _{c}}{n_{0}T_{\\perp i}a _{i}}=\\frac{\\theta _{\\perp }^{2}}\n\\theta _{\\parallel }}\\left( \\frac{T_{\\perp e}}{T_{\\parallel e}}-\\frac{3}{2\n\\right) >0.\\nonumber\n\\end{equation}\n\n\\noindent\n{\\it (ii) Equal anisotropies ($\\protect\\theta _{\\perp}=\\protect\\theta_{\\parallel }$)}\n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}(T_{\\perp i}+T_{\\perp e})\\left( 3a ^{2}-4a\n+1\\right)   \\nonumber \\\\\n&&-n_{0}(T_{\\perp i}+T_{\\perp e})\\left( 1-a \\right) =3a \\frac{B_{0}^{2\n}{8\\pi }>0.\\nonumber\n\\end{eqnarray}\n\n\\noindent\n{\\it (iii) Isotropic electron temperature:}\nThe coefficient\n$\\Lambda _{c}$ can be rewritten in the form \n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}(a _{i}-1)\\{T_{\\perp i}\\Big((3a _{i}-1)  \\nonumber\n\\\\\n&&+c_{i}\\Big[\\frac{1}{2}(1+c_{i})\\left( \\alpha _{i}-1\\right) +2-3a _{i\n\\Big]\\Big )  \\nonumber \\\\\n&&+T_{e}c_{e}\\Big[\\frac{1}{2}\\left( 1+c_{e}\\right) \\left( a _{i}-1\\right)\n+1\\Big]\\}+\\frac{B_{0}^{2}}{8\\pi }.  \\nonumber\n\\end{eqnarray\n   Furthermore, at  threshold, \n\\begin{equation}\n\\frac{1}{2}n_{0}(a _{i}-1)\\left[ T_{\\perp i}\\left( 2-c_{i}\\right)\n+T_{\\perp e}c_{e}\\right] =\\frac{B_{0}^{2}}{8\\pi }>0.\\nonumber\n\\end{equation\nHence, we simultaneously have two inequalities $a _{i}>1$ and $T_{\\perp\ne}c_{e}>T_{\\perp i}(c_{i}-2)$. Therefore, \n\\begin{eqnarray}\n\\Lambda _{c} &=&n_{0}(a _{i}-1)\\Big \\{T_{\\perp i}\\Big((3a _{i}-1) \n\\nonumber \\\\\n&&+c_{i}\\Big [\\frac{1}{2}(1+c_{i})\\Big(a _{i}-1\\Big)+2-3a _{i}\\Big\n\\Big)  \\nonumber \\\\\n&&+T_{e}c_{e}\\Big[\\frac{1}{2}(1+c_{e})(a _{i}-1)+1\\Big ]\\}  \\nonumber \\\\\n&&+\\frac{1}{2}n_{0}(a _{i}-1)\\left[ T_{\\perp i}\\left( 2-c_{i}\\right)\n+T_{\\perp e}c_{e}\\right]   \\nonumber \\\\\n&=&n_{0}(a _{i}-1)\\Big \\{T_{\\perp i}\\Big(3a _{i}(1-c_{i})  \\nonumber \\\\\n&&+c_{i}\\Big[\\frac{1}{2}(1+c_{i})\\left( a _{i}-1\\right) +\\frac{3}{2}\\Big\n\\Big)  \\nonumber \\\\\n&&+T_{e}c_{e}\\Big[2+\\frac{1}{2}\\left( 1+c_{e}\\right) \\left( a\n_{i}-1\\right) \\Big]\\Big \\},\\nonumber\n\\end{eqnarray\nwhich is positive, because $1-c_{i}= c_{e}=\n(1+\\theta _{\\parallel })^{-1}>0$ and $a _{i}>1$.\n\n\n\\begin{figure}[t]\n\\centerline{\n\\includegraphics[width=0.46\\textwidth]{fig1.eps}\n}\n\\caption{Fig. 1. Variation with $\\protect\\theta_\\|$ of the distance to\nthreshold $\\Gamma$ given by Eq. (\\protect\\ref{newthreshold}) (dashed line)\nand of the normalized nonlinear coupling coefficient $\\protect\\lambda$\n(solid line) evaluated from Eq. (\\protect\\ref{eqlambda}) for $\\protect\\theta_{\\perp}=1$\n, $\\protect a_i=1.1$ and $\\protect\\beta_{\\perp i} = 10$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centerline{\n\\includegraphics[width=0.46\\textwidth]{fig2.eps}\n}\n\\caption{Fig. 2. Variation with $\\protect\\beta_{\\perp i}$ of the minimum \n\\mathrm{min}\\,( \\protect\\lambda_c)$ of the normalized nonlinear coupling\ncoefficient taken in an interval of values of $a_p$ between $0$ and $a_{p1}\n\\protect\\beta_{\\perp i})$, defined such that the threshold is obtained for a\nvalue of $\\protect\\theta_\\|$ equal to $100$, for $\\protect\\theta_\\perp= 0.2$\n(solid line), $\\protect\\theta_\\perp=1$ (dotted line) and $\\protect\\thet\n_\\perp=5$ (dashed line).}\n\\label{fig2}\n\\end{figure}\n\n\n\\noindent\n{\\it (iv)  More general conditions:}\nA numerical approach was used in this case. Figure  1 displays, for typical values \nof the parameters (taken\nhere as $\\theta _{\\perp }=1$, $a _{i}=1.1$ and $\\beta _{\\perp i}=10$), the\ndistance to threshold $\\Gamma $ (dashed line) given by Eq. (\\ref\n{newthreshold}) and the non-dimensional nonlinear coupling coefficient \n\\lambda =\\Lambda \/(n_{0}T_{\\perp i})$ (solid line), where $\\Lambda $ is given by\nEq. (\\ref{eqlambda}), as a function of $\\theta _{\\Vert }$. This graph is\ntypical of the general behavior of these functions and shows that they are\nboth decreasing as $\\theta _{\\Vert }$ increases, with $\\lambda $ possibly\nreaching negative values, but only below threshold. In order to show that\nthe value $\\lambda _{c}$, given by Eq. (\\ref{eqlambda_c}), of $\\lambda $ \nat threshold is positive in a wider range of parameters, we display in Fig.\n2, as a function of $\\beta _{\\perp i}$ for $\\theta _{\\perp }=0.2$ (solid\nline), $\\theta _{\\perp }=1$ (dotted line) and $\\theta _{\\perp }=5$ (dashed\nline), the quantity $\\mathrm{{min}\\,(\\lambda _{c})}$ obtained after\nminimizing $\\lambda _{c}$ in an interval of values of $a_{p}$ between $0$\nand $a_{p1}(\\beta _{\\perp i})$. The latter quantity is arbitrarily defined\nsuch that the threshold is obtained for a value of $\\theta _{\\Vert }$ equal\nto $100$. This graph shows that $\\mathrm{min}(\\lambda _{c})$ varies little\nwith $\\theta _{\\perp }$ but is very sensitive to $\\beta _{\\perp i}$. As the\nlatter parameter is increased, $\\mathrm{{min}\\,(\\lambda _{c})}$ decreases\nbut remains always positive. Although \nthis numerical observation is  \nnot a rigorous proof, it convincingly shows that $\\Lambda>0 $\nin the parameter range  of physical interest.\n\n\\section{Stationary nonlinear structures}\n\nSubstituting the explicit expressions of the gyrotropic pressures\nin terms of the magnetic field amplitude\ngiven in the Section IV, within  the equation\nfor the balance of forces \n\\begin{eqnarray}\n&&-\\nabla \\Big( p_{\\perp }+\\frac{B^{2}}{8\\pi }\\Big) +\\Big[ 1+\\frac{4\\pi } \nB^{2}}( p_{\\perp }-p_{\\Vert }) \\Big] \\frac{(\\mathbf{B}\\cdot \\nabla )\\mathbf{B\n}{4\\pi }  \\nonumber \\\\\n&&\\qquad + \\mathbf{B}(\\mathbf{B}\\cdot\\nabla) \\Big (\\frac{p_\\perp -p_\\|}{B^2} \\Big ) \\, \n =0, \\label{equil_noFLR}\n\\end{eqnarray}\nleads to a closed system that seems overdetermined due to the divergenceless condition \n$\\nabla \\cdot {\\bf B}=0$. In fact, it can be checked, after some algebra\nusing the explicit expressions  (\\ref{Sperpi},\\ref{Sperpe}) and (\\ref{Sparali},\\ref{Sparale}),   \nthat the projection of Eq. \n(\\ref{equil_noFLR}) on the magnetic field vanishes identically, thus reducing the \nsystem to three equations for three unknowns.\nThese equations  can be useful  for finding, possibly numerically,  \nstationary profiles of three-dimensional finite-amplitude \nstationary mirror structures. Note that Eq. (\\ref{equil_noFLR}) differs from the \nGrad-Shafranov equation  \\cite{grad,shafra} in  that the parallel and perpendicular\npressures are here prescribed functions of the magnetic field amplitude.\nA main issue concerns the existence of \nstable subcritical solutions, a question that is beyond the scope of this \nletter and will be addressed in forthcoming works. \nSuch structures are reported by satellite observations \\cite{SLD07,Genot09} \nand are also expected from the subcritical character of the\nmirror instability \\cite{KPS07b}. Equilibrium solutions\nwere computed  in one-space dimension in \n\\cite{PRS06}, where they lead to discontinuous profiles. Their regularization would \nrequire that FLR corrections be retained. These  additional contributions are known \nfrom the linear kinetic theory but their extension to the finite-amplitude\ncase remains a challenging problem. \n\n\nThis work was supported by the CNRS PICS programme 6073 and RFBR grant\n12-02-91062-CNRS\\_a. T.P. and P.L.S. benefited from support from INSU-CNRS\n Programme National PNST. The work of\nE.K. was also supported by the RAS Presidium Program \"Fundamental problems\nof nonlinear dynamics in mathematical and physical sciences\",  Grant NSh\n7550.2006.2 and by the French Minist\\`ere de l'Enseignement Sup\\'erieur \net de la Recherche.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPoint cloud registration is a fundamental task in the fields of 3D computer vision, robotics, photogrammetry and remote sensing. It aims to align two point clouds by estimating the optimal rigid transformation (including the rotation and translation) between them, and has been broadly applied in 3D reconstruction~\\cite{henry2012rgb,choi2015robust,zhang2015visual}, object recognition and localization~\\cite{drost2010model,zeng2017multi,wong2017segicp,marion2018label}, SLAM~\\cite{zhang2014loam}, medical imaging~\\cite{audette2000algorithmic}, archaeology~\\cite{chase2012geospatial}, etc.\n\nThough Iterative Closet Point (ICP)~\\cite{besl1992method} is a popular registration method, its performance is still limited since it highly relies on the initial guess provided by users and is prone to converge to the local minima when the initialization is not good enough. To circumvent the need of initial guess, people tend to use 3D keypoint detecting and matching techniques (e.g. FPFH~\\cite{rusu2009fast}, ISS~\\cite{zhong2009intrinsic}, 3DSmoothNet~\\cite{gojcic2019perfect}) to establish correspondences between the point clouds, and then estimate the transformation based on these correspondences. However, 3D keypoint matching is much less robust and accurate than 2D keypoint matching (e.g. SIFT~\\cite{lowe2004distinctive}, SURF~\\cite{bay2006surf}), so it may easily generate a huge number of false matches, usually called \\textit{outliers}, among the putative correspondences. Consequently, robust estimation methods must be adopted to solve the correct transformation from the potentially abundant outliers.\n\nUnfortunately, many existing robust methods have their own nonnegligible limitations. RANSAC~\\cite{fischler1981random} is a well-known hypothesis-and-test consensus maximization paradigm for robust estimation, but it has exponentially growing computational cost w.r.t. the outlier ratio, thus unsuitable for handling high-outlier situations. Note that an outlier ratio of over 95\\% is common after 3D keypoint matching in real scenes~\\cite{bustos2017guaranteed}, so RANSAC is not a generally practical option. Branch-and-Bound (BnB)~\\cite{parra2014fast,horst2013global}, as another famous robust estimator, can yield the globally optimal solution, but it also suffers from the worst-case exponential runtime w.r.t. the problem size. Other robust methods include the non-minimal global solvers such as FGR~\\cite{zhou2016fast}, GNC~\\cite{yang2020graduated} and ADAPT~\\cite{tzoumas2019outlier} which are limited in robustness and generally cannot tolerate outlier ratios higher than 90\\%, the guaranteed outlier removal method GORE~\\cite{bustos2017guaranteed} which may also be too slow for use due to its probable internal use of BnB, and the certifiably optimal solver TEASER~\\cite{yang2019polynomial,yang2020teaser} which is also slow without using parallelism programming. Hence, we can see that almost all of these methods have limited performance in some ways.\n\nTherefore, our goal in this paper is to propose a new robust estimation approach, which can handle registration problems with high or extreme outliers in a time-efficient way.\n\n\\textbf{Our Contributions.} We present a specialized consensus maximization method to realize rapid robust estimation for point cloud registration with even extremely high outliers (e.g. up to 99\\%). \n\nFirst, we abandon the traditional time-consuming single-layered three-point sampling framework used in RANSAC, and present a more efficient strategy by smartly decomposing the three-point layer into: (i) a one-point sampling layer that serves as a raw outlier `filter' on the basis of the 3D-geometric rigidity constraint to diminish the correspondence size, and (ii) a two-point sampling layer that performs random sampling and minimal model estimating. This strategy can significantly reduce the computational cost for obtaining an all-inlier subset from the random samples.\n\nMoreover, for faster consensus maximization, we propose a compatibility-based consensus building strategy by introducing a novel stratified element-wise compatibility checking technique that is merely made up of very simple calculations and boolean conditions, in order to effectively cut down the time cost for the repetitive construction of consensus set as in RANSAC. \n\nThese two contributions lead to our robust solver DANIEL (Double-layered sAmpliNg with consensus maximization based on stratIfied Element-wise compatibiLity). Comprehensive experimental evaluation on various real-world datasets shows that the proposed solver is highly robust against over 99\\% outliers and is also rather fast in practice (running within 3 seconds for solving a 99\\%-outlier registration problem), outperforming existing state-of-the-art robust registration solvers.\n\n\n\n\\section{Related Work}\n\nThis section provides brief reviews on addressing the point cloud registration problem with correspondences.\n\nBefore the application of robust solvers, correspondences have to be first established between the point clouds. 3D keypoint detecting and matching with feature descriptors~\\cite{rusu2009fast,zhong2009intrinsic,zhong2009intrinsic} is a widely-used process for building the correspondences, based on which registration solvers are able to estimate the best transformation.\n\nIn the most ideal case, assuming that there is no outlier among these putative correspondences, closed-form estimators can efficiently solve the optimal transformation based on eigenvector~\\cite{horn1987closed} or Singular Value Decomposition (SVD)~\\cite{arun1987least}. More recent methods include the optimal BnB solver~\\cite{olsson2008branch} and the certifiable Semi-Definite relaxation method~\\cite{briales2017convex}. \n\nHowever, it is known to all that 3D keypoint matching is less robust than its 2D counterpart such as SIFT~\\cite{lowe2004distinctive} or SURF~\\cite{bay2006surf} and could easily generate outliers to corrupt the performance of these outlier-intolerant solvers in practical use. Even worse, the outliers may sometimes occupy a massive majority of the correspondences (as shown in~\\cite{bustos2017guaranteed} and Section~\\ref{Experiments}), making inliers (correct matches) fairly sparse. In this case, we need to apply robust estimation to reject outliers and find the true inliers to make reasonable estimates, where the robustness as well as efficiency of the robust methods could determine the registration results to a great extent.\n\nWe introduce several types of robust solvers as follows.\n\n\\subsection{Consensus Maximization}\n\nConsensus maximization consists in finding a model that can maximize the number of correspondences with residual errors lower than a certain inlier threshold w.r.t. this model. RANSAC~\\cite{fischler1981random} is a very common consensus maximization paradigm via the hypothesize-and-test model-fitting framework. Besides, further techniques, including local optimization~\\cite{chum2003locally,lebeda2012fixing} correspondence sorting~\\cite{chum2005matching}, etc, have been applied to improve the performance of RANSAC. However, the time cost of these RANSAC solvers generally increases exponentially with the outlier ratio. For instance, RANSAC should require more than 4.6$\\times$10$^6$ samples to select one all-inlier subset with 0.99 confidence if the outlier ratio is 99\\% in one registration problem. This would make RANSAC infeasible for use in realistic problems. \n\nAnother typical consensus maximization approach is BnB. It addresses the optimization problem globally optimally by searching in the parameter space (e.g. $SO(3)$ w.r.t. rotation or $SE(3)$ w.r.t. transformation). Unfortunately, BnB runs in exponential time and could not scale to problems with large correspondence numbers, but thousands of correspondences is quite common in reality, which limits the practicality of BnB.\n\nADAPT (Adaptive Trimming)~\\cite{tzoumas2019outlier} provides a non-minimal way to solve the consensus maximization problem. It can be directly in conjunction with standard non-minimal solvers to find the maximum consensus set through iterations, but it has the issue of limited robustness. In the registration problem, ADAPT could hardly tolerate more than 90\\% outliers.\n\nIn essence, our solver DANIEL is also a consensus maximization method based on the RANSAC paradigm. But much differently, DANIEL employs a smarter framework by decomposing the sampling into two layers and applying the stratified compatibility checking before the consensus building, so it exceeds RANSAC in time-efficiency by tens to tens of thousands of times.\n\n\\subsection{M-Estimation}\n\nM-estimation can actively decrease the weights of outliers by incorporating robust loss functions into the optimization process. Local M-estimators (e.g.~\\cite{agarwal2013robust,kummerle2011g,sunderhauf2012towards}) can be used to minimize the object function, but the problem is that they must require the initial guess and hence is liable to converge to local solutions if the initial guess is poor. FGR~\\cite{zhou2016fast} first applied Graduated Non-Convexity (GNC) in the registration problem, and more recently, GNC is incorporated with non-minimal solvers and extended to more robotics and computer vision problems~\\cite{yang2020graduated}. However, these solvers generally have confined robustness against outliers. For example, FGR and GNC-TLS would both become brittle once the outlier ratio exceeds 90\\%. \n\n\n\\subsection{Downside of RANSAC}\n\nWe now provide a more explicit analysis on the limitation of RANSAC. In traditional RANSAC or many of its variants, when we obtain one minimal subset in a certain iteration of random sampling, we must then: (i) estimate the coarse minimal model ($\\boldsymbol{R}^{*},\\boldsymbol{t}^{*}$), and (ii) build the consensus set with this model by computing the residual errors w.r.t. all the correspondences. This consensus maximization strategy has two apparent downsides: (i) when the outlier ratio is high, the probability to sample an all-inlier subset can be extremely low, thus making a great number of iterations necessary (exponentially growing with the outlier ratio), and (ii) it is time-consuming to build the consensus set in every iteration, especially when the problem size is large (e.g. thousands of correspondences). And these two limitations are the main issues that we aim to circumvent by using our solver.\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{ccc}\n\n\\footnotesize{(a) Correspondences with 99\\% outliers}\n&\n\\footnotesize{(b) Registration by DANIEL}\n&\n\\footnotesize{(c) Overview of DANIEL}\n\n\\\\\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{demo-corres-99-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.16\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{demo-res-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\n\n&\n\n\\begin{minipage}[t]{0.5\\linewidth}\n\\centering\n\\includegraphics[width=0.96\\linewidth]{demo-overview.pdf}\n\\end{minipage}\n\n\\end{tabular}\n\\vspace{-1mm}\n\\caption{Illustration of robust point cloud registration using our solver DANIEL. (a) An example of a registration problem with $N=1000$ correspondences and only 10 inliers. (b) DANIEL can robustly estimate the best transformation in only 2.597 seconds. (c) An intuitive overview of DANILE. Please see Section~\\ref{overview} for explicit descriptions.}\n\\label{demo-for-show}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\section{Our Method: DANIEL}\n\nIn this section, we present our robust point cloud registration method DANIEL.\n\n\\subsection{Problem Formulation: Consensus Maximization}\n\nFirst of all, we introduce the consensus maximization formulation for the registration problem.\n\nAssume that we have two sets of 3D points: $\\mathcal{X}=\\{\\boldsymbol{x}_i\\}_{i=1}^{N}$ and $\\mathcal{Y}=\\{\\boldsymbol{y}_i\\}_{i=1}^{N}$ where $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i\\,(\\boldsymbol{x}_i,\\boldsymbol{y}_i\\in\\mathbb{R}^3)$ makes up a putative correspondence. Point cloud registration aims to estimate the best transformation including rotation $\\boldsymbol{R}\\in SO(3)$ and translation $\\boldsymbol{t}\\in \\mathbb{R}^{3}$ that can align point set $\\mathcal{X}$ and $\\mathcal{Y}$. If correspondence $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i$ is an inlier, then we can have:\n\\begin{equation}\\label{prob-form1}\n\\boldsymbol{y}_i=\\boldsymbol{R}\\boldsymbol{x}_i+\\boldsymbol{t}+\\boldsymbol{\\epsilon}_i,\n\\end{equation}\nwhere $\\boldsymbol{\\epsilon}_i\\in\\mathbb{R}^3$ denotes the noise measurement, so that the following relation \n\\begin{equation}\n\\left\\|\\boldsymbol{R}\\boldsymbol{x}_i+\\boldsymbol{t}-\\boldsymbol{y}_i\\right\\|\\leq \\xi\n\\end{equation}\ncould be satisfied, where $\\xi\\geq\\|\\boldsymbol{\\epsilon}\\|$ denotes the inlier threshold; however, if $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i$ is an outlier, we can assume:\n\\begin{equation}\n\\left\\|\\boldsymbol{R}\\boldsymbol{x}_i+\\boldsymbol{t}-\\boldsymbol{y}_i\\right\\|>\\xi.\n\\end{equation}\nUsually, we assume the noise to be isotropic Gaussian with standard deviation $\\sigma$, so the inlier threshold can be typically set as: $\\xi=5\\sim 6\\sigma$.\n\nIn this case, letting set $\\mathcal{N}=\\{1,2,\\dots,N\\}$, the registration problem can be written as the following formulation:\n\\begin{equation}\\label{CM}\n\\begin{gathered}\n\\underset{\\mathcal{M}\\subset \\mathcal{N}}{\\max}\\, |\\mathcal{M}|, \\\\\ns.t. \\,\\|\\boldsymbol{R}_{\\mathcal{M}}\\boldsymbol{x}_i+\\boldsymbol{t}_{\\mathcal{M}}-\\boldsymbol{y}_i\\| \\leq \\xi,\\,(\\forall i\\in\\mathcal{M})\n\\end{gathered}\n\\end{equation}\nwhere $(\\boldsymbol{R}_{\\mathcal{M}},\\boldsymbol{t}_{\\mathcal{M}})$ represents a rigid transformation and $\\mathcal{M}$ is its corresponding consensus set. The goal of formulation~\\eqref{CM} is to find the transformation $(\\boldsymbol{R}_{\\mathcal{M}},\\boldsymbol{t}_{\\mathcal{M}})$ that maximizes the size of consensus $\\mathcal{M}$, the process of which is typically called \\textit{consensus maximization}. The goal of our solver is to solve~\\eqref{CM} in a computationally efficient way.\n\n\n\n\n\\subsection{Methodology Overview}\\label{overview}\n\nFig.~\\ref{demo-for-show} illustrates the effectiveness and the overview of our proposed solver DANIEL with descriptions given below.\n\nIn general, DANIEL maximizes the consensus set also by random sampling, but its main operation structure is significantly different from the traditional RANSAC solvers. To be specific, DANIEL consists of two random-sampling layers, where the latter is embedded into the main framework of the former. The first layer adopts one-point sampling in each iteration (in order to maximize the probability to obtain inliers) and then reduces the correspondence set size by using rigidity constraint. (See Section~\\ref{first-layer} for the first layer)\n\nIn one certain iteration of the first layer and after the reduction of the correspondence set, the second layer performs continuous random sampling of two points as well as the computing of the respective minimal models. In the meantime, a new stratified element-wise compatibility checking approach is proposed to conduct rapid compatibility checking between pairs of minimal models obtained, which serves as a prerequisite for consensus building in DANIEL. Only when we can obtain two minimal subsets (models) that are compatible with each other can we average the parameters of these two models and build the consensus set. Furthermore, we can derive probabilistic termination conditions (by computing maximum iteration numbers) for both of the two layers, and the maximized consensus in the second layer will be used to further maximize the consensus in the first layer. Finally, we can return the maximum consensus set in the first layer to further obtain the ultimate full inlier set. (See Section~\\ref{second-layer} for the second layer) \n\n\\subsection{First Layer: One-Point Sampling and Inlier Candidate Searching with Rigidity Constraint}\\label{first-layer}\n\nRigidity~\\cite{michel2017global,zach2015dynamic,quan2020compatibility,yang2019polynomial} (invariance of length) is a common pairwise constraint in 3D geometry. Specifically, for any two correspondences $\\boldsymbol{x}_i\\leftrightarrow\\boldsymbol{y}_i$ and $\\boldsymbol{x}_j\\leftrightarrow\\boldsymbol{y}_j$ that are both inliers, we can have the rigidity constraint such that\n\\begin{equation}\\label{rigidity}\n\\left|\\left\\|\\boldsymbol{y}_i-\\boldsymbol{y}_j\\right\\|-\\left\\|\\boldsymbol{x}_i-\\boldsymbol{x}_j\\right\\|\\right| \\leq 2\\xi,\n\\end{equation}\nwhich can be derived, based on triangular inequality and the property that norm is invariant to $\\boldsymbol{R}$, as follows:\n\\begin{equation}\\label{proof-of-rigidity}\n\\begin{gathered}\n\\left|{\\|\\boldsymbol{y}_i-\\boldsymbol{y}_j\\|}-{\\|\\boldsymbol{x}_i-\\boldsymbol{x}_j\\|}\\right| \\\\\n=\\left|{\\left\\|\\boldsymbol{R} (\\boldsymbol{x}_i-\\boldsymbol{x}_j+\\boldsymbol{R}^{\\top}\\boldsymbol{\\epsilon}_i-\\boldsymbol{R}^{\\top}\\boldsymbol{\\epsilon}_j)\\right\\|}-{\\|\\boldsymbol{x}_i-\\boldsymbol{x}_j\\|}\\right| \\\\\n\\leq \\left\\| (\\boldsymbol{x}_i-\\boldsymbol{x}_j)-({\\boldsymbol{x}_i-\\boldsymbol{x}_j})+\\boldsymbol{R}^{\\top}\\boldsymbol{\\varepsilon}_i-\\boldsymbol{R}^{\\top}\\boldsymbol{\\epsilon}_j \\right\\| \\\\ \n= {2\\|(\\boldsymbol{\\epsilon}_i-\\boldsymbol{\\epsilon}_j)\\|}\\leq 2\\xi.\n\\end{gathered}\n\\end{equation}\nIntuitively, this constraint indicates that the length between two points (both inliers) remains fixed before and after the rigid transformation, which underlies our strategy to reduce the correspondence set in our first sampling layer.\n\nIn the first layer of DANIEL, we perform one-point sampling, that is, to select only one random correspondence in each iteration, say correspondence $n\\in\\mathcal{N}$ ($\\mathcal{N}=\\{1,2,\\dots,N\\}$ denotes the full correspondence set). Then, we employ rigidity constraint to sift out all the eligible correspondences from $\\mathcal{N}$ that can satisfy~\\eqref{rigidity} with $n$, and then add them to set $\\mathcal{C}$, called the `\\textit{inlier candidate set}'. (See \\textbf{lines 3-9} in Algorithm~\\ref{DANIEL-algo})\n\nThe insight here lies in that if the sampled point $n$ is an inlier, then all the other inliers must fulfill condition~\\eqref{rigidity} with $n$ and must therefore lie within set $\\mathcal{C}$. In this way, we can establish a smaller correspondence set for our second layer. Note that this step could be rather effective especially when the outlier ratio is high, since a large portion of `raw' outliers can be swiftly removed by the rigidity constraint, exponentially increasing the probability of sampling an all-inlier subset later in the second layer.\n\n\\begin{algorithm}[t]\n\\caption{\\textit{CompatibilityStaircase}}\n\\label{Staircase}\n\\SetKwInOut{Input}{\\textbf{Input}}\n\\SetKwInOut{Output}{\\textbf{Output}}\n\\Input{two minimal models: ($\\boldsymbol{R}^*_{a},\\boldsymbol{t}^*_{a}$) and ($\\boldsymbol{R}^*_{b},\\boldsymbol{t}^*_{b}$); element-wise thresholds $\\theta^{r}$ and $\\theta^{t}$\\;}\n\\Output{boolean compatibility status $comp$\\;}\n\\BlankLine\n$\\boldsymbol{T}_{a}\\leftarrow{[{vec(\\boldsymbol{R}^*_{a})}^{\\top},{\\boldsymbol{t}^{*}_{a}}^{\\top}]}^{\\top}$, $\\boldsymbol{T}_{b}\\leftarrow{[{vec(\\boldsymbol{R}^*_{b})}^{\\top},{\\boldsymbol{t}^{*}_{b}}^{\\top}]}^{\\top}$, $\\boldsymbol{\\theta}\\leftarrow{[\\underbrace{\\theta^r,\\dots,\\theta^r}_{9\\times\\theta^r}, \\underbrace{\\theta^t,\\dots,\\theta^t}_{3\\times\\theta^t}]}^{\\top}$, and $comp\\leftarrow 0$\\;\n\\For{$rep=1:12$}{\n\\If{$\\left|\\boldsymbol{T}_{a}(rep)-\\boldsymbol{T}_{b}(rep)\\right|>\\boldsymbol{\\theta}(rep)$}{\n\\textbf{break}}\n\\If{$rep=12$}{\n$comp\\leftarrow 1$\\;\n}\n}\n\\Return boolean compatibility status $comp$\\;\n\\end{algorithm}\n\n\\subsection{Second Layer: Two-Point Sampling and Consensus Maximization based on Stratified Element-wise Compatibility}\\label{second-layer}\n\nIn the second layer of DANIEL, with the correspondence $n$ selected in the first layer, we only need to sample two more points to construct a three-point minimal subset for the estimation of the transformation, which requires much less computational cost in comparison with the traditional three-point RANSAC since: (i) the correspondence set is reduced from the original set $\\mathcal{N}$ to set $\\mathcal{C}$ now, and (ii) the sampling dimension is reduced from 3 to 2 (three-point to two-point). Subsequently, rigidity constraint~\\eqref{rigidity} can be applied once again for `raw' outlier removal. When we select a pair of random correspondences $\\{a,b\\}\\subset\\mathcal{C}$, knowing that both $(a,n)$ and $(b,n)$ have already satisfied rigidity in the first layer, we can further test $(a,b)$ with~\\eqref{rigidity}. If satisfied, the three-point set $\\{n,a,b\\}$ is entirely rigid (each line between any two points has fixed length after the transformation) and can be used to solve the transformation model $(\\boldsymbol{R}^*,\\boldsymbol{t}^*)$ minimally using Horn's triad-based method~\\cite{horn1987closed}. (See \\textbf{lines 12-14} in Algorithm~\\ref{DANIEL-algo})\n\nMore importantly, we depart from the traditional sampling-and-consensus-building technique as applied in RANSAC, and introduce a novel compatibility-based consensus maximization paradigm, which is partially inspired by~\\cite{sun2021ransic}.\n\nDifferent from the standard procedure of first making an estimate with a minimal subset and then establishing the consensus set in every single iteration in traditional RANSAC, when we compute a minimal model ($\\boldsymbol{R}^*,\\boldsymbol{t}^*$) with a random subset, we store its parameters and check the \\textit{mutual compatibility} between this model and every single existing model already in storage. (Intuitively, compatibility checking can be operated by measuring the mathematical error between the two models and then judging whether it is small enough.) Only when we have found two models that fulfill the mutual compatibility could we build the consensus set with these two models by computing the residual errors w.r.t. all the correspondences in set $\\mathcal{C}$. This strategy is inspired by the fact that models estimated with different true inliers should still be sufficiently similar (not equivalent due to the presence of noise), and it can tremendously curtail the time cost of the repeated but `redundant' consensus building in every iteration of sampling. (See \\textbf{lines 15-17} in Algorithm~\\ref{DANIEL-algo})\n\nHowever, when the outlier ratio is high, we may have to sample a large number of random subsets and store many minimal models so as to achieve two all-inlier subsets to satisfy this compatibility condition. Consequently, we further propose a series of \\textit{stratified element-wise compatibility tests} in order to ensure high efficiency in our compatibility checking process.\n\n\\subsubsection{Stratified Element-wise Model Compatibility}\n\nInstead of the holistic checking of compatibility, our idea is that two models must be mutually compatible as long as all their respective parameters are compatible.\nThe parameters of the model involved in the registration problem include 9 scalars from the rotation matrix $\\boldsymbol{R}\\in SO(3)$ plus 3 scalars from the translation vector $\\boldsymbol{t}\\in\\mathbb{R}^3$, which can be represented as $\\boldsymbol{R}=\\left[\\begin{array}{ccc} r_1, &r_4, & r_7 \\\\  r_2, &r_5, & r_8 \\\\  r_3, &r_6, & r_9  \\end{array}\\right]$ and $\\boldsymbol{t}=[t_1,t_2,t_3]^{\\top}$, respectively. \n\nFor a certain translation vector $\\boldsymbol{t}^*$ estimated from a minimal subset, it can be rewritten as:\n\\begin{equation}\n\\boldsymbol{t}^*=\\boldsymbol{t}_{gt}+\\boldsymbol{\\epsilon}^{\\boldsymbol{t}},\n\\end{equation}\nwhere $\\boldsymbol{t}_{gt}$ represents the ground-truth translation and $\\boldsymbol{\\epsilon}^{\\boldsymbol{t}}\\in\\mathbb{R}^3$ denotes the noise measurement on translation. Now assume that we have two such minimal translations, say $\\boldsymbol{t}^*_{a}$ and $\\boldsymbol{t}^*_{b}$. Then, we can derive the following compatibility condition if they are both inliers:\n\\begin{equation}\\label{t-comp}\n\\left\\|\\boldsymbol{t}^*_{a}-\\boldsymbol{t}^*_{b}\\right\\|=\\left\\|\\boldsymbol{\\epsilon}^{\\boldsymbol{t}_a}-\\boldsymbol{\\epsilon}^{\\boldsymbol{t}_b}\\right\\| \\leq 2\\xi^{\\boldsymbol{t}},\n\\end{equation}\nwhere $\\xi^{\\boldsymbol{t}}\\geq\\|\\boldsymbol{\\epsilon}^{\\boldsymbol{t}}\\|$ is the noise upper-bound for translation, and we usually set $\\xi^{\\boldsymbol{t}}=5\\sigma$. Moreover, since the three elements in translation are completely independent, compatibility~\\eqref{t-comp} can be easily decoupled into three errors in L1 norm: $|t^*_{a1}-t^*_{b1}|$, $|t^*_{a2}-t^*_{b2}|$ and $|t^*_{a3}-t^*_{b3}|$. We can then set a scalar threshold for each of them in order to fulfil condition~\\eqref{t-comp} such that\n\\begin{equation}\\label{t-comp-each}\n\\theta^{t}=\\left|{t}^*_{ap}-{t}^*_{bp}\\right| \\leq \\frac{2\\mu}{\\sqrt{3}}\\xi^{\\boldsymbol{t}},\\,(\\forall p\\in\\{1,2,3\\})\n\\end{equation}\n\n\n\\clearpage\n\\begin{algorithm*}[h]\n\\caption{DANIEL}\n\\label{DANIEL-algo}\n\\SetKwInOut{Input}{\\textbf{Input}}\n\\SetKwInOut{Output}{\\textbf{Output}}\n\\Input{correspondences $\\mathcal{P}=\\{(\\mathbf{x}_i,\\mathbf{y}_i)\\}_{i=1}^N$; noise $\\sigma$; minimum inlier number $I_{min}=\\max(5,0.01N)$\\;}\n\\Output{optimal $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$; inlier set $\\mathcal{N}^{\\star}$ \\;}\n\\BlankLine\n$\\mathcal{N}\\leftarrow [1,2,\\dots,N]$, $samp1\\leftarrow 0$, $\\mathcal{N}_{best}\\leftarrow\\emptyset$, $bestSize_1\\leftarrow0$, $maxItr_1\\leftarrow 459$, and get $\\theta^r$ and $\\theta^t$ with~\\eqref{r-comp-each} and~\\eqref{t-comp-each}\\;\n\\While{$samp_1\\leq maxItr_1$}{\nSelect a random $n\\in\\mathcal{N}$, $\\mathcal{C}\\leftarrow \\emptyset$, and $samp_1\\leftarrow samp_1+1$\\;\n\\For{\\textbf{\\textup{all}} $i\\in\\mathcal{N}$ and $i\\neq n$}{\n\\If{correspondence pair $(i,n)$ can satisfy condition~\\eqref{rigidity}}{\n$\\mathcal{C}=\\mathcal{C}\\cup{\\{i\\}}$\\;\n}\n}\n\\If{$|\\mathcal{C}|\\geq I_{min}$}{\n$bestSize_2\\leftarrow0$, $samp_2\\leftarrow 0$ and $\\mathcal{S}_{best}\\leftarrow\\emptyset$\\;\n\\While{$samp_2\\leq maxItr_2$}{\nSelect a random subset $\\{a,b\\}\\subset\\mathcal{C}$, and $samp_2\\leftarrow samp_2+1$\\;\n\\If{correspondence pair $(a,b)$ can satisfy condition~\\eqref{rigidity}}{\nEstimate $(\\boldsymbol{R}^{*},\\boldsymbol{t}^{*})$ minimally using Horn's method~\\cite{horn1987closed}\\;\n$\\mathcal{R}\\leftarrow\\mathcal{R}\\cup\\{\\boldsymbol{R}^{*}\\}$, and $\\mathcal{T}\\leftarrow\\mathcal{T}\\cup\\{\\boldsymbol{t}^{*}\\}$\\;\n\\For{$z=1:\\left(|\\mathcal{R}|-1\\right)$}{\n$comp\\leftarrow$\\textit{CompatibilityStaircase}($\\mathcal{R}_{(z)}$, $\\mathcal{T}_{(z)}$, $\\boldsymbol{R}^{*}$, $\\boldsymbol{t}^{*}$, $\\theta^r$, $\\theta^t$)\\;\n\\If{$comp=1$}{\n$\\boldsymbol{R}^{\\circ}\\leftarrow$\\textit{AverageRotPara}($\\mathcal{R}_{(z)}$, $\\boldsymbol{R}^{*}$), $\\boldsymbol{t}^{\\circ}\\leftarrow$\\textit{AverageTranPara}($\\mathcal{T}_{(z)}$, $\\boldsymbol{t}^{*}$), and $\\mathcal{S}\\leftarrow\\emptyset$\\;\n\\% Obtain the consensus set of the averaged model: $\\boldsymbol{R}^{\\circ}$ and $\\boldsymbol{t}^{\\circ}$ \\%\\\\\n\\For{\\textbf{\\textup{all}} $j\\in\\mathcal{C}$}{\n\\If{$\\|\\boldsymbol{R}^{\\circ}\\boldsymbol{x}_j+\\boldsymbol{t}^{\\circ}-\\boldsymbol{y}_j|\\leq 5\\sigma$}{\n$\\mathcal{S}\\leftarrow\\mathcal{S}\\cup\\{j\\}$\\;\n}\n}\n\\If{$|\\mathcal{S}|\\geq bestSize_2$}{\n$\\mathcal{S}_{best}\\leftarrow\\mathcal{S}$, $bestSize_2\\leftarrow |\\mathcal{S}|$, and update $maxItr_2$ with~\\eqref{max-itr-2}\\;\n}\n}\n}\n}\n}\n} \n\\If{$bestSize_2\\geq bestSize_1$}{\nSolve $(\\boldsymbol{R}^{\\dagger}, \\boldsymbol{t}^{\\dagger})$ non-minimally with $\\mathcal{S}_{best}$ using SVD~\\cite{arun1987least}, and $\\mathcal{N}_{best}\\leftarrow\\emptyset$\\;\n\\% Obtain the consensus set of current $\\boldsymbol{R}^{\\dagger}$ and $\\boldsymbol{t}^{\\dagger}$ \\%\\\\\n\\For{\\textbf{\\textup{all}} $j\\in\\mathcal{N}$}{\n\\If{$\\|\\boldsymbol{R}^{\\dagger}\\boldsymbol{x}_j+\\boldsymbol{t}^{\\dagger}-\\boldsymbol{y}_j\\|\\leq 5\\sigma$}{\n$\\mathcal{N}_{best}\\leftarrow\\mathcal{N}_{best}\\cup\\{j\\}$\\;\n}\n}\n$bestSize_1\\leftarrow |\\mathcal{N}_{best}|$, and update $maxItr_1$ with~\\eqref{max-itr-1}\\;\n}\n}\nSolve $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$ non-minimally with $\\mathcal{N}_{best}$ using SVD, and $\\mathcal{N}^{\\star}\\leftarrow\\emptyset$\\;\n\\% Obtain the consensus set with current $\\boldsymbol{R}^{\\star}$ and $\\boldsymbol{t}^{\\star}$ \\%\\\\\n\\For{\\textbf{\\textup{all}} $k\\in\\mathcal{N}$}{\n\\If{$\\|\\boldsymbol{R}^{\\star}\\boldsymbol{x}_k+\\boldsymbol{t}^{\\star}-\\boldsymbol{y}_k\\|\\leq 5\\sigma$}{\n$\\mathcal{N}^{\\star}\\leftarrow\\mathcal{N}^{\\star}\\cup\\{j\\}$\\;\n}\n}\nSolve $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$ non-minimally with $\\mathcal{N}^{\\star}$ using SVD\\;\n\\Return $(\\boldsymbol{R}^{\\star}, \\boldsymbol{t}^{\\star})$ and $\\mathcal{N}^{\\star}$\\;\n\\end{algorithm*}\n\\clearpage\n\n\\noindent where $\\mu \\geq1$. Note that if $\\mu=1$,  \\eqref{t-comp-each} $\\Rightarrow$ \\eqref{t-comp}. But we want a slightly more lenient threshold, so $\\mu$ can be set to slightly larger than 1. In practice, we can empirically set $\\mu=1.2$. \n\n\nFor rotation, the widely-used geodesic distance~\\cite{hartley2013rotation} is represented by angle and requires relatively complex computation including $\\arccos$ and $trace$, so it is difficult to be rewritten as element-wise. But according to~\\cite{hartley2013rotation}, the chordal distance, having natural relation to the geodesic distance, enables us to derive a series of element-wise compatibility conditions.\n\nAssume that we have two rotations $\\boldsymbol{R}^*_a$ and $\\boldsymbol{R}^*_b$ estimated from two minimal subsets. $d_{\\text{chord}}$ denotes the chordal distance between them that can be written as:\n\\begin{equation}\\label{chordal}\nd_{\\text{chord}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)=\\left\\|\\boldsymbol{R}^*_a-\\boldsymbol{R}^*_b\\right\\|_{F}=2\\sqrt{2}\\sin (\\frac{d_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)}{2}),\n\\end{equation}\nwhere $\\|\\cdot\\|_F$ denotes the Frobenius norm and $d_\\text{geo}(\\cdot,\\cdot)$ means the geodesic distance~\\cite{hartley2013rotation} such that\n\\begin{equation}\\label{geodesic}\nd_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)=\\left|\\arccos\\left(\\frac{trace({\\boldsymbol{R}_a^*}^{\\top}\\boldsymbol{R}^*_b)-1}{2}\\right)\\right|.\n\\end{equation}\nChordal distance~\\eqref{chordal} intuitively represents the square root of the sum of sqaures of all the 9 parameters in matrix $\\boldsymbol{R}^*_a-\\boldsymbol{R}^*_b$. Now our goal is to set a proper threshold for each of them as in~\\eqref{t-comp-each}. Similarly, a minimal rotation $\\boldsymbol{R}^*$ can be described by:\n\\begin{equation}\n\\boldsymbol{R}^*=\\boldsymbol{R}_{gt}\\cdot Exp\\left([\\boldsymbol{\\epsilon}^{\\boldsymbol{R}}]_{\\times}\\right),\n\\end{equation}\nwhere $\\boldsymbol{R}_{gt}$ is the ground-truth rotation, $\\boldsymbol{\\epsilon}^{\\boldsymbol{R}}\\in\\mathbb{R}^3$ is the noise measurement on rotation, $[\\,\\cdot\\,]_{\\times}$ denotes the skew-symmetric matrix for the size-3 vector, and $Exp(\\,\\cdot\\,)$ is the exponential map of matrix. Then we let $\\boldsymbol{\\epsilon}^{\\boldsymbol{R}}=\\mathit{S}\\cdot\\boldsymbol{e}$ where $\\boldsymbol{e}$ is a random vector of unit length and $\\mathit{S}$ denotes its scale, and according to~\\cite{barfoot2017state}, the geodesic error between $\\boldsymbol{R}^*$ and $\\boldsymbol{I}_3$ can be written as:\n\\begin{equation}\nd_{\\text{geo}}(\\boldsymbol{R}^*,\\boldsymbol{I}_3)=\\mathit{S},\n\\end{equation}\nand based on the triangular inequality of geodesic distances, we can then have that\n\\begin{equation}\nd_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b) \\leq d_{\\text{geo}}(\\boldsymbol{R}^*_a,\\boldsymbol{I}_3)+d_{\\text{geo}}(\\boldsymbol{R}_b^*,\\boldsymbol{I}_3)=2\\mathit{S},\n\\end{equation}\nso that the chordal distance between $\\boldsymbol{R}^*_a$ and $\\boldsymbol{R}^*_b$ satisfies:\n\\begin{equation}\nd_{\\text{chord}}(\\boldsymbol{R}^*_a,\\boldsymbol{R}^*_b)\\leq 2\\sqrt{2}\\sin \\left( \\mathit{S} \\right).\n\\end{equation}\n\nAfterwards, the threshold for the L1-norm distance of each of the 9 element in rotation should be:\n\\begin{equation}\\label{r-comp-each}\n\\theta^{r}=\\left|r^*_{aq}-r^*_{bq}\\right|\\leq \\mu\\cdot\\frac{2\\sqrt{2}\\sin \\left( \\mathit{S} \\right)}{3},\\,(\\forall q\\in\\{1,2,\\dots,9\\})\n\\end{equation}\nwhere $\\mu\\geq1$ and we can choose $\\mu=1.2$ for practical use, similar to~\\eqref{t-comp-each}. Note that in practice, if the maximum distances (diameters) of the point cloud in the 3 axes (X,Y and Z) are $D_X$, $D_Y$ and $D_Z$ and their mean value is $\\widetilde{D}$, we can provide an empirical choice for $\\mathit{S}$ such that\n\\begin{equation}\n\\mathit{S}=10\\frac{\\sigma}{\\widetilde{D}}.\n\\end{equation}\n\nUp to now, we can render our fast compatibility checking method that we name \\textit{CompatibilityStaircase} because it is operated like a staircase, only when the first condition is satisfied could we proceed to the second condition, as demonstrated in Algorithm~\\ref{Staircase}. (Note that we use $\\boldsymbol{V}(y)$ to denote the $y_{th}$ entry of the vector $\\boldsymbol{V}$.)\n\n\\textbf{Description of Algorithm~\\ref{Staircase}:} With two minimal models ($\\boldsymbol{R}^*_{a},\\boldsymbol{t}^*_{a}$) and ($\\boldsymbol{R}^*_{b},\\boldsymbol{t}^*_{b}$) estimated from two minimal subsets, we vectorize $\\boldsymbol{R}^*_{a}$ (and $\\boldsymbol{R}^*_{b}$) in column-wise and stack it with $\\boldsymbol{t}^*_{a}$ (and $\\boldsymbol{t}^*_{b}$) to form the size-12 vector $\\boldsymbol{T}_{a}$ (and $\\boldsymbol{T}_{b}$) (\\textbf{line 1}). Subsequently, we check the element-wise compatibility for the 12 pairs of elements based on condition~\\eqref{t-comp-each} and~\\eqref{r-comp-each} in sequence. If the $i_{th}$ element pair is not mutually compatible, then the rest $12-i$ element pairs are no longer required to be checked. This operation could be easily realized by \\textbf{lines 2-9}, where `$comp=1$' defines that the two models are compatible with each other whereas `$comp=1$' symbolizes their mutual incompatibility. Note that the operations mainly are subtracting, getting absolute values, and boolean conditions, which are all rather fast to implement.\n\n\n\\subsubsection{Parameter Averaging for Consensus Building}\n\nOnce two mutually compatible models (subsets) are found, implying that two different subsets are approximately pointing to the same model, it is reasonable and necessary to establish and evaluate the consensus set over these two models. Note that in traditional RANSAC, minimal models may be easily affected by noise, hence probably deviating from the ground-truth model to a great extent. But fortunately, since we have two models now, averaging them could reduce the influence of noise. Here, we prefer parameter averaging to applying the non-minimal solvers (e.g. SVD~\\cite{arun1987least}) because: (i) the former is more time-efficient, and (ii) the advantage of non-minimal solvers can be hardly shown here because the correspondence number is relatively small (no greater than 6). \n\nFor translation, we directly adopt the mean vector:\n\\begin{equation}\\label{t-average}\n\\boldsymbol{t}^{\\circ}=\\frac{\\boldsymbol{t}_a+\\boldsymbol{t}_b}{2},\n\\end{equation}\nwhile for rotation, we adopt the geodesic L2-mean~\\cite{hartley2013rotation} such that\n\\begin{equation}\\label{R-average}\n\\boldsymbol{R}^{\\circ}=\\boldsymbol{R}_a\\cdot Exp\\left(\\frac{log(\\boldsymbol{R}_a^{\\top}\\boldsymbol{R}_b)}{2}\\right),\n\\end{equation}\nwhere $log(\\,\\cdot\\,)$ denotes the logarithm of matrix. Here, we introduce functions: \\textit{AverageRotPara}($\\boldsymbol{R}_a,\\boldsymbol{R}_b$), \\textit{AverageTranPara}($\\boldsymbol{t}_a,\\boldsymbol{t}_b$), to represent the parameter averaging operations in~\\eqref{R-average} and~\\eqref{t-average}, respectively.\n\nSubsequently, we can compute the residual errors of the correspondences in set $\\mathcal{C}$ using $\\boldsymbol{R}^{\\circ}$ and $\\boldsymbol{t}^{\\circ}$ and then build the consensus set. (See \\textbf{lines 18-29} in Algorithm~\\ref{DANIEL-algo})\n\n\n\\begin{figure*}[t]\n\\centering\n\n\\begin{tabular}{cc}\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-demo-bunny-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-time-bunny-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-R-error-bunny-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.46\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Bench-t-error-bunny-eps-converted-to.pdf}\n\\end{minipage} \n\\end{tabular}\n\n\\caption{Standard benchmarking results in boxplot. We show the rotation and translation estimation accuracy as well as the runtime of different solvers over the `\\textit{bunny}' dataset~\\cite{curless1996volumetric} w.r.t. increasing outlier ratios from 20\\% up to 99\\%. The top-left image exemplifies the correspondences with 95\\% outliers in our experimental environment, where green lines denote the inliers while red lines denote the outliers.}\n\\label{Benchmarking}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Probabilistic Computation of Maximum Iteration Numbers}\\label{probability}\n\nThere exist two maximum iteration numbers involved in DANIEL, one for the one-point sampling in the first layer and the other for the two-point sampling in the second layer. \n\nIn the first layer, according to~\\cite{fischler1981random,chum2003locally}, given the size of the minimal subset $A_1=1$ (since we use one-point sampling), the so-far-the-best consensus set $\\mathcal{N}_{best}$, and the probability of sampling at least one all-inlier subset $P_1=0.99$ (sufficiently close to 1), we can compute the expected maximum number of (random sampling) iteration such that\n\\begin{equation}\\label{max-itr-1}\nmaxItr_1\\geq\\frac{\\log{(1-P_1)}}{\\log{\\left(1-{\\left(\\frac{|\\mathcal{N}_{best}|}{N}\\right)}^{A_1}\\right)}}.\n\\end{equation}\n\\noindent{At} the beginning of the algorithm, we are required to preset a minimum inlier number $I_{min}$ (usually we set $I_{min}=0.01N$ in practice), so replacing $|\\mathcal{N}_{best}|$ with $I_{min}$ in~\\eqref{max-itr-1}, we can have $maxItr_1=459$. During the sampling process, whenever a new consensus set is obtained, we update $maxItr_1$ with~\\eqref{max-itr-1}. (See \\textbf{line 42} in Algorithm~\\ref{DANIEL-algo})\n\nIn the second layer, we set $A_2=2$ (two-point sampling) and require to sample at least two all-inlier minimal subsets, so we can set $P_2=0.995$, and then compute $maxItr_2$ as\n\\begin{equation}\\label{max-itr-2}\nmaxItr_2\\geq2\\cdot\\frac{\\log{(1-P_2)}}{\\log{\\left(1-{\\left(\\frac{I_{min}}{|\\mathcal{C}|}\\right)}^{A_2}\\right)}}.\n\\end{equation}\nIn this case, the probability of sampling two all-inlier subsets to fulfill the mutual compatibility should be $0.995^2\\approx0.99$. Similarly, during the sampling process, when we construct a new consensus set after the compatibility of two subsets, we can update $maxItr_2$ with~\\eqref{max-itr-2}. (See \\textbf{lines 27} in Algorithm~\\ref{DANIEL-algo})\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{0pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{c|cccc|ccc}\n\n\\quad & \\,\\, & \\footnotesize{FPFH Correspondences} & \\footnotesize{Registration by DANIEL} & \\quad & \\,\\, & \\footnotesize{FPFH Correspondences} & \\footnotesize{Registration by DANIEL} \\\\\n\\hline \n&&&&&&&\n\\\\\n\\rotatebox{90}{\\scriptsize{bunny, $N=1401$, 92.78\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{bunny-cor-eps-converted-to.pdf}\n\\end{minipage}\n& \n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{bunny-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{armadillo, $N=876$, 89.81\\%}}\\,\n& &\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{armadillo-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.18\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{armadillo-res-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\rotatebox{90}{\\scriptsize{dragon, $N=1208$, 89.97\\%\\quad}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{dragon-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{dragon-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{cheff, $N=1500$, 93.09\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{cheff-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.205\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{cheff-res-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\rotatebox{90}{\\scriptsize{chicken, $N=1500$, 91.73\\%\\quad}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{chicken-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.205\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{chicken-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{rhino, $N=1500$, 93.88\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{rhino-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{rhino-res-eps-converted-to.pdf}\n\\end{minipage}\n\\\\\n\\rotatebox{90}{\\scriptsize{parasauro, $N=1500$, 92.98\\% \\quad}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{parasauro-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{parasauro-res-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\rotatebox{90}{\\scriptsize{T-rex, $N=1500$, 91.95\\%}}\\,\n& &\n\\begin{minipage}[t]{0.2\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{trex-cor-eps-converted-to.pdf}\n\\end{minipage}\n&\n\\begin{minipage}[t]{0.21\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{trex-res-eps-converted-to.pdf}\n\\end{minipage}\n\n\\end{tabular}\n\n\\caption{Qualitative registration results (boxplot) on multiple real point cloud datasets using our solver DANIEL. The first column shows the correspondences matched by FPFH~\\cite{rusu2009fast} where green lines denote the inliers while red lines denote the outliers. The second column displays the registration (projection) result using the transformation estimated by DANIEL.}\n\\label{qualit-partial}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Main Algorithm}\n\nThe pseudocode of the main algorithm of DANIEL is provided in Algorithm~\\ref{DANIEL-algo}. \n\n\\textbf{Description of Algorithm~\\ref{DANIEL-algo}:} After the initialization setup, we start the first layer of one-point random sampling with rigidity examining, as described in Section~\\ref{first-layer}. If the size of set $\\mathcal{C}$ exceeds $I_{min}$, meaning that $n$ is likely to be an inlier, we then move on to the second layer of two-point random sampling, as described in Section~\\ref{second-layer}. This layer has two sequential steps: (i) continuous sampling and model estimating with mutual compatibility checking, and (ii) parameter averaging to build the consensus set after a pair of compatible models are detected. In both sampling layers, we update the best consensus set (replacing the smaller consensus set with the larger one) as well as the maximum iteration numbers (according to Section~\\ref{probability}) iteratively. Note that the consensus updating and maximizing procedure in the second layer is conducted within set $\\mathcal{C}$, while in the first layer it uses the complete correspondence set $\\mathcal{N}$. Eventually, when the first layer converges\\footnote[1]{Here, convergence means that the actual iteration number reaches the current maximum iteration number computed with the best consensus set so far using formulation~\\eqref{max-itr-1} or~\\eqref{max-itr-2}. It is the stop condition of random sampling.}, the best consensus set can be applied to obtain the final inlier set and to compute the optimal transformation for this registration problem.\n\n\\textbf{`1+2$<$3'? Why is DANIEL Faster?} Our DANIEL uses a one-point sampling layer plus an inner two-point sampling layer, while traditional RANSAC adopts one single three-point sampling layer. Though it seems that 1+2 should be equal to 3, DANIEL can be in fact up to over 10 thousand times faster than RANSAC. The main reasons are given as follows.\n\nFirst, in the RANSAC paradigm, the theoretical maximum iteration number of finding the best consensus set should be\n\\begin{equation}\\label{max-itr}\nmaxItr^{\\star}=\\frac{\\log{(1-0.99)}}{\\log{\\left(1-{\\left(\\frac{N_{inlier}}{N}\\right)}^{A}\\right)}},\n\\end{equation}\nwhere $N_{inlier}$ is the true inlier number and $A$ is the subset size. The function of the first layer is two-fold: (i) to reduce $N$, which exponentially decrease $maxItr^{\\star}$, and (ii) to lower $A$ (from 3 to 2), which also exponentially decrease $maxItr^{\\star}$. And when we reach the second layer, where both $N$ and $A$ are reduced, the maximum iteration number has enormously declined. Besides, the first layer only involves the computation of norms and the judging of boolean conditions, so its own computational cost is fairly low. Using such a `cheap' operation in exchange for the exponential decrease of  iteration number is completely rewarding, and that is why DANIEL is much faster than RANSAC.\n\nSecond, another reason lies in our stratified compatibility checking method. Note that in each iteration, we replace the traditional repeated consensus building operation with the compatibility checking, where the latter also only consists of norm computing and boolean conditions, so the efficiency can be greatly enhanced. Generally, the larger the problem size $N$ is, the more apparent the speed superiority of DANIEL will be. In experiments, we further compare the runtime of DANIEL against that of RANSAC (Fig.~\\ref{RANSAC-vs-DANIEL}).\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{ccc}\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Partial-Reg-R-error-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Partial-Reg-t-error-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.32\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{Partial-Reg-time-eps-converted-to.pdf}\n\\end{minipage}\n\n\\end{tabular}\n\n\\caption{Quantitative registration results (boxplot) on multiple real point cloud datasets. The performances (estimation errors and runtime) of different solvers over different point clouds tested are shown.}\n\\label{Partial-reg}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\section{Experiments and Applications}\\label{Experiments}\n\nIn this section, we conduct multiple comprehensive experiments on the basis of various realistic datasets in order to fully evaluate the performance of DANIEL, in comparison with other state-of-the-art competitors. All the experiments are implemented in Matlab over a laptop with a 2.8Ghz CPU and a 16GB RAM without using any parallelism programming.\n\n\n\\subsection{Standard Benchmarking on Semi-Synthetic Data}\\label{sec-bench}\n\nWe first evaluate our DANIEL in the standard benchmarking experiment (the point cloud used is realistic while the outliers are artificially generated, thus called \\textit{semi-synthetic}), in comparison with the existing state-of-the-art solvers including: (i) three non-minimal solvers: FGR~\\cite{zhou2016fast}, GNC-TLS~\\cite{yang2020graduated} and ADAPT~\\cite{tzoumas2019outlier}, (ii) three RANSAC solvers: RANSAC~\\cite{fischler1981random}, LO-RANSAC~\\cite{chum2003locally} and FLO-RANSAC~\\cite{lebeda2012fixing} (also called LO$^+$-RANSAC), and (iii) the guaranteed outlier removal solver GORE~\\cite{bustos2017guaranteed} as well as its enhanced version GORE+RANSAC\\footnote[2]{Further using RANSAC to find the best consensus set after the guaranteed outlier removal of GORE.}. Note that the RANSAC solvers are all set with 10000 maximum iterations and 0.99 confidence, and the local optimization is set with 10 iterations. For all solvers, the inlier threshold is set to $\\xi=6\\sigma$. For quantitative evaluation of estimation errors, we use the geodesic error~\\eqref{geodesic} to represent rotation error in degrees such that\n\\begin{equation}\\label{geodesic}\nE_{\\text{rot}}(\\boldsymbol{R}_{gt}, \\boldsymbol{R}^{\\star})=\\left|\\arccos\\left(\\frac{trace({\\boldsymbol{R}_{gt}}^{\\top}\\boldsymbol{R}^{\\star})-1}{2}\\right)\\right|\\cdot\\frac{180}{\\pi}^{\\circ},\n\\end{equation}\nand use L2-norm to represent translation error in meters such that\n\\begin{equation}\\label{geodesic}\nE_{\\text{tran}}(\\boldsymbol{t}_{gt}, \\boldsymbol{t}^{\\star})=\\left\\|\\boldsymbol{t}_{gt}-\\boldsymbol{t}^{\\star}\\right\\|\\, m,\n\\end{equation}\nwhere $\\boldsymbol{R}_{gt}$ and $\\boldsymbol{t}_{gt}$ denotes the ground-truth transformation.\n\n\nWe adopt the `\\textit{bunny}' point cloud dataset from the Stanford 3D Scanning Repository~\\cite{curless1996volumetric}. This point cloud is downsampled to $N=1000$ and resized to be placed a $[-0.5,0.5]^3m$ box as our initial point set $\\mathcal{X}=\\{\\boldsymbol{x}_i\\}_{i=1}^{N}$. Then we transform $\\mathcal{X}$ with a random transformation such that $\\boldsymbol{R}\\in SO(3)$ and $\\boldsymbol{t}\\in\\mathbb{R}^3$ ($||\\boldsymbol{t}||\\leq 3$), and add random noise with $\\sigma=0.01m$ to the transformed point set $\\mathcal{Y}=\\{\\boldsymbol{y}_i\\}_{i=1}^{N}$. To create outliers simulating cluttered scenes, we artificially corrupt (replace) 20\\% to 99\\% of the points in $\\mathcal{Y}$ with completely random points inside a 3D sphere of radius 1. The boxplot benchmarking results are shown in  Fig.~\\ref{Benchmarking}, in which 50 Monte Carlo runs are implemented to obtain the result data.\n\nFrom Fig.~\\ref{Benchmarking}, we can clearly observe that non-minimal solvers (FGR, GNC-TLS and ADAPT) fail at 90-92\\% outliers, and the RANSAC solvers (RANSAC, LO-RANSAC and FLO-RANSAC) with 10000 maximum iterations break at over 95\\% outliers, meanwhile requiring very long computational time with high outlier ratios, whereas GORE, GORE+RANSAC and our DANIEL are robust against as many as 99\\% outliers. Furthermore, DANIEL is the fastest solver at all outlier ratios (from 20\\% to 99\\%) with (at least one of) the highest estimation accuracy, superior to all the other tested competitors in overall performance.\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}[t]{1\\linewidth}\n\\centering\n\\includegraphics[width=0.87\\linewidth]{RANSAC-DANIEL-time-eps-converted-to.pdf}\n\\end{minipage}\n\\caption{Comparison of mean runtime between DANIEL and RANSAC w.r.t. increasing outlier ratios.}\n\\label{RANSAC-vs-DANIEL}\n\\vspace{-1mm}\n\\end{figure}\n\n\\subsection{Runtime Comparison against RANSAC}\n\nSince that RANSAC has been a de-facto standard among all the robust solvers, we exclusively compare the runtime of our DANIEL with that of RANSAC, in order to more clearly observe DANIEL's performance in efficiency. Note that in this experiment, no maximum iteration number is imposed to RANSAC and we simply wait for it until convergence.\n\nWe also adopt the experimental setup in Section~\\ref{sec-bench} and demonstrate the mean runtime (based on 50 data points) of these two solvers w.r.t. increasing outlier ratios from 20\\% up to 95\\%. But when the outlier ratio is 99\\%, RANSAC would take over 7 hours per run, making it difficult to measure its mean runtime. Fortunately, we can reasonably deduce its runtime at 99\\% by computing the theoretical maximum iteration based on~\\eqref{max-itr}. Note that in each iteration of RANSAC, the operations are almost constant and fixed, that is, minimal sampling, model estimating and consensus building, so its runtime should be mainly dependent on its iteration number. For example, in our experiment, RANSAC's mean runtime at 95\\% is found to be 7.97 times greater than that at 90\\%, which fits well with the fact that the maximum iteration number required for 95\\% is 8 times greater than that required for 90\\% according to~\\eqref{max-itr}. As a result, since that the maximum iteration number of RANSAC at 99\\% outliers should be 125 greater than that at 95\\% which is about 206.37 seconds in average, the runtime of RANSAC at 99\\% can be computed as 206.37$\\times$125$\\approx$25,796 seconds.\n\nThus, according to Fig.~\\ref{RANSAC-vs-DANIEL}, we can see that DANIEL is 75 times faster than RANSAC at 90\\%, 357 times faster at 95\\%, and more than 10000 times faster at 99\\%, so that the efficiency superiority of DANIEL is extremely obvious.\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{0.1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{c|cc|ccccc}\n\n\\quad &\\,&\\,\\footnotesize{Correspondences}\\, &\\,&  \\footnotesize{GNC-TLS} & \\footnotesize{FLO-RANSAC} & \\footnotesize{GORE+RANSAC} & \\footnotesize{DANIEL} \\\\\n\n\\hline\n\n&&\\footnotesize{$N=534$, 97.94\\%}& & \\,\\footnotesize{$133.649^{\\circ},2.206m,0.065s$}\\, &\\,\\footnotesize{$90.351^{\\circ},2.177m,15.391s$}\\, &\\,\\footnotesize{$\\textbf{0.807}^{\\circ},\\textbf{0.024}m,0.998s$}\\,&\\,\\footnotesize{$\\textbf{0.807}^{\\circ},\\textbf{0.024}m,\\textbf{0.259}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-01, mug}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{01-mug-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=458$, 95.41\\%}& & \\,\\footnotesize{$101.597^{\\circ},1.632m,0.103s$}\\, &\\,\\footnotesize{$171.168^{\\circ},2.401m,18.079s$}\\, &\\,\\footnotesize{$\\textbf{0.187}^{\\circ},\\textbf{0.003}m,0.827s$}\\,&\\,\\footnotesize{$\\textbf{0.187}^{\\circ},\\textbf{0.003}m,\\textbf{0.291}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-02, box}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{02-box-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=689$, 97.68\\%}& & \\,\\footnotesize{$92.381^{\\circ},0.591m,0.055s$}\\, &\\,\\footnotesize{$110.968^{\\circ},2.199m,15.839s$}\\, &\\,\\footnotesize{${0.198}^{\\circ},\\textbf{0.003}m,1.329s$}\\,&\\,\\footnotesize{$\\textbf{0.139}^{\\circ},\\textbf{0.003}m,\\textbf{0.103}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-03, cap}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{03-cap-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=304$, 97.04\\%}& & \\,\\footnotesize{$150.055^{\\circ},1.566m,0.041s$}\\, &\\,\\footnotesize{$33.426^{\\circ},0.749m,12.562s$}\\, &\\,\\footnotesize{$\\textbf{1.969}^{\\circ},\\textbf{0.024}m,0.763s$}\\,&\\,\\footnotesize{$\\textbf{1.969}^{\\circ},\\textbf{0.024}m,\\textbf{0.118}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-04, mug}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{04-mug-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=773$, 97.41\\%}& & \\,\\footnotesize{$138.748^{\\circ},2.567m,0.132s$}\\, &\\,\\footnotesize{$152.764^{\\circ},2.566m,29.398s$}\\, &\\,\\footnotesize{$\\textbf{0.427}^{\\circ},\\textbf{0.011}m,1.321s$}\\,&\\,\\footnotesize{$\\textbf{0.427}^{\\circ},\\textbf{0.011}m,\\textbf{0.310}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-05, box}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{05-box-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\n\\end{tabular}\n\n\\caption{Object localization and pose estimation results on the RGB-D scenes dataset~\\cite{lai2011large}. The left-most column shows the raw FPFH correspondences where information regarding the correspondence number and outlier ratio is also provided, and the rest columns show the registration results (reprojecting the object back to the scene with the transformation estimated) using GNC-TLS, FLO-RANSAC, GORE+RANSAC and DANIEL, respectively. On top of the results of each solver, from left to right, we display the rotation error (in degrees), translation error (in meters) and runtime (in seconds), respectively. Best results are shown in \\textbf{bold} font.}\n\\label{obj-local1}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Reslistic Registration on Real Data}\n\nWe conduct realistic point cloud registration experiments for further comparative evaluation over more datasets, including the `\\textit{bunny}', `\\textit{armadillo}' and `\\textit{dragon}' from the Stanford 3D Scanning Repository~\\cite{curless1996volumetric}, as well as the `\\textit{cheff}', `\\textit{chicken}', `\\textit{rhino}', `\\textit{parasaurolophus}' and `\\textit{T-rex}' from Mian's dataset~\\cite{mian2006three,mian2010repeatability} (8 point cloud datasets in total). For each point cloud above, we first resize it to fit in a $[-50,50]^3m$ box, then manually divide its full scan into 10 random partial scans where each scan shares 30\\%-35\\% overlapping with the original full scan, generate random transformations ($\\boldsymbol{R}\\in SO(3)$ and $\\boldsymbol{t}\\in\\mathbb{R}^3$ where $||\\boldsymbol{t}||\\leq 100$) to transform these partial scans, and finally use FPFH~\\cite{rusu2009fast} (Matlab function: \\textit{extractFPFHFeatures}) to establish correspondences between each of the transformed partial scans and the initial full scan. When $N>1500$, we only preserve the first 1500 correspondences. Owing to serious partiality, the FPFH correspondences often contain huge amounts of outliers. Subsequently, these correspondences are fed to our DANIEL as well as the other solvers included in Section~\\ref{sec-bench} to robustly solve the registration problems (10 for each point cloud, hence 80 problems in total), where the noise is set to $\\sigma=0.1m$.\n\nThe qualitative registration examples (one example for each point cloud) are shown in Fig.~\\ref{qualit-partial}, where the average correspondence number and outlier ratio of each point cloud are labeled on the left side of the examples. In addition, the quantitative results in boxplot are displayed in Fig.~\\ref{Partial-reg}. We can find that our DANIEL (i) is the most robust and accurate solver, never yielding any single wrong estimate throughout the tests, and (ii) is fast in practice. Note that though the non-minimal solvers seem to show similar or even better (e.g. GNC-TLS) efficiency compared to DANIEL, they are prone to generate wrong results, unable to keep stably robust in all tests. Besides, the RANSAC solvers require relatively long runtime, and also occasionally return failure cases, while single-handed GORE has poor accuracy despite its promising runtime. GORE+RANSAC is the only solver that can have comparable robustness and accuracy with our DANIEL, but it is apparently slower than DANIEL. So overall, DANIEL manifests the best performance.\n\n\\begin{figure*}[t]\n\\centering\n\\setlength\\tabcolsep{0.1pt}\n\\addtolength{\\tabcolsep}{0pt}\n\\begin{tabular}{c|cc|ccccc}\n\n\\quad &\\,&\\,\\footnotesize{Correspondences}\\, &\\,&  \\footnotesize{GNC-TLS} & \\footnotesize{FLO-RANSAC} & \\footnotesize{GORE+RANSAC} & \\footnotesize{DANIEL} \\\\\n\n\\hline\n\n&&\\footnotesize{$N=186$, 95.16\\%}& & \\,\\footnotesize{$131.278^{\\circ},2.784m,0.032s$}\\, &\\,\\footnotesize{$\\textbf{0.261}^{\\circ},\\textbf{0.005}m,6.024s$}\\, &\\,\\footnotesize{$\\textbf{0.261}^{\\circ},\\textbf{0.005}m,0.604s$}\\,&\\,\\footnotesize{$\\textbf{0.261}^{\\circ},\\textbf{0.005}m,\\textbf{0.057}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-06, can}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{06-soda-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=214$, 95.33\\%}& & \\,\\footnotesize{$104.921^{\\circ},1.537m,0.031s$}\\, &\\,\\footnotesize{$\\textbf{0.544}^{\\circ},\\textbf{0.012}m,6.814s$}\\, &\\,\\footnotesize{$\\textbf{0.544}^{\\circ},\\textbf{0.012}m,0.744s$}\\,&\\,\\footnotesize{$\\textbf{0.544}^{\\circ},\\textbf{0.012}m,\\textbf{0.059}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-07, cap}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{07-cap-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=326$, 96.32\\%}& & \\,\\footnotesize{$102.869^{\\circ},1.769m,0.045s$}\\, &\\,\\footnotesize{$\\textbf{1.272}^{\\circ},\\textbf{0.017}m,12.510s$}\\, &\\,\\footnotesize{$\\textbf{1.272}^{\\circ},\\textbf{0.017}m,0.784s$}\\,&\\,\\footnotesize{$\\textbf{1.272}^{\\circ},\\textbf{0.017}m,\\textbf{0.093}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-11, can}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{11-soda-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=339$, 97.05\\%}& & \\,\\footnotesize{$140.652^{\\circ},2.390m,0.038s$}\\, &\\,\\footnotesize{$121.956^{\\circ},4.738m,18.808s$}\\, &\\,\\footnotesize{$\\textbf{0.582}^{\\circ},\\textbf{0.014}m,0.753s$}\\,&\\,\\footnotesize{$\\textbf{0.582}^{\\circ},\\textbf{0.014}m,\\textbf{0.119}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-12, mug}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{12-mug-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\n\\\\\n\n&&\\footnotesize{$N=382$, 96.07\\%}& & \\,\\footnotesize{$103.388^{\\circ},1.950m,0.047s$}\\, &\\,\\footnotesize{$\\textbf{0.640}^{\\circ},\\textbf{0.006}m,17.814s$}\\, &\\,\\footnotesize{$\\textbf{0.640}^{\\circ},\\textbf{0.006}m,0.910s$}\\,&\\,\\footnotesize{$\\textbf{0.640}^{\\circ},\\textbf{0.006}m,\\textbf{0.103}s$}\\,\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{Scene-14, bowl}}\\,}\\,\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-cor-eps-converted-to.pdf}\n\\end{minipage}\n\n& &\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\n\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\n\n&\n\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{14-bowl-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\n\\end{tabular}\n\n\\caption{Supplementary object localization and pose estimation results on the RGB-D scenes dataset~\\cite{lai2011large}. The explicit organization of the results is similar to Fig.~\\ref{obj-local1}. Best results are shown in \\textbf{bold} font.}\n\\label{obj-local2}\n\\vspace{-3mm}\n\\end{figure*}\n\n\\subsection{Real Application: Object Localization}\n\nWe evaluate DANIEL in the real-world 3D object localization and pose estimation application. We adopt the large-scale RGB-D Scenes dataset~\\cite{lai2011large}, where 3D object models (point clouds) in different shapes and different RGB-D scene scans are provided. For each pair of object and scene, we extract the point cloud of the object from the scene according to the ground-truth labels, and then change the pose of this object with a random transformation ($||\\boldsymbol{t}||\\leq 3$). Afterwards, we downsample the object and the scene and apply FPFH to build correspondences between the transformed object and the entire scene, which would easily trigger high outlier ratios. Then, we use one non-minimal solver: GNC-TLS, one RANSAC solver: FLO-RANSAC (with 10000 maximum iterations), one guaranteed outlier removal solver: GORE+RANSAC, and our solver DANIEL to estimate the relative pose (rigid transformation) between the object and scene, where noise is constantly set to $\\sigma=0.001m$.\n\nWe conduct 10 tests over 10 scenes with 5 objects, including the \\textit{coffee mug, cereal box, cap, soda can} and \\textit{bowl}, where all these tests are with extreme outlier ratios (ranging from 95.16\\% to 97.94\\%). We report the qualitative and quantitative object localization results in Fig.~\\ref{obj-local1} and~\\ref{obj-local2}, where each figure shows the results of 5 tests in 5 scenes (with possibly different objects). We can see that in such a high-outlier environment, GNC-TLS fails to render reasonable results in all tests, while FLO-RANSAC, mostly running in tens of seconds, could only succeed in 40\\% tests. GORE+RANSAC and our DANIEL are the two solvers that can successfully localize all the objects (to be specific, estimate the correct transformation and reproject the object back to the scene) in all the 10 tests. More importantly, DANIEL runs only in tens or hundreds of milliseconds and is significantly faster than GORE+RANSAC, showing the most outstanding performance.\n\n\n\\subsection{Scan Matching}\n\nScan matching, or called scene stitching, is another crucial application of point cloud registration, underlying the 3D reconstruction and SLAM technologies. We test DANIEL for scan matching based on the Microsoft 7-scenes dataset~\\cite{shotton2013scene} that provides realistic RGB images with associated depth images. We deliberately select 10 pairs of scans with overlapping scenes from the \\textit{red kitchen, office, chess, stairs} and \\textit{heads}. Since FPFH may generate too many outliers on RGB-D data, \n\n\\clearpage\n\n\\begin{figure*}[h]\n\\centering\n\\setlength\\tabcolsep{0pt}\n\\addtolength{\\tabcolsep}{-0.2pt}\n\\begin{tabular}{c|cc|c|c|c|c}\n\\quad &\\,\\footnotesize{Correspondences}\\, &\\,&  \\footnotesize{GNC-TLS} & \\footnotesize{FLO-RANSAC} & \\footnotesize{GORE+RANSAC} & \\footnotesize{DANIEL}\n\\\\\n\\hline\n\n & \\footnotesize{$N=1840$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.245s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.308,8.960s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.283},11.727s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.299,\\textbf{0.182}s$}\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-0-59-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-0-59-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\\\\n\n & \\footnotesize{$N=1273$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.118s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.352,34.662s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.343,8.657s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.335},\\textbf{1.190}s$}\n\n\\\\\n\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-55-110-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-55-110-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\\\\n\n & \\footnotesize{$N=900$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.093s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.438,12.606s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.335,7.440s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.314},\\textbf{0.177}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-100-140-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-100-140-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\\\\n\n & \\footnotesize{$N=1984$} && \\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.314,0.291s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.516,8.044s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.323,74.158s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.301},\\textbf{0.272}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-290-340-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-290-340-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=1823$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.199s$} &\\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,49.098s$}&\\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,63.692s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.278},\\textbf{0.781}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{red kitchen}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{redkitchen-450-520-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{redkitchen-450-520-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=2002$} && \\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.288,0.293s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.502,19.645s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.292,70.343s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.287},\\textbf{0.278}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{chess}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{chess-0-70-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{chess-0-70-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{chess-0-70-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=1822$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.262s$} &\\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,48.811$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.326,12.171s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.324},\\textbf{0.784}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{office}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{office-0-140-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-0-140-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-0-140-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=985$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.112s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.373,26.628$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.360,9.035s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.334},\\textbf{0.249}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{office}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{office-910-980-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{office-910-980-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{office-910-980-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=244$} && \\footnotesize{\\textcolor[rgb]{1,0,0}{Fail}$,\\verb|\\|,0.036s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.430},0.706$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.330,0.418s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.430},\\textbf{0.039}s$}\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{stair}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{stair-0-70-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{stair-0-70-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{stair-0-70-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\\\\\n\n & \\footnotesize{$N=2001$} && \\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,\\textbf{0.224},0.294s$} &\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.262,5.414$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,0.225,49.971s$}&\\footnotesize{\\textcolor[rgb]{0,0.8,0}{Succeed}$,{0.227},\\textbf{0.269}s$}\n\n\\\\\n\\rotatebox{90}{\\,\\,\\footnotesize{\\textit{heads}}\\,}\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.1\\linewidth}\n\\centering\n\\includegraphics[width=1\\linewidth]{heads-0-50-cor-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n& &\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GNC-TLS-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GNC-TLS-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-FLO-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-FLO-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GORE-RANSAC-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-GORE-RANSAC-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n&\n\\,\\,\n\\begin{minipage}[t]{0.19\\linewidth}\n\\centering\n\\includegraphics[width=.48\\linewidth]{heads-0-50-DANIEL-in-eps-converted-to.pdf}\n\\includegraphics[width=.48\\linewidth]{heads-0-50-DANIEL-eps-converted-to.pdf}\n\\end{minipage}\\,\\,\n\n\\end{tabular}\n\n\\caption{Scan matching results on the Microsoft 7-scenes dataset~\\cite{shotton2013scene}. The left-most column demonstrates the putative correspondences matched by SURF in cyan lines (already converted into the 3D space using depth and calibration information), and the rest columns show: (i) the inliers found and (ii) the qualitative scene stitching results, using FLO-RANSAC, GNC-TLS, GORE+RANSAC and DANIEL, respectively. On top of the results of each solver, from left to right, we display the stitching status (either \\textcolor[rgb]{1,0,0}{Fail} or \\textcolor[rgb]{0,0.5,0}{Succeed}), the RMSE and runtime (in seconds), respectively. Note that stitching with visibly distinguishable error should be regarded as \\textcolor[rgb]{1,0,0}{Fail}, and when the stitching is failed, we no longer display the RMSE since it would become meaningless. Best results are shown in \\textbf{bold} font. Best viewed when zoomed-in.}\n\\label{scan-matching}\n\\end{figure*}\n\\clearpage\n\n\\noindent we use SURF~\\cite{bay2006surf} (Matlab function: \\textit{detectSURFFeatures}) to match 2D keypoint correspondences across the two RGB images and then convert them into 3D correspondences by using depth data and camera intrinsic parameters. We also apply the four solvers: GNC-TLS, FLO-RANSAC, GORE+RANSAC and DANIEL, for comparative evaluation.\n\nBoth qualitative and quantitative scan matching results are displayed in Fig.~\\ref{scan-matching}, where we show the raw correspondences matched by SURF, and the inliers found as well as the scan matching results by the respective robust solvers in qualitative results, and show the registration status (\\textcolor[rgb]{1,0,0}{Fail} means that the scenes are wrongly stitched and \\textcolor[rgb]{0,0.5,0}{Succeed} means that the stitching is successful), RMSE (Root Mean Square Error) and runtime in quantitative results. Though 2D keypoint matching is used, we can observe that the inliers are merely a small minority of the putative correspondences. The results show that GNC-TLS is the least robust solver, failing in 70\\% tests, and FLO-RANSAC and GORE-RANSAC both generate a few failure cases but their runtime are too slow for practical use. Our DANIEL remains highly robust in all tests and also most often has the lowest RMSE and the fastest speed, which further validates the promising practicality of DANIEL.\n\n\n\n\\section{Conclusion}\n\nIn this paper, a novel consensus maximization method for correspondence-based robust point cloud registration with high or even extreme outliers is proposed. We design a double-layered random sampling framework with the smart application of 3D rigidity constraint, in order to tremendously reduce the computational cost for sampling pure inlier subsets even in high-outlier registration problems. We further render a stratified element-wise compatibility checking technique during the sampling process, which intends to circumvent the repeated consensus building procedure so as to accelerate the algorithm convergence. These two contributions enable both the high robustness and the high time-efficiency of the resulting solver DANIEL.  Extensive experiments over diverse datasets validate that DANIEL is superior to other existing state-of-the-art methods. To be specific, DANIEL is robust against extremely high outlier ratios as many as 99\\%, is up to four orders of magnitude faster than RANSAC and also faster than other solvers in most cases, and proves to be effective in practical application problems such as object pose estimation and scan matching. The source code of DANIEL is available at: \\url{https:\/\/github.com\/LeiSun-98\/Daniel}.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Acknowledgements}\nNicola Gigante and Luca Geatti acknowledge the support of the Free University of\nBozen-Bolzano, Faculty of Computer Science, by means of the projects TOTA\n(\\emph{Temporal Ontologies and Tableaux Algorithms}) and STAGE (\\emph{Synthesis\nof Timeline-based Planning Games}).\n\n\\bibliographystyle{eptcs}\n\n\\section{Controller synthesis}\n\\label{sec:games}\n\nIn this section we use the deterministic automaton constructed above to\nobtain a deterministic arena where we can solve a simple reachability game\nfor checking the existence of (and, in this case, to synthesize)\na controller for the corresponding timeline-based game.\n\n\\subsection{From the automaton to the arena}\n\nLet $G=(\\SV_C, \\SV_E, \\S, \\D)$ be a timeline-based game. We use the construction\nof the automaton explained in the previous section in order to obtain a game\narena. However, the automaton construction considers a planning problem with a\nsingle set of synchronization rules, while here we have to account for the roles\nof both $\\S$ and $\\D$. \n\nTo do that, let $A_\\S$ and $A_\\D$ be the deterministic automata built over the\ntimeline-based planning problem $P_\\S=(\\SV_C\\cup\\SV_E, \\S)$ and\n$P_\\D=(\\SV_C\\cup\\SV_E, \\D)$, respectively. We define the automaton $A_G$ as\n$\\overline{A_\\D}\\cup A_\\S$, \\ie the union of $A_\\S$ with the complement of\n$A_\\D$. Note that these are all standard automata-theoretic constructions over\nDFAs. Any accepting run of $A_G$ represents either a plan that violates the\ndomain rules or a plan that satisfies both the domain and the system rules, in\nconformance with \\cref{def:games:winning-strategy}. Note that $A_G$ is\ndeterministic and can be built from $A_\\D$ and $A_\\S$ with only a polynomial\nincrease in size.\n\nNow, the $A_G$ automaton is still not suitable as a game arena, because the\nmoves of the timeline-based game are not directly visible in the labels of the\ntransitions. In other words, the $A_G$ automaton reads events, while we need an\nautomaton that reads game \\emph{moves}. In particular, a single transition in\nthe automaton can correspond to different combinations of rounds, since the\npresence of $\\wait(\\delta)$ moves is not explicit in the transition. For\nexample, an event $\\event=(A, 5)$ can be the result of a $\\wait(5)$ move by\n\\charlie followed by a $\\play(5, A)$ move by $\\eve$, or by any $\\wait(\\delta)$\nmove with $\\delta>5$ followed by $\\play(5, A)$. Hence, we need to further adapt\n$A_G$ to obtain a suitable arena.\n\nLet $A_G=(Q,\\Sigma, q_0, F, \\tau)$ be the automaton built as described before.\nLet $\\event=(A,\\delta)$ be an event. If $\\delta>1$, this transition must have\nresulted from \\charlie playing a $\\wait(\\delta')$ move with $\\delta'\\ge\\delta$.\nHowever, if $A$ contains any $\\tokend(x,v)$ action with $x\\in\\SV_C$, this is for\nsure the result of more than one pair of starting\/closing rounds. In order to\nsimplify the construction below, we remove this possibility beforehand. More\nformally, we define a slightly different automaton $A_G'=(Q,\\Sigma,q_0,F,\\tau')$\nwhere $\\tau'$ is now a \\emph{partial} transition function (\\ie the automaton\nbecomes \\emph{incomplete}) that agrees with $\\tau$ on everything excepting that\ntransitions $\\tau(q,(\\actions,\\delta))$ is \\emph{undefined} if $\\delta>1$ and\n$\\actions$ contains any $\\tokend(x,v)$ action with $x\\in\\SV_C$. You can see an\nexample of this operation in \\cref{fig:constructions}, on the left. Note that\nthis removal does not change the plans accepted by the automaton because for\neach transition $\\tau(q,(\\actions,\\delta))=q'$ with $\\delta > 1$ there are two\ntransitions $\\tau(q,(\\emptyset,\\delta-1))=q''$ and $\\tau(q'',(\\actions,1))=q'$.\n\n\\begin{figure}\n  \\begin{tikzpicture}[state\/.style={fill, circle, minimum width=5pt}]\n    \\path (0,0) node[state] (n1) { }\n          (4,4) node[state] (n2) { }\n          (4,0) node[state] (n3) { };\n\n    \\path[draw,dashed,->] (n1) -- (n2) node[midway, above, sloped, font=\\scriptsize] { \n      $\\event=(\\set{\\tokend(x,v)}, 5)$\n    };\n\n    \\path[draw, ->] (n1) -- (n3) node[midway, above, font=\\scriptsize] { \n      $\\event'=(\\emptyset, 4)$\n    };\n    \\path[draw, ->] (n3) -- (n2) node[midway, above, sloped, rotate=180, font=\\scriptsize] { \n      $\\event''=(\\set{\\tokend(x,v)}, 1)$\n    };\n\n    \\begin{scope}[xshift=6cm]\n      \\path (0,0) node[state] (n1) { } node[above, outer sep=5pt] {$q$}\n            (9,4) node[state] (n2) { } node[above, outer sep=5pt] {$w$};\n\n      \\path[draw,dashed,->] (n1) -- (n2) node[midway, above, sloped, font=\\scriptsize] { \n        $\\event=(\\set{\\tokend(x,v_1),\\tokend(y,w_1),\\tokstart(x,v_2),\\tokstart(y,w_2)}, 5)$\n      };\n\n      \\path (3,0) node[state] (q6) { } node[above right] { }\n            (3,0.75) node[state] (q5) { } node[above right] { }\n            (3,-0.75) node[state] (q7) { } node[above right] { }\n            (3,-2.25) node[state] (q10) { } node[above right] { };\n\n      \\path[dashed] (q7) -- (q10);\n\n      \\path[draw,->] (n1) -- (q5) node[above, sloped, near end, font=\\scriptsize] { $\\wait(5)$ };\n      \\path[draw,->] (n1) -- (q6) node[above, sloped, near end, font=\\scriptsize] { $\\wait(6)$ };\n      \\path[draw,->] (n1) -- (q7) node[above, sloped, near end, font=\\scriptsize] { $\\wait(7)$ };\n      \\path[draw,->] (n1) -- (q10) node[above, sloped, near end, font=\\scriptsize] { $\\wait(10)$ };\n\n      \\path (6,0) node[state] (q'') { };\n      \\path[draw,->] (q5) -- (q'') node[above, sloped, midway, font=\\scriptsize] { $\\play\\left(5, \n        \\left\\{\\begin{array}{@{}c@{}}\n          \\tokend(x,v_1)\\\\\n          \\tokend(y,w_1)\n        \\end{array}\\right\\}\\right)$ };\n      \\path[draw,->] (q6) -- (q'');\n      \\path[draw,->] (q7) -- (q'');\n      \\path[draw,->] (q10) -- (q'') node[below, sloped, midway, font=\\scriptsize] { $\\play\\left(5, \n      \\left\\{\\begin{array}{@{}c@{}}\n        \\tokend(x,v_1)\\\\\n        \\tokend(y,w_1)\n      \\end{array}\\right\\}\\right)$ };\n\n      \\path (9,0) node[state] (q''') { };\n      \\path[draw,->] (q'') -- (q''') node[midway, above,font=\\scriptsize] {\n        $\\play(\\set{\\tokstart(x,v_2)})$\n      };\n      \\path[draw,->] (q''') -- (n2) node[midway, sloped,rotate=180, above,font=\\scriptsize] {\n        $\\play(\\set{\\tokstart(y,w_2)})$\n      };\n    \\end{scope}\n  \\end{tikzpicture}\n  \\caption{On the left, the removal of transitions $\\event=(A,\\delta)$ with\n  $\\delta>1$ and ending actions of controllable tokens in $A$. On the right, the\n  transformation of a transition of the $A_G$ into a sequence of transitions in\n  $A^a_G$, with $x\\in\\SV_C$, $y\\in\\SV_E$, and\n  $\\gamma_x(v_1)=\\gamma_y(w_1)=\\mathsf{u}$.}\n  \\label{fig:constructions}\n\\end{figure}\n\nNow we can transform the automaton in order to make the game rounds, and\nespecially $\\wait(\\delta)$ moves, explicit. Intuively, each transition of the\nautomaton is split into four transitions explicitating the four moves of the two\nrounds. Given the automaton $A_G'=(Q,\\Sigma, q_0, F, \\tau')$, we define the\nautomaton $A_G^a=(Q^a,\\Sigma^a, q_0^a, F^a, \\tau^a)$, which will be the arena of\nour game, as follows:\n\\begin{enumerate}\n  \\item $Q^a=Q\\cup\\set{q_\\delta\\mid 1\\le\\delta\\le d}\\cup\\set{q_{\\delta,A}\\mid 1\\le\\delta\\le d, A\\subseteq \\mathsf{A}}$ is the set of states;\n  \\item $\\Sigma^a=\\moves_C\\cup\\moves_E$, \\ie the alphabet is turned into the set\n    of moves of the two players;\n  \\item $q_0^a=q_0$ and $F^a=F$, \\ie initial and final states do not change;\n  \\item the (partial) transition function $\\tau^a$ is defined as follows. Let\n    $w=\\tau(q,\\event)$ with $\\event=(\\actions,\\delta)$. We distinguish the case where $\\delta=1$ or $\\delta>1$.\n    \\begin{enumerate}\n      \\item if $\\delta=1$, let $\\actions_C\\subseteq\\actions$ and\n      $\\actions_E\\subseteq\\actions$ be the set of actions in $\\actions$ playable\n      by \\charlie and by \\eve, respectively. Then:\n      \\begin{enumerate}\n        \\item $\\tau(q,\\play(\\actions_C^e))=q_{1,\\actions_C^e}$, where \n          $\\actions_C^e$ is the set of \\emph{ending} actions in $\\actions_C$;\n        \\item $\\tau(q_{1,\\actions_C^e},\\play(\\actions_E^e))=q_{1,\\actions_C^e\\cup\\actions_E^e}$, where \n          $\\actions_E^e$ is the set of \\emph{ending} actions in $\\actions_E$;\n        \\item $\\tau(q_{1,\\actions_C^e\\cup\\actions_E^e},\\play(\\actions_C^s))=q_{1,\\actions_C^e\\cup\\actions_E^e\\cup\\actions_C^s}$, where \n          $\\actions_C^s$ is the set of \\emph{starting} actions in $\\actions_C$;\n        \\item $\\tau(q_{1,\\actions_C^e\\cup\\actions_E^e\\cup\\actions_C^s},\\play(\\actions_E^s))=w$, where \n          $\\actions_E^s$ is the set of \\emph{starting} actions in $\\actions_E$;\n      \\end{enumerate}\n      where the mentioned states are added to $Q^a$ as needed.\n      \\item if $\\delta>1$, let $\\actions_C\\subseteq\\actions$ and\n      $\\actions_E\\subseteq\\actions$ be the set of actions in $\\actions$ playable\n      by \\charlie and by \\eve, respectively. Note that by construction,\n      $\\actions_C$ only contains \\emph{starting} actions. Then:\n      \\begin{enumerate}\n        \\item $\\tau(q,\\wait(\\delta_C))=q_{\\delta_C}$ for all \n          $\\delta\\le\\delta_C\\le d$;\n        \\item $\\tau(q_{\\delta_C},\\play(\\delta, \\actions_E^e))=q_{\\delta,\\actions_E^e}$\n          where $\\actions_E^e$ is the set of \\emph{ending} actions in\n          $\\actions_E$;\n        \\item $\\tau(q_{\\delta,\\actions_E^e},\\play(\\actions_C))=q_{\\delta,\\actions_E^e\\cup\\actions_C}$;\n        \\item $\\tau(q_{\\delta,\\actions_E^e\\cup\\actions_C},\\play(\\actions_E^s))=w$ where\n          $\\actions_E^s$ is the set of \\emph{starting} actions in $\\actions_E$;\n      \\end{enumerate}\n      where the mentioned states are added to $Q^a$ as needed.\n    \\end{enumerate}\n    All the transitions not explicitly defined above are undefined.\n\\end{enumerate}\n\nA graphical example of the above construction can be seen in\n\\cref{fig:constructions}, on the right. Note that the structure of the original\n$A_G$ automaton is preserved by $A^a_G$. In particular, one can see that for\neach $q\\in Q$ and event $\\event=(A,\\delta)$, any sequence of moves whose outcome\nwould append $\\event$ to the partial plan (see \\cref{def:games:round-outcome})\nreach from $q$ the same state $w$ in $A^a_G$ that is reached in $A_G$ by reading\n$\\event$. Hence, one can consider $A^a_G$ to also being able to \\emph{read}\nevent sequences, even though its alphabet is different. We denote as $[\\evseq]$\nthe state $q\\in Q^a$ reached by reading $\\evseq$ in $A^a_G$.\n\nMoreover, note that, with a minimal abuse of notation, any play $\\bar\\rho$ for\nthe game $G$ can be seen as a word readable by the automaton $A_G^a$. Hence, we\ncan prove the following.\n\\begin{theorem}\n  \\label{thm:arena-soundness}\n  If $G$ is a timeline-based game, for any play $\\bar\\rho$ for $G$, $\\bar\\rho$\n  is successful if and only if it is accepted by $A_G^a$.\n\\end{theorem}\n\n\\subsection{Computing the Winning Strategy}\n\nOnce built the arena, we can focus on computing the winning region $W_C$\nfor \\charlie, that is, the set of states of the arena from which \\charlie\ncan force the play to reach a final state of $A_G^a$, no matter of the\nstrategy of \\eve. These games are called \\emph{reachability\ngames}~\\cite{Thomas2008}.  If the winning region $W_C$ is not empty,\na winning strategy of \\charlie can be simply derived from $W_C$.  As\na consequence of \\cref{thm:soundness-completeness,thm:arena-soundness}, the\ncomputed winning strategy $\\sigma_C$ for $A_G^a$ respects\n\\cref{def:games:winning-strategy}.\n\nAs stated in \\cite[Theorem 4.1]{Thomas2008}, rechability games are\ndetermined, and the winning region $W_C$ along with the corresponding\npositional winning strategy $s$ are computable. Let $A_G^a=(Q^a,\\Sigma^a,\nq_0^a, F^a, \\tau^a)$ be the automaton built from $G$ as described in the\nprevious section. Note that, by construction, in any state $q\\in Q^a$ only\none of the players has available moves. Let $Q^a_C\\subseteq Q^a$ be the set\nof states \\emph{belonging} to \\charlie, \\ie states from which \\charlie can\nmove, and let $Q^a_E=Q^a\\setminus Q^a_C$. Moreover, let $E=\\set{ (q, q')\n\\in Q^a\\times Q^a \\mid \\exists \\event \\mathrel{.} \\tau^a(q, \\event) = q'}$,\n\\ie the set of all the edges of $A_G^a$. \n\nNow, for each $i\\ge0$, we can compute the $i$-th attractor of $F^a$, written $Attr^i_C(F^a)$, that is, the set of states from which \\charlie can win in at most $i$ steps. $Attr^i_C(F^a)$ is defined as follows:\n\\begin{align*} Attr^0_C (F^a) &= F^a \\\\\nAttr^{i+1}_C (F^a) &= Attr^i_C (F^a) \\\\\n&\\cup \\set{ q^a \\in Q^a_C \\, | \\, \\exists r \\big((q^a, r) \\in E \\land r \\in Attr^i_C (F^a)\\big) } \\\\\n&\\cup \\set{ q^a \\in Q^a_E \\, | \\, \\forall r \\big((q^a, r) \\in E \\implies r \\in Attr^i_C (F^a) \\big) }\n\\end{align*}\nAs remarked in \\cite{Thomas2008}, the sequence $Attr^0_C (F^a) \\subseteq\nAttr^1_C (F^a) \\subseteq Attr^2_C (F^a) \\subseteq \\ldots$ becomes\nstationary for some index $k \\leq \\lvert Q^a \\rvert$. Thus, we define\n$Attr_C (F^a) = \\bigcup^{\\lvert Q^a \\rvert}_{i=0} Attr^i_C(F^a)$. \nIn order to prove that $W_C = Attr_C(F^a)$, it suffices to use the proof of\n\\cite[Theorem 4.1]{Thomas2008} for showing that $Attr_C(F^a) \\subseteq W_C$\nand $W_C \\subseteq Attr_C(F^a)$.\n\nTo compute a winning strategy for \\charlie in the case that $q_0^a\\in W_C$,\nit is sufficient to define $s (q) = \\mu$ for any $\\mu$ such that\n$\\tau^a(q,\\mu)=q'$ with $q,q'\\in W_C$ (which is guaranteed to exist by\nconstruction of the attractor). Then, the strategy $\\sigma_C$ for \\charlie in $G$ (see \\cref{def:games:winning-strategy}) is defined as $\\sigma_C(\\evseq)\n= s([\\evseq])$. \n\n\\begin{theorem}\n    \\label{thm:winning-region-soundness-completeness}\n  Given $A_G^a=(Q^a,\\Sigma^a, q_0^a, F^a, \\tau^a)$, $q_0^a\\in W_C$ if and only\n  if \\charlie has a winning strategy $\\sigma_C$ for $G$.\n\\end{theorem}\n\\begin{proof}\n  We first prove \\emph{soundness}, that is, $q_0^a\\in W_C$ implies that\n  \\charlie has a winning strategy $\\sigma_C$ for $G$. If $q_0^a\\in W_C$,\n  then it means that there exists a positional winning strategy $s$ for\n  \\charlie for the reachability game over the arena $A_G^a$. By\n  \\cref{thm:arena-soundness} and by the definition of reachability game, we\n  know that each play generated by $s$ corresponds to a successful play for\n  game $G$. \n  Let $\\sigma_C(\\evseq)=s([\\evseq])$ be the winning strategy for \\charlie\n  in game $G$ as defined above. By construction of $\\sigma_C$ and by\n  \\cref{def:games:winning-strategy}, this means that $\\sigma_C$ is\n  a winning strategy of \\charlie for $G$.\n\n  To prove \\emph{completeness} (\\ie if \\charlie has a winning strategy\n  $\\sigma_C$ for $G$ then $q_0^a\\in W_C$), we proceed as follows. From\n  \\cref{def:games:winning-strategy} we know that a winning strategy\n  $\\sigma_C$ for \\charlie is a strategy such that for every admissible\n  strategy $\\sigma_E$ for \\eve, there exists $n \\ge 0$ such that the play\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ is successful.  From\n  \\cref{thm:arena-soundness}, we know that\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ is accepted by $A^a_G$.  Therefore,\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ reaches a state in the set\n  $F^a$ starting from $q^a_0$. By definition of reachability game, this\n  means that $q^a_0 \\in W_C$.\n\\end{proof}\n\n\n\\section{A deterministic automaton for timeline-based planning}\n\\label{sec:automaton}\n\nIn this section we encode a timeline-based planning problem into a\n\\emph{deterministic} finite state automaton (DFA) that recognises all and only\nthose event sequences that represent solution plans for such problem. This\nautomaton will form the basis for the game arena solved in the next section. The\nwords accepted by the automaton are \\emph{event sequences} representing solution\nplans.\n\nLet $P=(\\SV, S)$ be a timeline-based planning problem. To get a finite alphabet,\nwe define $d=\\max(L,U)+1$, where $L$ and $U$ are in turn the \\emph{maximum}\nlower and (finite) upper bounds appearing in any rule of $P$, and we account\nonly for event sequences such that the distance between two consecutive events\nis at most $d$. It can be easily seen that this assumption does not loose\ngenerality (for a proof, see Lemma~4.8 in \\cite{Gigante19}). Hence, the symbols\nof the alphabet $\\Sigma$ are \\emph{events} of the form $\\event = \\pair{A,\n\\delta}$, where $A \\subseteq \\actions_\\SV$ and $1\\le\\delta\\le d$. Formally,\n$\\Sigma=2^{\\actions_\\SV}\\times\\ar{d}$, where $\\ar{d}=\\set{1,\\ldots,d}$. Note\nthat the size of $\\Sigma$ is exponential in the size of the problem. Moreover,\nwe define the amount $\\window(P)$ as the product of all the non-zero\ncoefficients appearing as upper bounds in rules of $P$. Intuitively, $\\window(P)$\nis the maximum amount of time a rule of $P$ can \\emph{count} far away from the\noccurrence of the quantified tokens. For example, consider the following rule:\n\\begin{align*}\n  a_0[x_0=v_0]\\to{} &\\exists a_1[x_1=v_1] a_2[x_2=v_2] a_3[x_3=v_3] \\suchdot \\\\\n  &\\tokstart(a_1)\\before_{[4,14]}\\tokend(a_0)\n  \\land \\tokend(a_0)\\before_{[0,+\\infty]}\\tokend(a_2) \\land \\tokstart(a_2)\\before_{[0,3]}\\tokend(a_3) \n\\end{align*}\nIn this case, $\\window(P)$ (assuming this is the only rule of the problem),\nwould be $3\\cdot 14=42$. This means the rule can precisely account for what\nhappens at most $42$ time point from the occurrence of its quantified tokens.\nFor example, if the token $a_1$ appears at a given distance from $a_0$, it has\nto be at less than $42$ time points (less than $14$, in particular), and any\nmodification of the plan that changes such distance has the potential to break\nthe satisfaction of the rule. Instead, what happens further away from $a_0$ only\naffects the satisfaction of the rule \\emph{qualitatively}. Suppose the tokens\n$a_2$ and $a_3$ lie at $100$ time points from $a_0$ (at most $3$ time steps from\neach other). Changing this distance (while maintaining the qualitative order\nbetween tokens) cannot ever break the satisfaction of the rule. See\n\\cite{Gigante19} for a precise account of the properties of $\\window(P)$.\n\nA key observation underlying our construction is that every atomic temporal\nrelation $T \\before_{l,u} T'$ can be rewritten as the conjunction of two\ninequalities $T^\\prime - T \\leq u$ and $T - T^\\prime \\leq -l$, and that the\nclause \\clause of an existential statement \\E can be rewritten as a system of\ndifference constraints $\\nu(\\clause)$ of the form $T - T' \\leq n$, with $n \\in\n\\Z_{+\\infty}$. Then, the system $\\nu(\\clause)$ can be conveniently represented\nby a squared matrix $D$ indexed by terms, where the entry associated with $D[T,\nT']$ gives the upper bound on $T- T'$. Such matrices, which take the name of\n\\emph{Difference Bound Matrices} (DBMs)~\\cite{dill1989timing,peron2007abstract},\ncan be conveniently updated as the plan evolves to keep track of the\nsatisfaction of the atomic temporal relations among terms. In building a DBM for\nthe system of constraints $\\nu(\\clause)$, we augment the system with constraints\nof kind $\\tokstart(a_i)-\\tokend(a_i)\\leq-\\dmin^{x_i=v_i}$ and\n$\\tokend(a_i)-\\tokstart(a_i)\\leq\\dmax^{x_i=v_i}$, for any quantified token\n$a_i[x_i=v_i]$ of $\\E$. Moreover, if two different bounds $T - T'\\le n$ and\n$T-T'\\le n'$ with $n'<n$ belong to $\\nu(\\clause)$, we keep only $T-T'\\le n'$. As\nan example, the DBM for the existential graph of the rule above is the one in\n\\cref{fig:dbm}.\n\\begin{figure}\n  \\begin{equation*}\n    \\begin{array}{rcccccccc}\\toprule\n      &\\tokstart(a_0) & \\tokend(a_0) & \\tokstart(a_1) & \\tokend(a_1) &\n      \\tokstart(a_2) & \\tokend(a_2) & \\tokstart(a_3) & \\tokend(a_3) \\\\\\midrule\n      \\tokstart(a_0) \\\\\n      \\tokend(a_0) & & & -4 & & & \\\\\n      \\tokstart(a_1) & & 14 \\\\\n      \\tokend(a_1) \\\\\n      \\tokstart(a_2)  & & & & & & & & 3 \\\\\n      \\tokend(a_2) & & 0 \\\\\n      \\tokstart(a_3) \\\\\n      \\tokend(a_3)  & & & & & 0 & & & \\\\\\bottomrule\n    \\end{array}\n  \\end{equation*}\n  \\caption{DBM of an example synchronization rule. Missing entries are intended to be $+\\infty$.}\n  \\label{fig:dbm}\n\\end{figure}\nNote that, when the bounds of the temporal relations are translated into a DBM,\nthere is no longer a distinction between \\emph{lower} and \\emph{upper} bounds.\nHowever, for some of the entries we can retrieve their original meaning. Indeed,\nif $D[T,T'] < 0$, then such entry is the lower bound of a temporal relation\n$T\\before_{l,u}T'$, whereas, if $D[T, T'] > 0$, it is the upper bound of a\nrelation $T' \\before_{l,u} T$.\n\nOn top of DBMs, we define the concept of \\emph{matching structure}, a data\nstructure that allows us to manipulate and reason about partially matched\nexistential statements, \\ie existential statements of which only a part of the\nrequests has already been satisfied by the part of the word already read, while\nthe rest can be still potentially matched in the future.\n\n\\begin{definition}[Matching Structure]\n  \\label{def:matching-structure}\n  Let $\\E\\equiv \\exists a_1[x_1 = v_1] \\dots a_m[x_m = v_m] \\,.\\, \\clause$ be\n  the existential statement of a synchronisation rule $\\Rule \\equiv a_0[x_0 =\n  v_0] \\rightarrow \\E_1 \\lor \\dots \\lor \\E_k$ over the set of state variables\n  \\SV.\n\n  The \\emph{matching structure} for $\\E$ is a tuple $\\M_{\\E} = (V, D, M, t)$\n  where:\n  \\begin{itemize}\n  \\item $V$ is the set of terms $\\tokstart(a)$ and $\\tokend(a)$ for\n    $a\\in\\set{a_0, \\dots, a_m}$;\n  \\item $D \\in \\Z_{+\\infty}^{|V|^2}$ is a DBM indexed by terms of $V$ where\n    $D[T,T']=n$ if $(T-T'\\le n) \\in \\nu(\\clause)$, $D[T,T']=0$ if $T=T'$, and $D[T,T']=+\\infty$ otherwise;\n  \\item $M \\subseteq V$ and $0\\le t \\le \\window(P)$.\n  \\end{itemize}\n\\end{definition}\n\nThe set $M$ contains the terms of $V$ that the matching structure has correctly\nmatched over the event sequence read so far. With $\\overline{M} = V \\setminus M$\nwe denote the actions that we have yet to see. Then, we say that a matching\nstructure $\\M$ is \\emph{closed} if $M = V$, it is \\emph{initial} if $M =\n\\emptyset$ and it is \\emph{active} if it is not closed and $\\tokstart(a_0) \\in\nM$. Intuitively, a matching structure is \\emph{active} if its trigger has been\nmatched over the word the automaton is reading. Then, when all the terms have\nbeen matched over the word, the matching structure becomes \\emph{closed}. The\ncomponent $t$ is the time elapsed since $\\tokstart(a_0)$ has been matched.\nWhen time flows, a matching structure can then be updated as follows.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{definition}[Time shifting]\n  \\label{def:time-shift}\n  Let $\\delta > 0$ be a positive amount of time, and $\\M = (V, D, M, t)$ be a\n  matching structure. The result of shifting $\\M$ by $\\delta$ time units,\n  written $\\M + \\delta$, is the matching structure $\\M^\\prime = (V, D^\\prime, M,\n  t')$ where:\n  \\begin{itemize}\n  \\item for all $T, T' \\in V$:\n    \\begin{equation*}\n      D^\\prime[T,T'] =\n      \\begin{cases}\n        D[T,T'] + \\delta &\\text{if } T \\in M \\text{ and } T' \\in\n        \\overline{M}\\\\% D[Tj] \u00e8 lower bound\n        D[T,T'] - \\delta &\\text{if } T \\in \\overline{M} \\text{ and } T' \\in\n        M\\\\% D[ij] \u00e8 upper bound\n        D[T,T'] &\\text{otherwise}\n      \\end{cases}\n    \\end{equation*}\n  \\item and\n    \\[\n      t' =\n      \\begin{cases}\n        t+\\delta & \\text{if } \\M \\text{ is \\emph{active}}\\\\\n        t & \\text{otherwise}\n      \\end{cases}\n    \\]\n  \\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Matching]\n  \\label{def:matching}\n  Let $\\M = (V, D, M, t)$ be a matching structure and $I \\subseteq \\overline{M}$\n  a set of matched terms. A matching structure $\\M^\\prime = (V, D, M^\\prime, t)$\n  is the result of matching the set $I$, written $\\M \\cup I$, if $M^\\prime = M\n  \\cup I$.\n\\end{definition}\n\nBeside updating the reference $t$ to the trigger occurrence of an active\nmatching structure, \\Cref{def:time-shift} dictates how to update the entries of\nthe DBM. In particular, the distance bounds between any pair of terms $T$ and\n$T'$ where one is in $M$ and the other is not are tighten by the elapsing of\ntime: when $T\\in M$ and $T'\\in\\overline{M}$, $D[T,T']$ is a lower bound loosen\nby adding the elapsed time $\\delta$, when $T\\in\\overline{M}$ and $T'\\in M$,\n$D[T,T']$ is an upper bound tighten by subtracting $\\delta$. For example,\nconsider the DBM in \\cref{fig:dbm} and consider the pair of terms\n$\\tokstart(a_1)$ and $\\tokend(a_0)$. $D[\\tokstart(a_1),\\tokend(a_0)]=-4$,\nmeaning that $\\tokend(a_0)-\\tokstart(a_1)\\le 4$ must hold. Suppose\n$\\tokstart(a_1)\\in M$ (\\ie it has been matched), and $\\tokend(a_0)\\not\\in M$ (it\nstill has to). Now, if $1$ time point passes, the entry in the DBM is\nincremented and updated to $-4+1=-3$, which corresponds to the constraint\n$\\tokend(a_0)-\\tokstart(a_1)\\le 3$. This reflects the fact that to be able to\nsatisfy the constraint, $\\tokend(a_0)$ has now only $3$ time steps left before\nit is too late.\n\\Cref{def:matching} tells us how to update the set $M$ of a matching structure.\n\nTo correctly match an existential statement while reading an event sequence, a\nmatching structure is updated only as long as no violations of temporal\nconstraints are witnessed. As such, an event is classified from the standpoint\nof a matching structure as \\emph{admissible} or not.\n\n\\begin{definition}[Admissible Event]\n  An event $\\event = (A, \\delta)$ is \\emph{admissible} for a matching structure\n  $\\M_{\\E} = (V, D, M, t)$ if and only if, for every $T \\in M$\n  and $T' \\in \\overline{M}$, $\\delta \\leq D[T',T]$, \\ie the elapsing of\n  $\\delta$ time units does not exceed the upper bound of some term $T'$ not yet\n  matched by $\\M_{\\E}$.\n\\end{definition}\n\nEach admissible event $\\event$ read from the word can be matched with a subset\nof the terms of the matching structure. There are usually more than one way to\nmatch events and terms. The following definition makes this choice explicit.\n\n\\begin{definition}[$I$-match Event]\\label{def:match-event\n  Let $\\M_{\\E} = (V, D, M, t)$ be a matching structure and  $I \\subseteq\n  \\overline{M}$. An $I$\\emph{-match event} is an admissible event $\\event = (A,\n  \\delta)$ for $\\M_{\\E}$ such that:\n  \\begin{enumerate}\n  \\item for all token names $a \\in \\mathsf{N}$ quantified as $a[x = v]$ in $\\E$\n    we have that:\\label{def:match-event:good-match}\n    \\begin{enumerate}\n    \\item if $\\tokstart(a) \\in I$, then $\\tokstart(x, v) \\in A$;\n      \\label{def:match-event:good-match:start}\n    \\item $\\tokend(a) \\in I$ if and only if $\\tokstart(a) \\in M$ and $\\tokend(x,v) \\in\n      A$;\\label{def:match-event:good-match:end}\n    \\end{enumerate}\n  \\item and for all $T \\in I$ it holds that:\\label{def:match-event:relations}\n    \\begin{enumerate}\n    \\item \\label{def:match-event:preceding-terms} for every other term $T' \\in\n      V$, if $D[T',T] \\leq 0$, then $T' \\in M \\cup I$;\n    \\item \\label{def:match-event:lower-bounds} for all $T' \\in M$, $\\delta \\geq\n      -D[T',T]$, \\ie all the lower bounds on $T$ are satisfied;\n    \\item \\label{def:match-event:zero-no-bounds} for each other term $T' \\in I$,\n      either $D[T',T] = 0$, $D[T,T'] = 0$, or $D[T',T] = D[T, T'] = +\\infty$.\n    \\end{enumerate}\n  \\end{enumerate}\n\\end{definition}\n\nIntuitively, an event is an $I$-match event if the actions in the event\ncorrectly match the terms in $I$. \\Cref{def:match-event:good-match} ensures that\neach term is correctly matched over an action it represents and, most\nimportantly, that the endpoints of a quantified token correctly identify the\nendpoints of a token in the event sequence. \\Cref{def:match-event:relations}\nensures that matching the terms in $I$ does not violate any atomic temporal\nrelation. In particular, \\Cref{def:match-event:preceding-terms} deals with the\nqualitative aspect of an ``happens before'' relation, while \\Cref{%\n  def:match-event:lower-bounds,%\n  def:match-event:zero-no-bounds%\n} deal with the quantitative aspects of the lower bounds of these relations.\nNote that an $\\emptyset$-event is admitted.\n\nLet $\\matchstructs_P$ be the set of all the matching structures for a planning\nproblem $P$. By \\Cref{def:match-event}, a single event can represent several\n$I$-match events for a matching structure, hence a matching structure can evolve\ninto several matching structures, one for each $I$-match event. Such\nevolution is defined as a ternary relation $S \\subseteq\\matchstructs_P \\times\n\\Sigma \\times \\matchstructs_P$ such that $(\\M, (A, \\delta), \\M^\\prime) \\in {S}$\nif and only if $(A, \\delta)$ is an $I$-match event for $\\M$ and $\\M^\\prime = (\\M\n+ \\delta) \\cup I$. To deal with the nondeterministic nature of this relation,\nstates of the automaton will comprise sets of matching structures collecting all\nthe possible outcomes of $S$, so that suitable notation for working with sets of\nmatching structures, denoted by $\\Upsilon$ hereafter, is introduced. We define\n$\\Upsilon^\\Rule_t\\subseteq\\Upsilon$ as the set of all the \\emph{active} matching\nstructures $\\M\\in\\Upsilon$ with timestamp $t$, associated with any existential\nstatement of $\\Rule$. Intuitively, matching structures in $\\Upsilon^\\Rule_t$\ncontribute to the fulfilment of the same triggering event for the rule \\Rule\n(because they have the same timestamp), regardless of the existential statement\nthey represent. We also define $\\Upsilon_\\bot\\subseteq\\Upsilon$ as the set of\n\\emph{non active} matching structures of $\\Upsilon$. A set $\\Upsilon$ is\n\\emph{closed} if there exists $\\M \\in \\Upsilon$ such that $\\M$ is \\emph{closed}.\nLastly, a function $\\step_\\event$ extends the relation $S$ to sets of matching\nstructures: $\\step_\\event(\\Upsilon) = \\set{\\M' | (\\M,\\event,\\M^\\prime)\\in S,\n\\text{ for some } \\M \\in \\Upsilon}$.\n\nWe are now ready to define the automaton. If $\\E$ is an existential statement,\nlet $\\mathbb{E}_\\E$ be the set of all the existential statements of the same\nrule of $\\E$. Let $\\mathbb{F}_P$ be the set of functions mapping each existential\nstatement of $P$ to a set of existential statements, and let $\\mathbb{D}_P$ be the\nset of functions mapping each existential statement to a set of matching\nstructures. A simple automaton $\\TV_P$ that checks the transition function and\nduration functions of the variables is easy to define. Then, given a\ntimeline-based planning problem $P=(\\SV,S)$, the corresponding automaton is\n$A_P=\\TV_P\\cap\\S_P$ where $\\S_P$, the automaton that checks the satisfaction of\nthe synchronization rules, is defined as $\\S_P = (Q, \\Sigma, q_0, F, \\tau)$,\nwhere:\n\\begin{enumerate}\n\\item $Q = 2^\\matchstructs \\times \\mathbb{D} \\times \\F \\cup \\set{\\bot}$ is the\n  finite set of states, \\ie states are tuples of the form $\\langle \\Upsilon,\n  \\Delta, \\Phi \\rangle\\in2^\\matchstructs \\times \\mathbb{D} \\times \\F$, plus a\n  sink state $\\bot$;\n\\item $\\Sigma$ is the input alphabet defined above;\n \n \n\\item the initial state $q_0 = \\langle \\Upsilon_0, \\Delta_0, \\Phi_0 \\rangle$ is\n  such that $\\Upsilon_0$ is the set of initial matching structures of the\n  existential statements of $P$ and, for all existential statements $\\E$ of $P$,\n  we have $\\Delta_0(\\E) = \\emptyset$ and $\\Phi_0(\\E) = \\mathbb{E}_\\E$;\n \n \n \n \n\\item $F \\subseteq Q$ is the set of final states defined as:\n  \\[\n    F = \\Set{ \\langle \\Upsilon, \\Delta, \\Phi \\rangle \\in Q |\n      \\begin{gathered}\n        \\M \\text{ is not \\emph{active} for all } \\M \\in\n        \\Upsilon\\\\\n        \\text{and }\\Delta(\\E)=\\emptyset\\text{ for all }\\E\\text{ of } P\n      \\end{gathered}}\n  \\]\n\\item $\\tau : Q \\times \\Sigma \\rightarrow Q$ is the transition function that\n  given a state $q=\\langle \\Upsilon, \\Delta, \\Phi \\rangle$ and a symbol $\\event\n  = (A, \\delta)$ computes the new state $\\tau(q,\\event)$. Let\n  $\\step^\\E_\\event(\\Upsilon^\\Rule_t)=\\set{\\M_\\E \\mid\n  \\M_\\E\\in\\step_\\event(\\Upsilon^\\Rule_t)}$. Moreover, let $\\Psi^\\Rule_t = \\set{ \\E |\n  \\M_{\\E} \\in \\step_\\event(\\Upsilon^\\Rule_t)}$. Then, the updated components of\n  the state are based on what follows, where $W = \\window(P)$:\n  \\begin{align*}\n    \\Upsilon' &= \\step_\\event(\\Upsilon_\\bot) \\cup \\bigcup \\Set{\n      \\step_\\event(\\Upsilon^\\Rule_t) |\n      \\text{$t<W-\\delta$ and \n      $\\step_\\event(\\Upsilon^\\Rule_t)$ is not \\emph{closed}}} \\\\\n    \\Delta'(\\E) &=\\begin{cases}\n        \\step^\\E_\\event(\\Upsilon^\\Rule_t) & \\text{where $t$ is the minimum such that $t\\ge W-\\delta$ and $\\step^\\E_\\event(\\Upsilon^\\Rule_t)\\ne\\emptyset$} \\\\\n        \\step_\\event(\\Delta(\\E)) & \\text{if such $t$ does not exist}\n      \\end{cases}\\\\\n    \\Phi'(\\E) &= \\begin{cases}\n      \\mathbb{E}_\\E\\quad\\text{if $\\E\\in\\Psi(\\E')$ for some $\\E'$ such that \n      $\\Delta'(\\E')$ is \\emph{closed}}  \\\\\n      \\Phi(\\E) \\setminus \n        \\set{\n          \\E'\\mid \\exists t\\ge W-\\delta \\suchdot \\E'\\in\\Psi^\\Rule_t \n          \\land \\E\\not\\in\\Psi^\\Rule_t\n        } \\quad \\text{otherwise}\n    \\end{cases}\n  \\end{align*}\n\n  Let $\\Delta''(\\E)=\\Delta'(\\E)$ unless there is an $\\E'$ with $\\E\\in\\Phi'(\\E')$\n  such that $\\Delta'(\\E')$ is \\emph{closed}, in which case\n  $\\Delta''(\\E)=\\emptyset$. Then, $\\tau(q,\\event)=\\seq{\\Upsilon', \\Delta'',\n  \\Phi'}$ if the following holds:\n  \\begin{enumerate}\n  \\item for every $\\Upsilon^\\Rule_t$, $\\step_\\event(\\Upsilon^\\Rule_t) \\neq\n    \\emptyset$, and \\label{dfa:delta:no-failed-step}\n  \\item for every synchronisation rule $\\Rule \\equiv a_0[x_0=v_0] \\rightarrow\n    \\E_1 \\lor \\dots \\lor \\E_n$ in $S$, if $\\tokstart(x_0, v_0) \\in A$, then\n    there exists $\\M_{\\E_i} = (V,D,M,0) \\in \\Upsilon'$, \n    with $i \\in \\{1\\dots n\\}$, such that $\\tokstart(a_0) \\in M$;\\label{dfa:delta:trigger-capture}\n  \\end{enumerate}\n  Otherwise, $\\tau(q,\\event)=\\bot$.\n\\end{enumerate}\n\nLet us explain what is going on. The first component $\\Upsilon$ of an automaton\nstate $q=(\\Upsilon, \\Delta,\\Phi)$ is a set of matching structures that keeps\ntrack of what have been tracked so far. Intuitively, the automaton precisely\nkeeps track of what happened to the last $\\window(P)$ time points, and only\nsummarizes what happened before that window, which is what allows us to keep the\nsize under control. Any matching structure in $\\Upsilon$ has $t<\\window(P)$.\nMatching structures in $\\Upsilon$ evolve following the $\\step$ function, until\nthey are closed or the $t$ component reaches $\\window(P)$. Matching structures\nthat reach $\\window(P)$ are promoted to a new role. Their new task is to record\nthe pieces of existential statements that still have to be matched in order to\nsatisfy all the trigger events of $\\Rule$ that no longer fit into the\n\\emph{recent history} of the event sequence (\\ie the last $\\window(P)$ time\npoints). These matching structures are not stored in $\\Upsilon$ though, they are\nsummarized by the function $\\Delta$ that maps each existential statement $\\E$ of\na rule $\\Rule$ to the set of matching structures for $\\E$ with $t=\\window(P)$.\n\nWhen a set $\\Upsilon^\\Rule_t$ exceeds the bound $\\window(P)$, the $\\Delta$\nfunction must be updated by merging the information of $\\Upsilon^\\Rule_t$ to the\ninformation already present in $\\Delta$. Now, it has to be noted that, by\nclosing a set $\\Delta(\\E)$, we can not conclude that every event that triggered\n$\\Rule$ actually satisfies $\\Rule$. Indeed, there can be sets $\\Delta(\\E)$ and\n$\\Delta(\\E')$ that are in charge of the satisfaction of the same rule $\\Rule$,\nbut for different trigger events, and closing $\\Delta(\\E)$ does not imply that\n$\\Rule$ has been satisfied. The opposite case may also arise, in which\n$\\Delta(\\E)$~and~$\\Delta(\\E')$ contribute to the fulfilment of the same trigger\nevents and closing either set suffices to satisfy $\\Rule$. To overcome the\ninformation lost when a set of matching structures gets added to the $\\Delta$\nfunction, the $\\Phi$ function (the third component of the automaton states) maps\neach existential statement $\\E$ to the set of existential statements $\\E'$ such\nthat $\\Delta(\\E')$ tracks the fulfilment of the same trigger events of the set\n$\\Delta(\\E)$. We use $\\Phi$ as follows: when a set $\\Delta(\\E)$ gets closed, we\ncan discard its matching structures and all the matching structures of the sets\n$\\Delta(\\E')$, with $\\E' \\in \\Phi(\\E)$.\n\nOne can prove the soundness and completeness of our construction.\n\n\\begin{theorem}(Soundness and completeness)\n  \\label{thm:soundness-completeness}\n  Let $P=(\\SV, S)$ be a timeline-based planning problem and let $\\A_P$ be the\n  associated automaton. Then, any event sequence $\\evseq$ is a solution plan for\n  $P$ if and only if $\\evseq$ is accepted by $\\A_P$.\n\\end{theorem}\n\nRecall that we assumed the timestamp of each event of event sequences to be\nbounded, but since events can have an empty set of actions,\n\\cref{thm:soundness-completeness} can actually deal with arbitrary event\nsequences, after adding suitable empty events. Now, let us look at the size of\nthe automaton. Let $E$ be the overall number of existential statements in $P$,\nwhich is linear in the size of $P$. It can be seen that $\\abs{\\mathbb{D}_P} \\in\n\\O({(2^{\\abs{\\matchstructs_P}})}^E)= \\O(2^{E\\cdot\\abs{\\matchstructs_P}})$, \\ie\nthe number of $\\Delta$ functions is doubly exponential in the~size~of~$P$. Then,\nobserve that $\\lvert\\mathbb{F}_P\\rvert \\in \\mathcal{O}({(2^E)}^E) =\n\\mathcal{O}(2^{E^2})$. Then, $\\abs{\\S_P} \\in \\O(\\abs{\\Sigma}\\cdot\n2^{\\abs{\\matchstructs_P}})$, that is, the size of $\\S_P$ is at most exponential\nin the number of possible matching structures. To bound this number, let $N$ be\nthe largest finite constant appearing in $P$ as bound in any atom or value\nduration function and let $L$ be the length of the largest existential prefix of\nan existential statement occurring inside a rule of $P$. Notice that $N$ is\nexponential in the size of $P$, since constants are expressed in binary, while\n$L \\in \\O(\\abs{P})$. Then, the entries of a DBM for $P$, of which there is a\nnumber quadratic in $L$, are constrained to take values within the interval\n$\\ar{-N,N}$ (excluding the infinitary value $+\\infty$), whose size is linear in\n$N$. By \\Cref{def:matching-structure}, it follows that, for the planning problem\n$P$, $\\abs{\\matchstructs_P} \\in \\O(N^{L^2} \\cdot 2^L \\cdot \\window(P))$, \\ie the\nnumber of matching structures is at most exponential in the size of $P$. Hence,\nwe proved the following:\n\\begin{theorem}[Size of the automaton]\n  Let $P=(\\SV, S)$ be a timeline-based planning problem and let $\\A_P$ be the\n  associated automaton. Then, the size of $A_P$ is at most doubly-exponential in\n  the size of $P$.\n\\end{theorem}\nNote that this is the same size as the automaton built by\nDella~Monica~\\etal~\\cite{DellaMonicaGMS18}, but their automaton was\n\\emph{nondeterministic}, while ours is by construction deterministic, essential \nfor its use as a game arena.\n\n\n\\section{Timeline-based games}\n\\label{sec:preliminaries}\n\nIn this section, we introduce timeline-based games, as defined in\n\\cite{GiganteMOCR20}. \n\n\\subsection{State variables, event sequences, synchronization rules}\nThe first basic concept is that of \\emph{state variable}.\n\n\\begin{definition}[State variable]\n  \\label{def:timelines:state-variable}\n  A \\emph{state variable} is a tuple $x=(V_x,T_x,D_x,\\gamma)$, where:\n  \\begin{itemize}\n  \\item $V_x$ is the \\emph{finite domain} of $x$;\n  \\item $T_x:V_x\\to2^{V_x}$ is the \\emph{value transition function} of $x$,\n    which maps each value $v\\in V_x$ to the set of values that can immediately\n    follow it;\n  \\item $D_x:V_x\\to\\N\\times\\N$ is the \\emph{duration function} of $x$, mapping\n    each value $v\\in V_x$ to a pair $(\\dmin, \\dmax)$ specifying respectively the\n    minimum and maximum duration of any interval where $x=v$;\n  \\item $\\gamma:V_x\\to\\set{\\mathsf{c},\\mathsf{u}}$ is the \n    \\emph{controllability tag}, that, for each value $v\\in V_x$, specifies whether it\n    \\emph{controllable} ($\\gamma(v)=\\mathsf{c}$) or \\emph{uncontrollable}\n    ($\\gamma(v)=\\mathsf{u}$).\n  \\end{itemize}\n\\end{definition}\n\nIntuitively, a state variable $x$ takes a value from a finite domain and\nrepresents a simple finite-state machine, whose transition function is $T_x$.\nThe behaviour over time of a set of state variables $\\SV$ is defined by a \nset of \\emph{timelines}, one for each variable.\nInstead of reasoning about timelines directly, though, in this paper we follow \nthe approach outlined in \\cite{GiganteMOCR20} and represent the whole execution of a \nsystem, modeled by means of a set of state variables, with a single word, called \\emph{event \nsequence}.\n\n\\begin{definition}[Event sequence \\cite{GiganteMOCR20}]\n  \\label{def:event-sequence}\n  %\n  Let $\\SV$ be a set of state variables. Let $\\actions_\\SV$ be the set of all\n  the terms, called \\emph{actions}, of the form $\\tokstart(x,v)$ or\n  $\\tokend(x,v)$, where $x\\in\\SV$ and $v\\in V_x$.\n\n  An \\emph{event sequence} over $\\SV$ is a sequence\n  $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$ of pairs $\\event_i=(A_i,\\delta_i)$,\n  called \\emph{events}, where $A_i\\subseteq\\actions_\\SV$ is a set of actions,\n  and $\\delta_i\\in\\N_+$, such that, for any $x\\in\\SV$:\n  \\begin{enumerate}\n  \\item \\label{def:event-sequence:start}\n        for all $1\\le i\\le n$, if $\\tokstart(x,v)\\in A_i$, for some $v\\in V_x$,\n        then there is no $\\tokstart(x,v')$ in any $\\event_j$ before the\n        closest $\\event_k$, with $k > i$, such that $\\tokend(x,v)\\in A_k$ (if\n        any);\n  \\item \\label{def:event-sequence:end}\n        for all $1\\le i\\le n$, if $\\tokend(x,v)\\in A_i$, for some $v\\in V_x$,\n        then there is no $\\tokend(x,v')$ in any $\\event_j$ after the\n        closest $\\event_k$, with $k < i$, such that $\\tokstart(x,v)\\in A_k$ (if\n        any);\n  \\item \\label{def:event-sequence:gaps-right}\n        for all $1\\le i < n$, if $\\tokend(x,v)\\in A_i$, for some $v\\in V_x$, then\n        $\\tokstart(x,v')\\in A_i$, for some $v'\\in V_x$;\n  \\item \\label{def:event-sequence:gaps-left}\n        for all $1< i \\le n$, if $\\tokstart(x,v)\\in A_i$, for some $v\\in V_x$,\n        then $\\tokend(x,v')\\in A_i$, for some $v'\\in V_x$.\n  \\end{enumerate}\n\\end{definition}\n\nIntuitively, an event sequence represents the evolution over time of the state\nvariables of the system by representing the \\emph{start} and the \\emph{end} of\n\\emph{tokens}, \\ie a sequence of adjacent\nintervals where a given variable takes a given\nvalue. An event $\\event_i=(A_i,\\delta_i)$ consists of a set $A_i$ of actions\ndescribing the start or the end of some tokens, happening $\\delta_i$ time steps\nafter the previous one. In an \\emph{event sequence}, events are collected to\ndescribe a whole plan.\n\n\\Cref{def:event-sequence} intentionally implies that a started token is not\nrequired to end before the end of the sequence, and a token can end without the\ncorresponding starting action to have ever appeared before. In this case we say\nthe event sequence is \\emph{open} (on the right or on the left, respectively).\nOtherwise, it is said to be \\emph{closed}. An event sequence closed on the left\nand open on the right is also called a \\emph{partial plan}. Note that the empty\nevent sequence is closed on both sides for any variable. Moreover, on closed\nevent sequences, the first event only contains $\\tokstart(x,v)$ actions and the\nlast event only contains $\\tokend(x,v)$ actions,\none for each variable $x$.\nGiven an event sequence $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$ over a set of\nstate variables $\\SV$, with $\\event_i=(A_i,\\delta_i)$, we define\n$\\delta(\\evseq)=\\sum_{1<i\\le n}\\delta_i$, that is, $\\delta(\\evseq)$ is the time\nelapsed from the start to the end of the sequence (its duration). The amount\nof time spanning a subsequence, written as $\\delta_{i,j}$ when $\\evseq$ is clear\nfrom context, is then $\\delta(\\slice\\evseq_{i,j})=\\sum_{i<k\\le j}\\delta_k$.\nFinally, given an event sequence $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$, for\neach $1<i\\le n$, we define $\\evseq_{<i}$ as $\\seq{\\event_1,\\ldots,\\event_{i-1}}$. \n\nIn timeline-based games, the controller plays to satisfy a set of\n\\emph{synchronization rules}, which describe the desired behavior of the system.\nSynchronization rules relate tokens, possibly belonging to different timelines,\nthrough temporal relations among token endpoints. Let \\SV be a set of state variables\nand $\\toknames = \\set{a,b,\\ldots}$ be an arbitrary set of \\emph{token names}.\nMoreover, let an \\emph{atomic temporal relation}, or simply \\emph{atom}, be an\nexpression of the form $\\production{term} \\before_{l,u} \\production{term}$, \nwhere $l\\in\\N$, $u\\in\\N\\cup\\set{\\infty}$, and a \\emph{term} is either $\\tokstart(a)$ \nor $\\tokend(a)$, for some $a\\in\\toknames$.\nA synchronisation rule \\Rule takes the following form:\n\\begin{equation*}\\label{eq:synchronisation-rules}\n  \\begin{array}{rcl}\n    \\Rule&\\bydef&a_0[x_0=v_0] \\implies \\E_1 \\lor \\E_2 \\lor \\dots \\lor \\E_k\\quad where\\\\[0.4em]\n    \\E_i&\\bydef&\\exists a_1[x_1=v_1] a_2[x_2=v_2]\\ldots a_n[x_n=v_n] \\suchdot \\clause_i\n  \\end{array}\n\\end{equation*}\nwhere $a_0, \\ldots, a_n \\in \\toknames$, $x_0,\\ldots,x_n \\in \\SV$, $v_0,\\ldots,\nv_n$ are such that $v_i \\in V_{x_i}$, for all $0 \\leq i\\leq n$, and\n$\\clause_i$ is a conjunction of atomic temporal relations (a clause). The elements\n$a_i[x_i=v_i]$ are called \\emph{quantifiers} and the quantifier $a_0[v_0=x_0]$\nis called the \\emph{trigger}. The disjuncts in the body are called\n\\emph{existential statements}.\n\nWe say that a token $\\tau = (x, v, d)$ \\emph{satisfies} a quantifier\n$a_i[x_i=v_i]$ if $x = x_i$ and $v = v_i$. The semantics of a synchronisation\nrule \\Rule states that for every token satisfying the trigger, at least one of\nthe existential statements is satisfied. Each existential statement $\\E_i$\nrequires the existence of some tokens, satisfying the quantifiers in its prefix,\nsuch that the clause $\\clause_i$ is satisfied. When a token satisfies the\ntrigger of a rule, it is said to \\emph{trigger} such a rule.\n\nFor space concerns, we do not provide all the details of the semantics of synchronization rules. The reader can find them in \\cite{GiganteMOCR20}. Intuitively, each time there is a token that satisfies the trigger of a rule, one of its existential statements must be satisfied as well. The existential statements in turn assert the existence of other tokens that satisfy a conjunction of atoms.\n\nIf $a$ and $b$ are two token names, then examples of atomic relations are $\\tokstart(a)\\before_{3,7}\\tokend(b)$ and $\\tokstart(a)\\before_{0,+\\infty}\\tokstart(b)$. Intuitively, a token name $a$ refers to a specific token, that is, a pair of $\\tokstart(x,v)$ and $\\tokend(x,v)$ actions in an event sequence, and $\\tokstart(a)$ and $\\tokend(a)$ to its endpoints. Then, an atom such as $\\tokstart(a)\\before_{l,u}\\tokend(b)$ constrains $a$ to start before the end of $b$, with the distance between the two endpoints to be comprised between the lower and upper bounds $l$ and $u$.\n\nExamples of synchronization rules\nare the following,\nwhere the relations $=$ and $\\before$ are respectively syntactic sugar for\n$\\before_{0,0}$ and $\\before_{0,+\\infty}$:\n\\begin{align*}\n  a[x_s=\\mathsf{Comm}] \\implies {}\n  & \\exists b[x_g=\\mathsf{Available}] \\suchdot\n    \\tokstart(b) \\before \\tokstart(a) \\land \\tokend(a) \\before \\tokend(b)\\\\\n  a[x_s=\\mathsf{Science}] \\implies {}\n  & \\exists b[x_s=\\mathsf{Slewing}] c[x_s=\\mathsf{Earth}] d[x_s=\\mathsf{Comm}]\n    \\suchdot {}\\\\\n  & \\tokend(a) = \\tokstart(b) \\land \\tokend(b) = \\tokstart(c) \\land\n    \\tokend(c) = \\tokstart(d)\n\\end{align*}\nwhere the variables $x_s$ and $x_g$ represent respectively the state of a\nspacecraft and the visibility of the communication ground station. The first\nrule requires the satellite and the ground station to synchronise their\ncommunications, so that when the satellite is transmitting the ground station\nis available for reception. The second rule instructs the system to transmit\ndata back to Earth after every measurement session, interleaved by the required\nslewing operation. A rule whose trigger is empty ($\\true$), called \n\\emph{triggerless rule}, can be used to state the \\emph{goal} of the system.\nAs an example, they allow one to force the spacecraft to\nperform some scientific measurement at all:\n\\begin{equation*}\n  \\true \\implies\\exists a[x_s=\\mathsf{Science}]\n\\end{equation*}\n\nTriggerless rules have a trivial universal quantification, which means they only\ndemand the existence of some tokens, as specified by the existential statements.\nAlthough triggerless rules are meant to specify the goals of a planning problem,\nthey can be regarded as syntactic sugar on top of the syntax described above.\nIndeed, triggerless rules can be translated into triggered\nrules~\\cite{GiganteMOCR20}, and thus we do not consider them from here onwards.\n\nFinally, even though our focus is on timeline-based games, we conclude the section \nby formally defining \\emph{timeline-based planning problems}.\n\\begin{definition}[Timeline-based planning problem]\n  A \\emph{timeline-based planning problem} is a pair $P=(\\SV,\\S)$, where $\\SV$ is\n  a set of state variables and $\\S$ is a set of synchronization rules over\n  $\\SV$. An event sequence $\\evseq$ over $\\SV$ is a solution plan for $P$ if all\n  the rules in $\\S$ are satisfied by $\\evseq$.\n\\end{definition}\n\n\\subsection{The game arena}\n\nWe are now ready to introduce \\emph{timeline-based games}. Their definition is quite\ninvolved, as their structure has been designed with the goal of being strictly\nmore general than \\emph{timeline-based planning with\nuncertainty}~\\cite{CialdeaMayerOU16} while being able to capture its semantics\nprecisely. For space concerns, we keep the exposition quite terse, but the\nreader can refer to \\cite{GiganteMOCR20} for details.\n\n\\begin{definition}[Timeline-based game]\n  \\label{def:games:game}\n  A \\emph{timeline-based game} is a tuple $G=(\\SV_C,\\SV_E,\\S,\\D)$,\n  where $\\SV_C$ and $\\SV_E$ are the sets of \\emph{controlled}\n  and \\emph{external} variables, respectively, and $\\S$ and $\\D$ are the sets of\n  \\emph{system}  and \\emph{domain} synchronisation rules, respectively, both \n  involving variables from $\\SV_C$ and $\\SV_E$.\n\\end{definition}\n\nA partial plan for $G$ is a partial plan over the state variables\n$\\SV_C\\cup\\SV_E$. Let $\\partialplans_G$ be the set of all possible partial plans\nfor $G$, simply $\\partialplans$ when there is no ambiguity.\n\nSince $\\epsilon$ is a closed event sequence and $\\delta(\\epsilon)=0$, the\n\\emph{empty} partial plan $\\epsilon$ is a good starting point for the game.\nPlayers incrementally build a richer partial plan, starting from $\\epsilon$, by\nplaying actions that specify which tokens to start and\/or to end, adding an\nevent that extends the event sequence, or complementing the existing\nlast event of the sequence. We partition all the available actions into those\nthat are playable by either of the two players.\\fitpar\n\n\\begin{definition}[Partition of player actions]\n  \\label{def:games:actions-partition}\n  %\n  Let $\\SV=\\SV_C\\cup\\SV_E$. The set $\\actions$ of available actions over $\\SV$\n   is partitioned into the sets $\\actions_C$ of \\charlie's actions and  \n   $\\actions_E$ of \\eve's actions, which are defined as follows:\\fitpar\n  \\begin{align}\n    \\actions_C = {} &\n      \\underbrace{%\n        \\set{\\tokstart(x,v)\\suchthat x\\in\\SV_C,\\; v\\in V_x}%\n      }_{\\text{start tokens on \\charlie's timelines}}\\;\\cup\\;\n      \\underbrace{%\n        \\set{\\tokend(x,v)\\suchthat x\\in\\SV,\\; v\\in V_x,\\; \\gamma_x(v)=\\ctag}%\n      }_{\\text{end controllable tokens}}\\\\\n    \\actions_E = {} &\n      \\underbrace{%\n        \\set{\\tokstart(x,v)\\suchthat x\\in\\SV_E,\\; v\\in V_x}%\n      }_{\\text{start tokens on \\eve's timelines}}\\;\\cup\\;\n      \\underbrace{%\n        \\set{\\tokend(x,v)\\suchthat x\\in\\SV,\\; v\\in V_x,\\; \\gamma_x(v)=\\utag}%\n      }_{\\text{end uncontrollable tokens}}\n  \\end{align}\n  %\n\\end{definition}\n\nHence, players can start tokens for the variables that they own, and end the\ntokens that hold values that they control. Actions are combined into\n\\emph{moves} that can start\/end multiple tokens at once.\n\n\\begin{definition}[Moves]\n  \\label{def:games:moves}\n  %\n  A \\emph{move} $\\move_C$ for \\charlie is a term of the form $\\wait(\\delta_C)$\n  or $\\play(A_C)$, where $\\delta_C\\in\\N^+$ and $\\emptyset\\ne\n  A_C\\subseteq\\actions_C$ is  either a set of \\emph{starting} actions or \n  a set of \\emph{ending} actions.\n\n  A \\emph{move} $\\move_E$ for \\eve is a term of the form $\\play(A_E)$ or\n  $\\play(\\delta_E,A_E)$, where $\\delta_E\\in\\N^+$ and $A_E\\subseteq\\actions_E$ is\n  either a set of \\emph{starting} actions or a set of \\emph{ending} actions.\n  %\n\\end{definition}\n\nWe denote by $\\moves_C$ and $\\moves_E$ the set of moves playable by \\charlie\nand $\\eve$, respectively. Moves such as $\\play(A_C)$ and\n$\\play(\\delta_E,A_E)$ can play either $\\tokstart(x,v)$ actions only or\n$\\tokend(x,v)$ actions only. A move of the former kind is called a\n\\emph{starting} move, while a move of the latter kind is called an \\emph{ending}\nmove. We consider $\\wait$ moves as \\emph{ending} moves. Moreover, Starting and\nending moves have to be alternated during the game.\n\n\\begin{definition}[Round]\n  \\label{def:games:round}\n  %\n  A \\emph{round} $\\round$ is a pair\n  $(\\move_C,\\move_E)\\in\\moves_C\\times\\moves_E$ of moves such that:\n  \\begin{enumerate}\n  \\item \\label{def:games:round:alternation}\n        $\\move_C$ and $\\move_E$ are either both \\emph{starting} or both\n        \\emph{ending} moves;\n  \\item \\label{def:games:round:paring}\n        either $\\round=(\\play(A_C),\\play(A_E))$, or\n        $\\round=(\\wait(\\delta_C),\\play(\\delta_E,A_E))$, with\n        $\\delta_E\\le\\delta_C$;\n  \\end{enumerate}\n  %\n\\end{definition}\n\nA \\emph{starting} (\\emph{ending}) round is one made of starting (ending) moves.\nNote that since \\charlie cannot play empty moves and $\\wait$ moves are\nconsidered ending moves, each round is unambiguously either a starting or an\nending round. Also note that since $\\play(\\delta_E,A_E)$ moves are played only\nin rounds together with $\\wait(\\delta_C)$, and $\\wait(\\delta_C)$ is always an\nending move, then any $\\play(\\delta_E,A_E)$ must be an ending move. We can now\ndefine how a round is applied to the current partial plan to obtain the new one.\nThe game always starts with a single \\emph{starting round}.\n\n\\begin{definition}[Outcome of rounds]\n  \\label{def:games:round-outcome}\n  %\n  Let $\\evseq=\\seq{\\event_1,\\ldots,\\event_n}$ be an event sequence, with\n  $\\event_n=(A_n,\\delta_n)$ or $\\event_n=(\\emptyset,0)$ if $\\evseq=\\epsilon$.\n  Let $\\round=(\\move_C,\\move_E)$ be a round, let $\\delta_E$ and $\\delta_C$ be\n  the time increments of the moves, with $\\delta_C=\\delta_E=1$ for $\\play(A)$\n  moves, and let $A_E$ and $A_C$ be the set of actions of the two moves ($A_C$\n  is empty if $\\move_C$ is a $\\wait$ move).\n\n  The \\emph{outcome} of $\\round$ on $\\evseq$ is the event sequence\n  $\\round(\\evseq)$ defined as follows:\n  %\n  \\begin{enumerate}\n  \\item \\label{def:games:round-outcome:starting}\n        if $\\rho$ is a starting round, then $\\rho(\\evseq)=\\evseq_{< n}\\event_n'$,\n        where $\\event_n'\\nobreak=\\nobreak(A_n\\cup A_C\\cup A_E,\\delta_n)$;\n  \\item \\label{def:games:round-outcome:ending}\n        if $\\rho$ is an ending round, then $\\rho(\\evseq)=\\evseq\\event'$, where\n        $\\event'=(A_C\\cup A_E,\\delta_E)$;\n  \\end{enumerate}\n  %\n  We say that $\\round$ is \\emph{applicable} to $\\evseq$ if:\n  \\begin{enumerate}[label=\\alph*)]\n  \\item \\label{def:games:round-outcome:integrity}\n        the above construction is well-defined, \\ie $\\round(\\evseq)$ is a\n        valid event sequence by \\cref{def:event-sequence};\n  \\item \\label{def:games:round-outcome:alternation}\n        $\\round$ is an ending round if and only if $\\evseq$ is open for all\n        variables.\n  \\end{enumerate}\n  %\n\\end{definition}\n\nWe say that a single move by either player is applicable to $\\evseq$ if there is\na move for the other player such that the resulting round is applicable to\n$\\evseq$.\n\nThe game starts from the empty partial plan $\\epsilon$, and players play in\nturn, composing a round from the move of each one, which is applied to the\ncurrent partial plan to obtain the new one.\n\nIt is now time to define the notion of \\emph{strategy} for each player, and of\n\\emph{winning strategy} for \\charlie.\n\n\\begin{definition}[Strategies]\n  \\label{def:games:strategies}\n  %\n  A \\emph{strategy for Charlie} is a function\n  $\\strategy_C:\\partialplans\\to\\moves_C$ that maps any given partial plan\n  $\\evseq$ to a move $\\move_C$ applicable to $\\evseq$.\n  A \\emph{strategy for Eve} is a function\n  $\\strategy_E:\\partialplans\\times\\moves_C\\to\\moves_E$ that maps a partial\n  plan $\\evseq$ and a move $\\move_C\\in\\moves_C$ applicable to $\\evseq$, to a\n  $\\move_E$ such that $\\round=(\\move_C,\\move_E)$ is applicable to\n  $\\evseq$.\n  %\n\\end{definition}\n\nA sequence $\\rounds=\\seq{\\round_0,\\ldots,\\round_n}$ of rounds is called a\n\\emph{play} of the game. A play is said to be \\emph{played according to} some\nstrategy $\\strategy_C$ for \\charlie, if, starting from the initial partial plan\n$\\evseq_0=\\epsilon$, it holds that $\\round_i=(\\strategy_C(\\Pi_{i-1}),\n\\move_E^i)$, for some $\\move_E^i$, for all $0<i\\le n$, and to be played\naccording to some strategy $\\strategy_E$ for \\eve if $\\round_i=(\\move_C^i,\n\\strategy_E(\\Pi_{i-1},\\move_C^i))$, for all $0<i\\le n$. It can be seen that for\nany pair of strategies $(\\strategy_C,\\strategy_E)$ and any $n\\ge0$, there is a\nunique run $\\rounds_n(\\strategy_C,\\strategy_E)$  of length $n$ played according\nboth to $\\strategy_C$ and $\\strategy_E$.\n\nThen, we say that a partial plan $\\evseq$, and the play $\\rounds$ such that\n$\\evseq=\\rounds(\\epsilon)$, are \\emph{admissible}, if the partial plan satisfies\nthe domain rules, and are \\emph{successful} if the partial plan satisfies the\nsystem rules.\n\n\\begin{definition}[Admissible strategy for \\eve]\n  \\label{def:games:admissible-strategy}\n  %\n  A strategy $\\strategy_E$ for \\eve is \\emph{admissible} if for each strategy\n  $\\strategy_C$ for \\charlie, there is $k\\ge 0$ such that the play\n  $\\rounds_k(\\strategy_C,\\strategy_E)$ is admissible.\n  %\n\\end{definition}\n\n\\charlie wins if, \\emph{assuming} domain rules are respected, he manages to\nsatisfy the system rules no matter how \\eve plays.\n\n\\begin{definition}[Winning strategy for \\charlie]\n  \\label{def:games:winning-strategy}\n  %\n  Let $\\strategy_C$ be a strategy for \\charlie. We say that $\\strategy_C$ is a\n  \\emph{winning strategy} for \\charlie if for any \\emph{admissible} strategy\n  $\\strategy_E$ for \\eve, there exists $n\\ge0$ such that the play\n  $\\rounds_n(\\strategy_C,\\strategy_E)$ is successful.\n  %\n\\end{definition}\n\nWe say that \\charlie \\emph{wins} the game $G$ if he has a winning strategy,\nwhile \\eve \\emph{wins} the game if a winning strategy for \\charlie does not\nexist.\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper, we completed the picture about timeline-based games by providing\nan effective procedure for controller synthesis, whereas before only a proof of\nthe complexity of the strategy existence problem was known. Previous approaches\neither provided a deterministic concurrent game structure which was however not\nbuilt effectively, or an effectively built automata which was, however,\nnondeterministic and thus unsuitable for use as a game arena without a costly\ndeterminization. Our approach surpasses the limits of both previous ones by\nproviding a deterministic construction, of optimal asymptotic size, suitable to\nbe used as a game arena. Then, we solve the reachability game on the arena with\nstandard methods to effectively compute the winning strategy for the game, if it\nexists.\n\nThis work paves the way to interesting future developments. On the one hand, the\neffective procedure shown here can be finally implemented, bringing\ntimeline-based games from theory to practice. On the other hand, developing an\neffective system based on such games requires to answer many interesting\nquestions, from which concrete modeling language to adopt, to which algorithmic\nimprovements are needed to make the approach feasible. For example, it can be\nforeseen that, to solve the fixpoint computation that leads to the strategy with\nreasonable performance, the application of \\emph{symbolic techniques} would be\nneeded.\n\\section{Introduction}\n\\label{sec:introduction}\n\nIn the timeline-based approach to planning, the world is viewed as a system made\nof a set of independent but interacting components whose behaviour over time\n(the timelines) is governed by a set of temporal constraints, called\n\\emph{synchronization rules}. Timeline-based planning has been originally\nintroduced in the space industry~\\cite{Muscettola94}, with timeline-based\nplanners developed and used by space agencies on both sides of the\nAtlantic~\\cite{CestaCFOP06,CestaCDDFOPRS07,FrankJ03,BernardiniS07,ChienRKSEMESFBST00},\nboth for short- to long-term mission planning~\\cite{ChienRTTDNASVGA15} and\non-board autonomy~\\cite{FratiniCORD11}.\n\nWhile successful in practice, only recently timeline-based planning has been\nstudied from a theoretical perspective. The formalism has been at first compared\nwith traditional action-based languages \\ala STRIPS, proving that they can be\nexpressed by means of timeline-based languages~\\cite{GiganteMMO16}. Then, the\ncomplexity of the timeline-based plan existence problem has been studied: the\nproblem is \\EXPSPACE-complete~\\cite{GiganteMMO17} over discrete time in the\ngeneral case, and \\PSPACE-complete with qualitative\nconstraints \\cite{DellaMonicaGTM20}. On dense time, the problem goes\nfrom being \\NP-complete to undecidable, depending on the syntactic restrictions\napplied~\\cite{BozzelliMMPW20}. The expressiveness of timeline-based languages\nhas also been studied from a logical perspective~\\cite{DellaMonicaGMSS17}, and\nan automata-theoretic point of view~\\cite{DellaMonicaGMS18}.\n\nTraditional timeline-based planning systems excel at the integration of planning\nwith \\emph{execution} by treating explicitly the concept of \\emph{temporal\nuncertainty}: the exact timings of the events under control of the environment\nneed not to be precisely known in advance. However, general nondeterminism,\nwhere the environment can also decide \\emph{what} to do (instead of only\n\\emph{when} to do it) is usually not handled by these systems. To overcome this\nlimitation,  the concept of \\emph{timeline-based games} has been recently\nintroduced~\\cite{GiganteMOCR20}. In these games, the state variables are\npartitioned between the controller and the environment, and the latter has the\nfreedom to play arbitrarily as long as a set of \\emph{domain} rules, that define\nthe game arena, are satisfied. The controller plays to satisfy his set of\n\\emph{system} rules. A  strategy for controller is winning if it allows him\/her \nto win independently from the choices of the environment.\n\nEstablishing whether a winning strategy exists for these games has been proved to be\n\\EXPTIME[2]-complete~\\cite{GiganteMOCR20}. However, no concrete way to\nsynthesize a controller implementing such strategies is known. The proof technique\nof the aforementioned complexity result involves the construction of\na huge (doubly exponential) \\emph{concurrent game structure}, which is used to\nmodel check some Alternating-time Temporal Logic (ATL) formulas~\\cite{AlurHK02}.\nWhile this structure is deterministic and can be in principle used as an arena\nto solve a reachability game and synthesize a controller, its construction is\nbased on theoretical nondeterministic procedures which have no hope to\nbe ever concretely implemented. On the other hand, the automata-theoretic\napproach by Della~Monica~\\etal~\\cite{DellaMonicaGMS18} provides a concrete and\neffective construction of an automaton that accepts a word if and only if the\noriginal planning problem has a solution plan. However, the automaton is\n\\emph{nondeterministic} and already doubly exponential, and the determinization\nneeded to use it as an arena would result into a further blow up and a\nnon-optimal procedure.\n\nIn this paper, we provide a\nconcrete and computationally optimal approach\nto controller synthesis for timeline-based games. We overcome the limitations\nof both the above-mentioned approaches by devising a direct construction for a\n\\emph{deterministic} finite-state automaton that recognizes solution plans,\nwhich is doubly exponential in size (thus not requiring the determinization of a\nnondeterministic automaton). This automaton is then used as the arena of a\nreachability game for which plenty of controller synthesis techniques are known\nin the literature.\n\nThe paper is structured as follows. In \\cref{sec:preliminaries} we introduce the\nneeded background on timeline-based planning and timeline-based games. Then,\n\\cref{sec:automaton} provides the core technical contribution of the paper,\nnamely the construction of the deterministic automaton recognizing solution\nplans. \\Cref{sec:games} uses this automaton as the game arena to solve the\ncontroller synthesis problem. Last, \\cref{sec:conclusions} summarizes the\nmain contributions of the work and discuss future developments.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\footnotetext[1]{The data presented herein were obtained at the\n  W.M. Keck Observatory, which is operated as a scientific partnership\n  among the California Institute of Technology, the University of\n  California and the National Aeronautics and Space\n  Administration. The Observatory was made possible by the generous\n  financial support of the W.M. Keck Foundation.}\n  \\footnotetext[2]{\n  Based on observations obtained with MegaCam, a joint\n  project of CFHT and CEA\/DAPNIA, at the Canada-France-Hawaii\n  Telescope (CFHT) which is operated by the National Research Council\n  (NRC) of Canada, the Institute National des Sciences de l'Univers of\n  the Centre National de la Recherche Scientifique of France, and the\n  University of Hawaii.}\n\n\nRoughly 70\\% of bright galaxies observed at redshift $z=0$ possess\nstellar discs (e.g. \\citealt{hammer05,park07,choi07} and\n\\citet{dserrano10}), including our own Galaxy and its two largest\nneighbours, M31 and M33. From this, we can infer that spiral\nmorphologies are the dominant configuration for galaxies viewed at the\npresent epoch. Under the formalism of hierarchical structure\nformation, galaxies are believed to evolve into their present forms\nvia the accretion of, and mergers with, smaller systems. The effect of\nthis process on the seemingly dynamically stable stellar discs we see\nin Milky Way (MW) type galaxies is still largely uncertain. The\nability of these fragile objects to survive a ``major merger'' event\n(i.e. a merger with a system $>1\/3$ of the host's mass) is something\nthat is still debated. Such major mergers are thought to be\ncosmologically common, with $\\sim70\\%$ of all galaxies with a halo\nmass of $M\\sim10^{12}{\\rm\\,M_\\odot}$ having experienced at least one major\ninteraction within the past 8 Gyrs \\citep{stewart08,purcell09}. Thus\nit has been argued that galaxies that possess thin stellar discs at\n$z=0$ could not have experienced a major merger within the last 10 Gyr\nwithout the disc being destabilized\n\\citep{toth92,walker96,stewart08,purcell09}. This poses a significant\nchallenge to our understanding of the formation of disc galaxies like\nthe MW and M31. Recently, several authors have argued that these thin\ndiscs could survive such an event if the merging system is\nsufficiently gas rich\n\\citep{robertson06,brook07,hopkins09,stewart09,brooks09}, although the\ndisc would still undergo heating, resulting in a thicker disc than\nthat observed presently in the MW.  In addition to the effect of major\nmergers on the structure of discs, galaxies viewed at the present\nepoch have undergone (and are still undergoing) many smaller ``minor''\nmergers which are not sufficiently massive to destroy thin stellar\ndiscs, but are thought to kinematically heat them, causing them to\nflare outwards and create a second, thick disc component\n\\citep{quinn93,robin96,walker96,velazquez99,chen01,sales09,villalobos09,purcell10}.\nOther physical processes are also thought to heat up and thicken the\nthin disc, including the accretion of a satellite on a radial orbit\nabout its host \\citep{abadi03,read08}, internal heating within the\ndisc from massive star clusters, interactions with spiral arms,\netc. (\\citealt{villumsen85,carlberg87,sellwood02,hanninen02,benson04,hayashi06,kazantzidis07,roskar08,schonrich09a,loebman10}). Thick\ndiscs may also have formed thick, with significant star formation\noccurring above the mid-plane of the galaxy or with large initial\nvelocity dispersions \\citep{brook04,kroupa02}. In recent work by\n\\citet{roskar10}, they suggest that in-situ formation\ncould also occur if the stellar disc is misaligned with the hot, gaseous\nhalo. This misalignment results in a significant warping of the\nouter disc, and subsequent star formation within this warp results in a low\nmetallicity thick disc. Finally, it is also possible that a number of\nthese mechanism will act in conjunction. In particular, it has been\nsuggested by a number of authors that secular growth from internal\nheating may be significantly enhanced by minor merger events via swing\namplification (e.g. \\citealt{sellwood98,dubinski08}), as these\nprocesses often occur simultaneously. As such, it makes little sense\nto treat these two scenarios as separate processes. \n\nWith so many potential mechanisms capable of producing thickened\nstellar discs, just how common are thick discs in spiral galaxies at\nthe present epoch?  \\citet{dalcanton02} claim that thick disc\nformation is a universal feature of disc formation, and as such should\nbe observed in all spiral galaxies. Whether such discs are formed\npredominantly via one mechanism, or a mixture of them is still\nuncertain, and disentangling the various formation scenarios from one\nanother in present data sets has proven difficult.\n\nIn the MW, the existence of a thick disc has long been known, and was\nfirst identified by \\citet{gilmore83}. Subsequent spectroscopic\nstudies of this component have shown it to be kinematically distinct\nfrom the thin stellar disc, with the thick disc lagging behind the\nthin disc by $\\sim50{\\rm\\,km\\,s^{-1}}$ \\citep{carollo10} and having a larger\nvelocity dispersion than the thin disc. This thick component also\nseems to be composed of older, more metal deficient stars\n(e.g. \\citealt{chiba00,wyse06}). However the observed properties of\nthe thick disc, such as scale height, length and velocity dispersion,\ntend to vary depending on the survey sample and tracer population used\n\\citep{juric08,ivezic08,carollo10,dejong10}. As such, the origin of\nthe MW thick disc is still a subject of great debate in the\nliterature. Thick discs have also been observed in a number of edge on\nspiral galaxies\n(e.g. \\citealt{burstein79,tsikoudi79,vanderkruit84,shaw89,vandokkum94,dalcanton02,elmegreen06,yoachim06,yoachim08a,yoachim08b}),\nand spectroscopic observations of these objects also show the thick\ndiscs to be composed of older stars than their corresponding thin\ndiscs. However, as these galaxies are all located at distances greater\nthan $\\sim$10 Mpc from the MW, one cannot obtain spectra for\nindividual stars, and must instead rely on the integrated spectral\nproperties of RGB stars. Obtaining spectra with a high enough\nsignal-to-noise (S:N) to discern velocity dispersion profiles and\nreliable metallicities is also challenging, making it impossible to\ndistinguish between the various formation mechanisms for these\nstructures.\n\n\nIf thick stellar discs are universal amongst spiral galaxies, and are\nformed by mergers with, or accretions of, satellites, one might expect\nto see such a structure in M31. This neighbouring galaxy is considered\nto be a ``typical'' spiral galaxy when compared with other local\nexternal disc galaxies \\citep{hammer07}. It is thought to have had an\nactive merger history, and a recent panoramic photometric survey by\nthe Pan-Andromeda Archaeological Survey collaboration (PAndAS,\n\\citealt{mcconnachie09}) has shown the halo of this galaxy to be\nlittered with tidal streams from interactions with in-falling\nsatellites. These include the Giant Southern Stream (GSS,\n\\citealt{ibata01b,gilbert09}) and streams A, B, C, D and E\n\\citep{ibata07,chapman08,mcconnachie09}. The outer disc of M31 is very\nperturbed \\citep{ferguson02,richardson08}, suggestive of some tidal\ninteraction. M31 is also host to 25 known dSph and 4 dE satellites, at\nleast 2 of which (NGC 205 and M32) show evidence for significant tidal\ninteraction \\citep{choi02,mcconnachie04a,geha06,howley08}. Therefore\nthe possibility of numerous interactions between the disc of M31 and\nits satellite population seems highly likely.  \\citet{mcconnachie09}\nalso present evidence for a recent interaction between M31 and its\nneighbouring spiral galaxy, M33, which could have significantly\ndistorted and heated the M31 disc, giving rise to a thick disc\ncomponent or substantial substructure in the outer disc.  Other groups\nhave postulated links between the formation of bulges and thick discs\nin spiral galaxies \\citep{melendez08,hopkins08,bournaud09,bensby10},\nand as M31 is known to have a reasonably massive bulge\n\\citep{saglia10}, it is an interesting candidate for hosting a thick\ndisc. Despite its high inclination to us along the line of sight\n(77$^\\circ$, \\citealt{walterbos88}), M31 is not seen sufficiently\nedge-on to allow us to look for such a population using photometry.\nTherefore to look for evidence of a thick disc in M31, we must search\nfor it via its kinematic signature, using spectroscopy. Given its\nproximity to us (785 kpc, \\citealt{mcconnachie05a}), M31 is an ideal\ntarget for spectroscopic observations as we are able to resolve and\nobtain reliable velocities for individual Red Giant Branch (RGB)\nstars, and it has an advantage over our own galaxy as we are afforded\na panoramic view, whereas in the MW we are hampered along various\nlines of sights by confusion from the disc and the bulge.\n\nSince 2002 our group has been conducting a systematic spectroscopic\nsurvey of M31, including the disc, halo and regions of substructure\nusing the DEIMOS instrument mounted on the Naysmyth focus of Keck II\n(I05,\n\\citealt{chapman05,chapman06,chapman07,chapman08,collins09,collins10}). The\ndata from this survey gives us an ideal opportunity to identify a\nthick disc if present. In fact, in their study of M31's extended disc\nusing this same data set, I05 identified a population lagging behind\nthe thin disc which they excluded from their study that they termed a\n`thick disc-like' population and \\citet{chapman06} briefly examined\nthis component as a function of radius but were unable to comment on\nits global properties. In this work, we discuss the results from an in\ndepth study of this population, analysing its kinematics and\nchemistry, comparing them to M31's thin and extended discs, the thin\nand thick discs in the MW, and those seen in other galaxies, and\ncomment on possible formation scenarios for this component. The paper\nis set out as follows; in \\S2 we discuss the known structure of M31,\n\\S3 focuses on our spectroscopic survey of M31 and discusses field\nselection and analysis techniques. We present our results in \\S4 and\ndiscuss their implications in \\S5. We conclude our findings in \\S6.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{map_thick.eps}\n\\caption{Map showing the location of fields within our M31\n  survey. Fields selected for study in this work are labelled and\n  highlighted in red. The outer ellipse shows a segment of a 55 kpc\n  radius ellipse flattened to c\/a= 0.6, and the major and minor axis\n  are indicated with straight lines out to this ellipse.  The inner\n  ellipse corresponds to a disk of radius 2$^\\circ$, (27 kpc), with the\n  same inclination as the main M31 disk.  }\n\\label{map}\n\\begin{flushleft}\n\n\\end{flushleft}\n\\end{center}\n\\end{figure}\n\n\\section{The bulge, discs and halo of M31}\n\nThe first recorded observation of M31 was made in 964 AD in \nthe `Book of constellations and fixed stars' written by the Persian astronomer, \nAbd al-Rahman al-Sufi, who described it as `a small cloud' in the night sky. \nIn the centuries that have passed since, M31 has been a \npopular target for astronomers, and much has been learned \nabout its structure. M31 is a spiral galaxy of SA(s)b type, \nwith a significant bulge, a classical thin stellar disc, \na vast extended stellar disc and a metal poor halo. In \nthis section, we outline the properties of each of these \ncomponents. \n\nFirst we discuss the bulge component. Numerous studies have show that\nM31 possesses a classical bulge, with a Sersic index of $\\sim2$ and an\neffective radius of 1.93 kpc \\citep{kormendy99,seigar08}. It is\nlargely supported by random motions, although recent work by\n\\citet{saglia10} has found evidence for rotation in the innermost\nregions. Saglia et al.  also find the bulge to be dominated by an old\nstellar population (age $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 12$~Gyrs) of roughly solar metallicity,\nwith a large velocity dispersion of 166-170${\\rm\\,km\\,s^{-1}}$. This component is\ndominant out to about $8'$ ($\\sim2{\\rm\\,kpc}$), at which point the disc\nbegins to dominates the surface brightness profile of the galaxy\n\\citep{saglia10}, however according to \\citet{merrett06} the bulge can\nbe traced out as far as 10 effective radii, equivalent to $\\sim15{\\rm\\,kpc}$\nmeaning that some of our innermost disc fields may be subjected to\nminor contamination from this component.\n\nStudies of the thin stellar disc of M31 have been performed by a\nnumber of authors (e.g. \\citealt{walterbos88,ferguson01}; I05;\n\\citealt{ibata07,mcconnachie09}), and have challenged our previous\nnotions of the structure of classical discs. With a scale length of\n5.9 kpc (\\citealt{walterbos88}, corrected for assumed distance to M31\nof 785 kpc, and \\citealt{merrett06}), it is more extensive than that\nof our Galaxy, and also appears to be forming stars at a lower rate\n\\citep{walterbos94}.  And while it is a characteristic feature of the\nsurface brightness profiles of stellar discs to steeply decline at\n3--4 scale lengths \\citep{vanderkruit81,pohlen00}, which corresponds\nto 18--24 kpc in M31, a spectroscopic study by \\citet{ibata05}\nuncovered a vast, extended disc component that can be traced out to\ndistances of $\\sim40$~kpc ($\\sim8$ scale lengths) from the centre of the\ngalaxy that has an exponential surface density profile that is very\nsimilar to the inner disc. While this structure is rather clumpy, on\naverage it appears to follow on smoothly from the classical inner\ndisc, although perhaps with a slightly larger scale length of\n6.6$\\pm0.6{\\rm\\,kpc}$ and with a slight lag behind circular velocities at\nlarge radii ( $\\langle\\Delta v\\rangle=20{\\rm\\,km\\,s^{-1}}$, I05). It is dynamically\ncold, with a velocity dispersion ranging from 20--40${\\rm\\,km\\,s^{-1}}$, leading\nI05 to conclude that it is likely not a thickened disc. Whether this\nextended component is truly separate from the classical thin disc, or\nmerely an extension of it that shows some evidence of heating and\nwarping at larger radii where the disc is more sensitive to\nperturbations from mergers and interactions, is unclear. Owing to the\nsimilarity of the thin and extended discs, we will refer to them both\nas the `thin disc' throughout this paper. Where we wish to make a\ndistinction between the two, we shall use the terminologies\n`classical' and `extended' disc.\n\nThe presence of a smooth, pressure supported metal-poor halo in M31 eluded\ndetection until very recently. In 2006, two groups\n\\citep{chapman06,kalirai06} independently identified such a component\nusing the DEIMOS instrument on the Keck II telescope.  Centred on the\nsystemic velocity of M31, with a central velocity dispersion of\n152${\\rm\\,km\\,s^{-1}}$, and showing no strong evidence of rotation, both groups\nfound this component to be metal poor with an average metallicity of\n${\\rm[Fe\/H]}=-1.4\\pm0.2$ \\citep{chapman06}. \\citet{kalirai06} were able to\ntrace this component out as far as 165 kpc from the centre of the\ngalaxy although there is an inevitable confusion with the halo of M33\nat these large distances \\citep{ibata07,koch08, mcconnachie09}.\n\nThe halo of M31 is also a known host to a number of kinematic\nsubstructures, such as the GSS, the tangential streams that cross the\nSE minor axis, the western shelf and a wealth of substructure in the\nNE of the galaxy that is thought to be linked to the GSS. In the\nfollowing analysis, we will carefully consider the kinematics of all\nthese components to ensure any thick disc sample that we define is\nfree from contamination by any of these sources. We shall discuss this\nin greater detail in \\S3.\n\n\\section{Observations and field selection}\n\nA detailed description of the observational methodology and target\nselection employed in the survey is given in I05, which we briefly\nsummarize here. Using Colour-Magnitude Diagrams (CMDs) from both the\nCanada France Hawaii (CFHT) and Isaac Newton (INT) telescopes\n\\citep{ferguson02,ibata07,mcconnachie09}, we selected targets for\nobservation by prioritising Red Giant Branch (RGB) stars in M31 with\n$20.5\\leq i\\leq21.2$ and colours $1.0\\leq (V-i)_0\\leq4.0$ (priority\nA), then filling the remainder of the masks with stars with $I\\le22.0$\nthat are unsaturated (priority B), where the V and I colours are\ntransformed from their native $g$ and $i$ colours using the relations\ndescribed in \\citet{mcconnachie04b} and \\citet{ibata07}.  We used a\ncombination of standard DEIMOS multislit mode for low density fields,\nsuch as the halo, and our own minislitlet approach which allowed us to\ntarget $>600$ stars per mask in more crowded regions, such as the\ndisc. Our observational setup covers the range of the Calcium Triplet\n(Ca II) lines at 8498, 8542 and 8662$\\; \\buildrel \\circ \\over {\\mathrm A}$, a prominent absorption\nfeature that can be used both to measure radial velocities, and as a\nmetallicity indicator. To obtain velocities, we cross-correlate all\nobserved stars with a template Ca II spectrum. We estimate the errors\non our velocities by following the procedures of \\citet{simon07} and\n\\citet{kalirai10}. First, we make an estimate of our velocity\nuncertainties for each observed star using a Monte Carlo method,\nwhereby noise is randomly added to each pixel in the spectrum,\nassuming a Poisson distribution for the noise and the velocity is\nrecalculated using the same cross correlation technique described\nabove. This procedure is repeated 1000 times, and then the error is\ncalculated to be the square root of the variance of the resulting mean\nvelocity. We combine this error with a systematic error, $\\epsilon$,\nwhich contains information on any errors we may not have accounted for\n(for example, wavelength calibration error, misalignment of the mask\netc.). For the fields observed with the 600 line\/mm grating, we\nevaluate this error directly by using repeat measurements in fields\n231Dis and 232Dis, a total of 332 stars. We define the normalised\nerror, $\\sigma_N$ as:\n\n\\begin{equation}\n\\sigma_N=\\frac{v_1-v_2}{(\\sigma_1^2+\\sigma_2^2+2\\epsilon^2)^{1\/2}}\n\\end{equation}\n\n\\noindent where $v_1$ and $v_2$, $\\sigma_1$ and $\\sigma_2$ are the\nvelocities and errors of each measurement pair, and $\\epsilon$ is the\nadditional random error required in order to reproduce a unit Gaussian\ndistribution with our data (shown in Fig.~\\ref{errors}) This gives us\na systematic error for this setup of $\\epsilon=5.6{\\rm\\,km\\,s^{-1}}$, slightly\nlower than the value of $\\epsilon=6.2{\\rm\\,km\\,s^{-1}}$ derived by\n\\citet{collins10} for the same setup, though we note that their\nmeasurement was based on repeat observations of 47 stars, compared\nwith our much larger data set of 332 stars. The typical uncertainties\nfor these measurements above a threshold of S:N = 3, are 5-10${\\rm\\,km\\,s^{-1}}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.8\\hsize]{n2nvars.eps}\n\\caption{A histogram showing the normalized error distribution for\n  repeat measurements of the same stars in two of our fields that were\n  observed with the 600 lines\/mm grating. The normalized error,\n  $\\sigma_N$ incorporates the velocity differences between the repeat\n  measurements (v$_1$ and v$_2$), and the Monte-Carlo uncertainties\n  calculated for each observation ($\\sigma_1$ and $\\sigma_2$). In\n  order to reproduce a unit Gaussian distribution for our\n  uncertainties, we also include an additional error term, $\\epsilon$,\n  which accounts for any systematic uncertainties we have not\n  included. We find $\\epsilon$=5.6 kms$^{-1}$ for this setup. For\n  fields using the 1200 lines\/mm setup, we using the \\citet{simon07}\n  value of $\\epsilon=2.2{\\rm\\,km\\,s^{-1}}$.}\n\\label{errors}\n\\end{center}\n\\end{figure}\n\n\nThe Ca II features also provide us with a method for measuring the\nspectroscopic metallicity of our observed sample. Following the\nprocedure of \\citet{rutledge97} and \\citep{battaglia08}, we fit\nGaussian functions to the three Ca II peaks to estimate their\nequivalent widths (EWs), and calculate [Fe\/H] using equation (1)\n\n\\begin{equation}\n[Fe\/H]=-2.66+0.42[\\Sigma Ca+0.64(V_{RGB}-V_{HB})]\n\\end{equation}\n\n\n\\noindent where $\\Sigma$Ca=0.5EW$_{8498}$+EW$_{8542}$+0.6EW$_{8662}$,\n$V_{RGB}$ is the magnitude (or, if using a composite spectrum, the\naverage magnitude) of the RGB star, and $V_{HB}$ is the mean\n$V$-magnitude of the horizontal branch (HB). Using $V_{HB}-V_{RGB}$\nremoves any strong dependence on distance or reddening in our\ncalculated value of [Fe\/H], and gives us the Ca II line strength at\nthe level of the HB. For M31, we set this value to be 25.17\n\\citep{holland96}. We note that this assumed value is sensitive to age\nand metallicity effects, see \\citealt{chen09} for a discussion,\nhowever owing to the large distance of M31, small differences in this\nvalue within the disc of M31 will have a negligable effect on\nmetallicity calculations. For individual stars, these measurements\ncarry large errors ($\\gta0.4$ dex), but the errors are significantly\nreduced when stacking the spectra into a composite in order to measure\nan average metallicity for a given population.\n\n\n\\subsection{Field selection and sample definition}\n\n\\begin{table*}\n\\begin{center}\n\\caption{Properties of fields analysed in this work}\n\\label{fprops}\n\\begin{tabular}{lcccccccc}\n\\hline\nField  & Date observed & $\\alpha_{J2000}(hh:mm:ss)$  & $\\delta_{J2000}(^\\circ:':'')$ & Grating & P.A. & Exp. time (s) & No. targets & R$_{proj}$ (kpc)\\\\\n\\hline\n228Dis & 23\/09\/2006 & 00:40:50.56 & 40:43:54.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 301 & 9.8\\\\\n227Dis & 23\/09\/2006 & 00:39:37.40 & 40:50:42.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 312 & 15.6\\\\\n166Dis & 03\/10\/2005 & 00:39:17.89 & 40:42:18.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 209 & 15.8\\\\\n106Dis & 30\/08\/2005 & 00:39:10.00 & 40:39:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 257 & 16.1\\\\\n105Dis & 30\/08\/2005 & 00:39:00.00 & 40:28:12.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 271 & 16.2\\\\\n224Dis & 25\/09\/2006 & 00:38:50.00 & 40:20:30.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 265 & 16.6\\\\\n232Dis & 05\/10\/2006 & 00:38:50.00 & 40:20:00.0 & 600 lines\/mm  & 90$^\\circ$ & 16200s& 184 & 16.6\\\\\n104Dis & 30\/08\/2005 & 00:38:50.00 & 40:20:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 271 & 16.7\\\\\n220Dis & 22\/09\/2006 & 00:38:00.00 & 40:06:12.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 322 & 20.3\\\\\n213Dis & 22\/09\/2006 & 00:38:11.60 & 40:06:12.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 155 & 20.5\\\\\n102Dis & 30\/08\/2005 & 00:38:00.00 & 40:00:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 268 & 21.5\\\\\n231Dis & 05\/10\/2006 & 00:38:00.00 & 40:00:00.0 & 600 lines\/mm  & 90$^\\circ$ & 16200s & 185 & 21.6\\\\\n223Dis & 25\/09\/2006 & 00:37:12.00 & 39:57:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 304 & 23.2\\\\\n101Dis & 30\/08\/2005 & 00:38:00.00 & 39:54:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 275 & 23.5\\\\\n222Dis & 22\/09\/2006 & 00:37:12.49 & 39:48:06.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 298 & 24.9\\\\\n221Dis & 22\/09\/2006 & 00:37:11.97 & 39:45:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 303 & 25.8\\\\\n50Disk & 16\/09\/2004 & 00:37:35.29 & 39:33:55.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 216 & 30.1\\\\\n107Ext & 30\/08\/2005 & 00:35:28.00 & 39:36:19.1 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 265 &31.0\\\\\nw11old & 30\/09\/2002 & 00:35:27.02 & 39:37:15.3 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 95  &31.0\\\\\n167Hal & 03\/10\/2005 & 00:34:30.24 & 39:23:58.7 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 205 &34.2\\\\\n148Ext & 04\/10\/2005 & 00:37:07.23 & 39:12:00.0 & 1200 lines\/mm & 90$^\\circ$ & 3600s & 211 & 39.6\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{rotcurve.eps}\n\\caption{Here we show a number of HI rotation curves for\n  M31. Throughout this work, we use results based on the\n  \\citet{chemin09} work, shown as filled black circles. We also show\n  rotation curves from \\citet{corbelli10} and I05, which differ\n  slightly from the Chemin et al. 2009 curve in the outermost\n  regions. We find that using these curves vs. the \\citet{chemin09}\n  results do not affect our results.}\n\\label{rotcurve}\n\\end{center}\n\\end{figure}\n\nIn order to detect a thick disc component in M31 kinematically, we\nneed to measure the velocities of stars within our sample relative to\nsome model for the velocity of stars within the thin disc of the\ngalaxy. If a thick disc is present, we should observe a population\nthat lags behind the thin disc in terms of its rotational\nvelocity. The component is also expected to have a larger velocity\ndispersion than the thin disc.  This is observed in the MW, where the\nthick disc lags the thin by between 20-50 kms$^{-1}$\n\\citep{chiba00,soubiran03} and has an average rotational velocity\ndispersion of $\\sigma_{V_{\\phi}}$=57 kms$^{-1}$ \\citep{carollo10}. For\nthe purposes of this work, we shall use an updated version to the disc\nmodel of I05. In this model, we assume circular orbits for all stars\nabout the centre of M31 and interpolate their velocities from the HI\nrotation curve of \\citet{chemin09}, which is shown in\nFig.~\\ref{rotcurve} as the solid black points. This rotation curve\ndiffers from that adopted by I05, particularly in the\noutermost regions. They used a compilation of CO data\nfrom \\citet{klypin02} and HI data from \\citet{brinks84}, which we also\nshow in Fig.~\\ref{rotcurve} as red triangles. We also show the HI rotation curve derived by\n\\citet{corbelli10} from a WRST survey \\citep{braun09} as blue\nsquares. Using either of these rotation curves as opposed to that of\nChemin et al. leads to differences in our interpolated velocities of\norder a few--$20{\\rm\\,km\\,s^{-1}}$, however there is a negligible effect on the\ndispersions within particular populations, and so the adoption of any\nof these curves would give us consistent results when analysing the\nglobal properties of the stellar discs in this work. We assume an\ninclination for M31 of 77$^\\circ$ \\citep{walterbos88}, and adopt\nparameters for the thickness of the disc identical to those used in\nI05, with a constant thickness for the disc of 350 pc (which is\nroughly the thickness of the MW disc, \\citet{ivezic08}) out to 16 kpc,\nat which point we assume the stellar disc begins to flare with a scale\nheight that increases linearly with radius. We set a maximum scale\nheight of 1.69 kpc at radius of 30.5 kpc, beyond which we assume that\nthe disc has constant thickness. We then integrate along the line of\nsight through this flaring exponential disc and project the velocities\nof objects on circular orbits about M31 onto the line of sight. This\nproduces an average velocity map for the disc of M31, which we display\nin Fig.~\\ref{velmap}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{rotationmap_fin.eps}\n\\caption{A contour map of the expected velocities of stars in circular\n  orbits in the disc of M31. This was constructed using our simple\n  model as discussed in \\S3.  }\n\\label{velmap}\n\\end{center}\n\\end{figure}\n\nOnce our disc model has been constructed, we need to select\na sample of fields from our DEIMOS survey that will\nprovide the most reliable kinematic comparison with respect to the\nvelocity of the disc. As the disc of M31 is not infinitesimally thin,\nbut possesses some unknown scale height, any line of sight taken\nthrough the galaxy traverses a significant depth. Given the\ninclination of M31, some lines of sight will traverse larger depths\nthan others, which could have the effect of smearing out the\nvelocities of objects with respect to the disc model. This is\nillustrated in Fig.~8 of I05. They find that objects along the major\naxis of M31 are less susceptible to this effect than those that are\nlocated off the major axis, and therefore we limit our initial study to\nfields along the major axis.\n\nA further complication in field selection arises from MW\ncontamination. Our colour selection criteria means that we inevitably\nobserve Galactic K dwarf stars within our sample, as these lie\ncoincident with M31 RGB stars in the CFHT and INT CMDs. Eliminating\nthese stars from our sample is straightforward in the South West (SW)\nregion of M31, as the disc of Andromeda and the halo of the MW occupy\ndistinct positions in heliocentric velocity space. Assuming the\nBesancon model is a good description of the foreground populations in\nthe direction of M31, it can be shown that the Galactic population\npeaks at v$_{hel}$=-61kms$^{-1}$, and the contribution of MW K dwarfs\nto our sample at v$_{hel}\\leq-100$kms$^{-1}$ is very low\n(\\citealt{robin04}, I05). Given that the average rotational velocity\nin the SW of M31 is less than -300 kms$^{-1}$ (I05), we are able to\ncleanly separate M31 stars from MW field stars. However, in the North\nEast (NE) of M31, the average heliocentric velocity of the M31 disc\ntypically ranges between -100 kms$^{-1}$ and -200 kms$^{-1}$,\nresulting in a significant overlap between Galactic and M31\npopulations, making it difficult to distinguish between the two. While\nit is possible to remove some of this contamination by examining the\nstrength of the Sodium doublet (NaI), located at a rest wavelength of\n$\\sim8190\\; \\buildrel \\circ \\over {\\rm A}$, this is not a perfect discriminator. One can also\neliminate some foreground contamination via a comparison of\nphotometric and spectroscopic metallicities \\citep{gilbert06}, but\ngiven the uncertainties on the individual spectroscopic [Fe\/H] of our\nobserved stars (discussed above), we still retain a significant\npopulation of contaminants within our sample. There is also\ncontamination in the NE from M31 substructure\n(I05,\\citealt{chapman06,richardson08}) which can be difficult to\nseparate from the M31 disc in the NE. For these reasons, we limit our\ninitial study to the SW major axis. We hope to analyse the NE\npopulation in a future paper. These criteria leave us with a sample of\n21 fields along the SW major axis, highlighted in red in\nFig.~\\ref{map}. The positions and properties of these fields can be\nfound in Table~\\ref{fprops}. Two of these fields (231Dis and 232Dis)\nwere observed as part of our ultra-deep M31 disc survey (Chapman et\nal. in prep.) and were integrated for 4.5 hours with the 600 line\/mm\ngrating, which allowed us to make more robust measurements of\nindividual stellar metallicities than for our other fields.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{contam.eps}\n\\caption{Histograms for both heliocentric (top) and disc lag (bottom)\n  velocities of all stars within our sample of 21 fields. Regions\n  expected to be inhabited by thin disc (light blue), thick disc\n  (red), halo (green) and MW foreground (grey) are highlighted. }\n\\label{contam}\n\\end{center}\n\\end{figure} \n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{velcuts_new.eps}\n\\caption{Our initial sample of fields, selected for their position\n  along the South West major axis as described in the text, are shown\n  in order of increasing (projected) radius. Gaussian fits\n  indicating the thin and (where applicable) thick disc are shown as\n  magenta and blue curves respectively. Our selection criteria are\n  overlaid, with the dashed lines representing our 2$\\sigma$ cuts and\n  the solid lines representing our Gaussian cuts both of which sample\n  roughly the same region of velocity space.  }\n\\label{fields}\n\\end{center}\n\\end{figure*} \n\nTo isolate a potential thick disc population in our sample of 21\nfields, we must first define a statistical measure of what constitutes\na lagging population with respect to the thin and extended discs. We\nmust also ensure that our definition is able to distinguish between\nthis population and that of the metal-poor M31 halo which, as it is a\nnon-rotating component \\citep{chapman06,kalirai06}, also lags behind\nthe disc. In Fig.~\\ref{contam}, we display two histograms, one with\nthe heliocentric velocities ($v_{hel}$) of the stars in our 21 fields,\nand one with their velocities with respect to the disc ($v_{lag}$) and\nwe highlight the regions we expect each of these components to\ninhabit, along with where we expect to see contamination from halo\nK-dwarfs in the MW.\n\n\n We do this using two separate methods. The first is to fit Gaussians\n to both a disc component, located on or around\n v$_{lag}$=0~kms$^{-1}$, and a broad halo component centered on or\n around v$_{lag}$=-300~kms$^{-1}$. We then define a thick disc\n population to encapsulate anything that lags the thin disc by\n $>2\\sigma$ of the thin disc peak value and we implement a lower cut\n on this population by requiring the contribution from the halo would\n be $<$1 star per velocity bin (20 kms$^{-1}$), thus minimizing the\n contamination. In fields where there is no obvious halo component to\n fit to, we use a Gaussian centered on -300 kms$^{-1}$ with a\n dispersion of 90~kms$^{-1}$ \\citep{chapman06}, and normalize it with\n respect to the thin disk by assuming that the halo contributes\n $\\sim$10\\% to the total number of stars within the field (a\n conservative estimate, given that the stellar halo contributes\n $<<10$\\% to the total stellar light in disc galaxies). The second\n method is to fit multiple component Gaussians to each of the\n fields. We apply a Gaussian Mixture Modelling (GMM) technique, which\n allows the number of Gaussians to vary freely between 1 and\n 7 components. To discern which model best fits the data, we apply a\n likelihood ratio test (LRT) to the resulting probabilities of the\n fits.  The use of the LRT in astronomy was popularised by\n \\citet{cash79}, and is often used in the literature to determine\n whether the properties of an observed stellar population can be well\n described by single vs. multiple Gaussian components\n (e.g. \\citealt{ashman94,carollo10}). The LRT compares the likelihoods\n of nested models (in our case, a mixture of Gaussian components) to\n determine whether applying a model with additional parameters\n produces a significantly better fit than a simpler model. This is\n done by calculating the LRT statistic, $-2\\rm ln( \\mathcal{L}_1\/\n \\mathcal{L}_2)$, where $ \\mathcal{L}_1$ and $ \\mathcal{L}_2$\n represent the likelihoods of the simple and complex model\n respectively, and comparing it with a $\\chi^2$ distribution with\n degrees of freedom equal to the difference between the number of\n parameters in the two models (3 in our case). For a model with\n additional parameters to be accepted as a statistically better fit,\n this ratio must be greater than 7.82 which corresponds to a P-value\n of $<0.05$. In general, this technique converges on fits with three\n components (a thin disc, a halo and a thick population) though there\n are a few exceptions. We shall discuss these fits in greater detail\n in the following section. Where this technique converges on fits that\n identify a lagging component that is distinct from both the thin disc\n and halo, we define a sample of highly probable thick disc stars by\n applying a standard Bayesian classification scheme to assign each\n star a probability of being a member of the thin disc $P$(thin),\n thick disc, $P$(thick), or halo, $P$(halo), population based on their\n velocity, and the properties of the Gaussian fits to each population\n on a field by field basis. We define a star as being a highly\n probable member of the thick disc if $P$(thick)$\\geq0.997$. The\n results of both these techniques can be seen in\n Fig.~\\ref{fields}. The velocity cuts for stars selected using our\n 2$\\sigma$ technique are shown as dashed lines, and the range of\n velocities selected using the Bayesian classification technique are\n marked with solid lines. It can be seen that both techniques isolate\n a very similar population. In Fig.~\\ref{cmd}, we plot a CMD showing\n the V-I colours of the thin (blue points) and thick (magenta points)\n populations, and we overlay Dartmouth isochrones \\citep{dart08} of\n ${\\rm[\\alpha\/Fe]}=+0.2$ and an age of 8 Gyrs (in line with the estimated range of\n ages for the thin disc of 4--8 Gyrs, \\citealt{brown06}) with\n metallicities ranging from ${\\rm[Fe\/H]}=-0.4$ to ${\\rm[Fe\/H]}= -1.5$. Both thin and\n thick populations inhabit roughly the same region in this CMD, which\n we shall discuss in more detail in \\S4.3.\n\nFinally, we note that in both selection methods, we expect some cross\ncontamination between the thin and thick disc components, as the two\npopulations significantly overlap. However, we assume this\ncontamination will be lower in our cuts based on the Gaussian fits, as\nthese are more conservative. Therefore, we use these cuts\npredominantly in this paper when referring to clean thin and thick\ndisc samples. We have also assumed both components have symmetric,\nGaussian distributions in velocity, which may not be the case, and\nthis could cause further contamination if the populations are\nskewed. We also expect some contamination from the halo, however,\ngiven that the disc is the dominant population in all our fields, we\nexpect this contamination to be negligible in comparison to the cross\ncontamination between the discs.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{cmd.eps}\n\\caption{CMD for our thin (blue points) and thick (magenta points)\n  populations in standard Landolt V and I colours. Dartmouth\n  isochrones with ${\\rm[\\alpha\/Fe]}=+0.2$ and an age of 8 Gyrs are overlaid with\n  metallicities from left to right of [Fe\/H]=-1.5, -1.0, -0.5,\n  -0.4. The colours of the remaining stars in our DEIMOS survey are\n  also plotted in light blue.}\n\\label{cmd}\n\\end{center}\n\\end{figure}\n\n\\subsection{Testing the statistical significance of our sample}\n\nBefore we analyse our sample, we test the significance of our thick\npopulation to ensure it is not merely consistent with noise above a\nthin disc population plus smooth halo component. To do this, we fit a\nsingle Gaussian to the disc and halo components, as described above,\nthen calculate the deviation of the data from the fit for all\nvelocities greater than the peak disc velocity (i.e. the right hand\nside of the disc fit), normalizing it to the expected contribution\nfrom the Gaussian in this region. We then define the noise to be 1.5\ntimes the median absolute deviation of this sample. We repeat this\nexercise for all velocities less than the thin disc peak and greater\nthan -200~${\\rm\\,km\\,s^{-1}}$ in the lag frame, in this case comparing to the\nexpected contribution from both thin disc and halo fits. This allows\nus to work out the significance of our thick disc population,\n$\\sigma_{conf}$. In all cases where the GMM identified a thick disc\ncomponent, we find that our excess above the thin disc plus halo model\nhas a significance of $>3\\sigma$ (see table~\\ref{kprops}). For the\nfields where the GMM converged on a 2 Gaussian fit (232Dis, 222Dis,\n107Ext and w11old), we find $\\sigma_{conf}<3\\sigma$. We also identify an\nadditional 3 fields (101Dis, 166Dis and 227Dis), where\n$\\sigma_{conf}<3\\sigma$. Two of these fields are located at radii of\n$\\sim15{\\rm\\,kpc}$, where there may be residual contamination from the bulge\ncomponent. This may also explain the large dispersions (of order\n50${\\rm\\,km\\,s^{-1}}$) seen in our innermost fields. Excluding these, we are left\nwith 14 of our 21 ($2\/3$) fields where we confidently detect a\nthick component. We shall focus on these fields in the remainder of\nour analysis, but we shall discuss the significance of the\nnon-detections in \\S5.\n\n\\section{Results}\n\n\n\\subsection{Kinematic and structural properties of the thin and thick discs}\n\nIn this section we present measurements for the kinematic and\nstructural properties of the thin and thick discs. Properties of\nindividual fields can be found in Table~\\ref{kprops}, while the\naverage properties for both components can be found in\nTable~\\ref{avprops}.\n\n\\subsubsection{Velocity lag and dispersion profiles}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.9\\hsize]{vel_summary.eps}\n\\caption{{\\bf Top panel:} The difference in velocity, $\\Delta v$,\n  between the thin disc and thick component as a function of projected\n  radius. This lag appears to be approximately constant as a function\n  of radius, with an average lag of $46.0\\pm3.9{\\rm\\,km\\,s^{-1}} $. There appears\n  to be a slight in crease in lag in the outermost part, however this\n  is largely driven by fields that lie off the major axis of M31, and\n  therefore the velocities are less reliable. {\\bf Middle panel:\n  }Dispersion, $\\sigma_v$, of both thin (black squares, solid line)\n  and thick (red triangles, dot-dashed line) components are plotted as\n  a function of projected radius. The thin disc appears to maintain a\n  constant dispersion of $\\sigma_{thin}$=35.7$\\pm1.0{\\rm\\,km\\,s^{-1}}$, however the\n  thick component appears to decrease somewhat at larger radii. {\\bf\n    Bottom panel: }Average spectroscopic metallicity of thin and thick\n  components as a function of projected radius. Neither component\n  evolves with radius.}\n\\label{summary}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.9\\hsize]{slength_thin.eps}\n\\includegraphics[angle=0,width=0.9\\hsize]{slength.eps}\n\\caption{{\\bf Top panel:} Plot of the density of stars\n  (N$_*$\/arcmin$^2$) in the thin disc against R$_{proj}$. The\n  densities are calculated by first separating our sample by their\n  target prioritisation (A or B, see \\S3), then counting all stars\n  with $v_{thin}>v_{lag}>v_{thin}+2\\sigma_{thin}$ and multiplying\n  these values by 2 (i.e. assuming the distributions are symmetric)\n  for both prioritisations. We then calculate\n  $\\rho_*==n_sn_t\/n_o-n_b$, and combine these results from priority A\n  and B. Fitting an exponential profile to these points we deduce\n  $h_r=7.3\\pm1.1$~kpc. Solid line represents the best fit to the data\n  from a weighted-least-squares routine, and the shaded region\n  indicates the 1$\\sigma$ errors from the fit. {\\bf Bottom panel:} As\n  above, for the thick disc. Here we count all stars with\n  $v_{thick}-2\\sigma_{thick}>v_{lag}>v_{thick}$ and multiply by two\n  again. Fitting an exponential profile to these points we deduce\n  $h_r=8.0\\pm1.2$~kpc.}\n\\label{scale}\n\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.3\\hsize]{scalerel.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{thinscalerel.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{thickscalerel.eps}\n\\caption{Here we compare the scale lengths of the thin and thick discs\n  in M31 to those of other galaxies with observed thick discs in order\n  to infer their scale heights. {\\bf Left panel: }Here we plot thin disc\n  scale lengths against thick disc scale lengths for 34 external\n  galaxies (YD06) as black points, and overplot the same measurements\n  for the MW (blue triangle). We fit a linear relation to these points\n  with a gradient of 1.3. We plot our result for M31 as a red square,\n  and it is in excellent agreement with this relation. {\\bf Centre panel:\n  }Here we plot scale height, $z_0$, against scale height for the thin\n  disc of the YD06 sample plus the MW and derive a best-fit linear\n  function with a gradient of 0.17. From this, we can estimate a scale\n  height for the M31 thin disc of 1.1$\\pm0.2$~kpc, which we overplot\n  on the relation as a red square. {\\bf Right panel: } We now plot the\n  same, but for the thick disc and find a best fit gradient of 0.35,\n  and therefore infer a scale height for the thick disc in M31 of\n  2.8$\\pm0.6$~kpc, which is overplotted as a red square.}\n\\label{height}\n\\end{center}\n\\end{figure*}\n\nIn this first section, we initially address the thin and extended\ndiscs of M31. In I05, the extended disc was identified as a stellar\ndisc that, while appearing in many respects to be similar to the\nclassical thin disc, was a separate entity that was clumpy in terms of\nits structure, and lagged behind the classical disc in terms of its\nkinematics. As we are limiting our study to one slice down the major\naxis of M31, we do not attempt to comment on the global `clumpiness'\nof this extended disc, but we return to the issue of the velocity lag\nand distinction from the thin stellar disc. As we have analysed the\ndisc frame velocities for all our fields using a rotation curve that\ndiffers from the one used in I05, it is useful for us to determine\nwhether the increasing lag with respect to the classical thin disc is\nseen here also. In I05, they split their sample of 21 fields into an\ninner (with R$_{proj}<20{\\rm\\,kpc}$) and outer (with 20$<$R$_{proj}<30{\\rm\\,kpc}$)\nsample to determine the average properties of the disc and extended\ndisc. For their inner (classical) disc sample, they calculated an\naverage velocity for the disc in the disc lag frame of\n$v_{lag}=-17.0{\\rm\\,km\\,s^{-1}}$ and a dispersion of $\\sigma_v=50.0{\\rm\\,km\\,s^{-1}}$. In the\nouter (extended) sample they calculated an average velocity of\n$v_{lag}=-16.0{\\rm\\,km\\,s^{-1}}$ and a dispersion of $\\sigma_v=51.0{\\rm\\,km\\,s^{-1}}$. If we\nperform the same analysis for our study, we find an average lag of\n$v_{lag}=-14.8{\\rm\\,km\\,s^{-1}}$ for our inner fields and $v_{lag}=-25.5{\\rm\\,km\\,s^{-1}}$ for\nour outer fields. However, we note that this value is calculated with\nthe inclusion of fields 107Ext, w11old and 167Hal, which have very large lags of\n$v_{lag}<-55{\\rm\\,km\\,s^{-1}}$ compared with the other fields. We note that these \nfields are located slightly off the semi major axis (see Fig.~\\ref{map}) \nwhere our interpolated disc-frame velocities are subject to larger uncertainties. If\nwe exclude these fields, we find an average lag of $v_{lag}=-14.9{\\rm\\,km\\,s^{-1}}$,\nvery similar to our inner sample. We therefore conclude that there is\na negligible difference in the lags of the classical and extended disc\nbehind circular velocities. For these samples we also calculate\naverage dispersions of $\\sigma_v=42.7{\\rm\\,km\\,s^{-1}}$ and $\\sigma_v=30.0{\\rm\\,km\\,s^{-1}}$,\nimplying that the extended disc has a lower dispersion than the\nclassical disc. However, in our inner sample, we are more likely to see \nresidual contamination from the bulge and we also have a large\nproportion of fields for which we could not cleanly isolate the thick\ndisc ($\\sim40$\\% cf. $\\sim20$\\% in the outer sample). These factors may \ncause us to overestimate the dispersion of the disc in these\nregions. From these results, we therefore find no concrete reason to\nassume that the extended disc is a separate component from the\nclassical disc and we treat these two components as one thin stellar\ndisc in the remainder of our analysis.\n\nBy using the information from our Gaussian fits to the thin and thick\ncomponents, we can comment on their global kinematic properties, and\ndiscuss any variation of these properties with radius. In\nTable~\\ref{kprops} we show the peak velocities and velocity\ndispersions of both thin and thick (where applicable) components in\neach field, with associated errors from the GMM fits. Where both thin\nand thick components are detected, we compute the lag between the two\ncomponents, $\\Delta v=v_{thin}-v_{thick}$, and plot this lag as a\nfunction of radius in Fig.~\\ref{summary}.  The 14 fields for which a\nthick disc component is reliably detected cover a range of radii from\n15.2 to 39.6 kpc. In the top panel of Fig.~\\ref{summary}, we can see\nthat the lag between the two components does not appear to increase\nwith distance from the centre of M31, and shows an average lag of\n$\\langle\\Delta v\\rangle=46.0{\\rm\\,km\\,s^{-1}}$.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.3\\hsize]{feh_spec_new10.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{feh_spec_new8.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{feh_spec_new5.eps}\n\\caption{Here we display the spectroscopic MDF for all stars in our\n  thin and thick components (shown as blue hatched and red solid\n  histograms respectively) In the left panel, we show the MDF using\n  all stars for which metallicities can be reliably measured (i.e.\n  S:N$\\ge10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, and the middle and right panels apply lower\n  quality cuts (S:N$\\ge8$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ and S:N$\\ge5$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$). For our\n  lower S:N cuts, we note that the median [Fe\/H] values for both\n  populations remain similar, and the dispersions (inter-quartile\n  range) begin to increase, losing some of the detail of the shape of\n  the MDF. In all cases the median [Fe\/H] of the thick disc is more\n  metal poor than the thin by $\\sim0.1$ dex.}\n\\label{spechist}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{sumspec_prob.eps}\n\\caption{Composite spectra of both the thin disc and thick component,\n  using both 2$\\sigma$ (top panel) and Gaussian (lower panel) cuts to\n  isolate the thick component. The composite spectra are constructed\n  from stars within the selection regions that possess a S:N $\\geq$\n  3.0$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$.  We find that the average metallicity for the thin\n  component is more metal rich than the thick by 0.2--0.3 dex. These\n  results are consistent with composites formed from spectra with S:N\n  $\\leq$ 3$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ and with our field-by-field metallicity estimates\n  (Fig.~\\ref{summary}). We also show the locations of a number of Fe I\n  lines present in these spectra.}\n\\label{spectra}\n\\end{center}\n\\end{figure}\n   \nWe also plot the dependence of velocity dispersion, $\\sigma_{thin}$ and\n$\\sigma_{thick}$, for both components with radius in the middle panel\nof Fig.~\\ref{summary}. For the thin disc, we fit both a constant\nrelation and a single power law to the data. The linear power law\nsuggests a decrease in dispersion with radius, with a gradient of\n-0.87${\\rm\\,km\\,s^{-1}}{\\rm\\,kpc}^{-1}$, however this fit is not statistically better\nthan a constant fit, with an average dispersion of\n$\\sigma_{thin}=31.6\\pm1.1{\\rm\\,km\\,s^{-1}}$ (reduced $\\chi^2$ of 5.6 vs. 5.2). We\ndo the same for our thick disc results, and we find that a linearly\ndecreasing profile where\n$\\sigma_{thick}=-0.8(\\pm0.2)R_{proj}+66.1(\\pm5.8)$ has a marginally better\nfit to the data than a fit with no evolution, however the difference\nin negligible (reduced $\\chi^2$ of 1.2 vs. 1.4), and deemed\ninsignificant in a $\\chi^2$ significance test. Even if we were to\naccept this fit as preferred, we note that the two outermost fields\nsituated at 34.2 and 39.6~kpc, are perhaps the driving force in the decreasing\ndispersion seen in our thick disc component. As these field are the\nfurthest out in our survey, they also suffers from the greatest chance\nof halo contamination in our sample, and therefore could be\nunreliable. If we exclude these final points from the fit, we find that\n$\\sigma_{thick}$ is best fit with no evolution as a\nfunction of radius, with an average dispersion of 50.8$\\pm1.9{\\rm\\,km\\,s^{-1}}$. We\ntherefore conclude that our data cannot tell us anything reliable\nabout the dependence of these kinematic properties with radius, and\nallow us to merely calculate the average kinematics of both\ncomponents.\n\n\n\\subsubsection{Scale length of the thin and thick disc}\n\nTo determine the scale lengths of our two disc components, we need to\ncalculate the number density of thin and thick disc stars within our\nDEIMOS field of view. There are 2 complications we must consider\nbefore we proceed. Firstly, the two components are not completely\ndistinct from one another, and in all fields, we observe some\noverlap. Secondly, owing to our selection criteria (discussed in\n\\S~3), we prioritise stars of certain colours and magnitudes above\nothers, and this must be considered when calculating densities on a\nfield-by-field basis.\n\n\nWe determine the number of stars associated with the extended thin\ndisc, $n_s$, in each of our fields by integrating the Gaussian we have\nfit to this component. To determine the density of stars contained in\nthe thin disc, we multiply $n_s$ by the total number of available\ntarget stars within our DEIMOS field that fall within our selection\ncriteria, $n_t$, and divide this by the total number of stars that\nwere observed with our DEIMOS mask, $n_o$. We then subtract the\ndensity of background stars $n_b$, which is computed from a number of\nfields on the edge of our survey region;\ni.e. $\\rho_*=n_sn_t\/n_o-n_b$. To account for our prioritised selection\ntechnique, we perform this calculation separately for our priority A\nand priority B stars, then combine these measurements. We repeat this\ncalculation for the thick disc. We plot the results in\nFig.~\\ref{scale}, where we apply a weighted least-squares exponential\nfit to our data points, and determine $h_r=7.3\\pm1.1$~kpc for the thin\ndisc and $h_r=8.0\\pm1.2$~kpc for the thick. Comparing this to previous\ncalculations for the scale length of the thin and extended discs, we\nfind that the extended disc has a larger scale length than the\nexponential thin disc, ($5.1\\pm0.1$~kpc, I05). The value of 7.3~kpc\nthat we derive is slightly higher than that derived in I05 of\n6.6$\\pm0.4$~kpc, and with much larger error bars but the two are\nconsistent within their $1\\sigma$ uncertainties. The difference\nbetween the two values can be attributed to two factors. Firstly, in\nI05, they included fields from the NE of the galaxy, plus fields\nlocated away from the major axis, where we have have sampled fields\nsolely from the SW major axis. Secondly, in I05 they did not fully\naddress any biases that may have been introduced by our two-tiered\nprioritisation system. Finally, we note that the thick disc appears to\nbe more radially extended than either the thin or extended disc,\nalthough it is consistent with the scale length of the extended disc\nwithin its $1\\sigma$-errors.\n\n\nIn previous work \\citet{yoachim06} (hereafter YD06) measured\nthe scale lengths of 34 edge-on disc galaxies using a photometric\nfitting technique, and found that the scale lengths of the thick discs\nwere larger than those of the thin discs by a factor of$\\sim1.3$. We\nplot their results in the left panel of Fig.~\\ref{height}, and overlay\na linear relation with a gradient of 1.3. We add to this our results\nfor M31, using an average value for the thin disc from the range of\nscale lengths derived for the thin and extended discs (5.1--7.3 kpc)\nof 6.3~kpc, and our calculated value of 8.0~kpc. We also overplot the\nresult for the MW (using \\citealt{juric08} values of 2.6 and 3.6 kpc\nfor thin and thick discs respectively). and note that M31 sits in\nexcellent agreement with this relation.\n\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Kinematic properties of thin and thick disc components}\n\\label{kprops}\n\\begin{tabular}{lccccccc}\n\\hline\nField & v$_{thin,lag}$ (${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thin} ({\\rm\\,km\\,s^{-1}})$ & v$_{thick, lag}$ (${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thick} ({\\rm\\,km\\,s^{-1}})$ & $\\sigma_{conf}$ & [Fe\/H]$_{thin}$ & [Fe\/H]$_{thick}$\\\\\n\\hline\n228Dis &  6.8$\\pm5.0$  & 55.2$\\pm3.2$ & -141.9$\\pm7.5$ & 41.2$\\pm11.8$ & 37.0 & -0.8$\\pm$0.1 & -1.0$\\pm$0.1\\\\\\\n227Dis & -3.7$\\pm10.8$ & 68.7$\\pm3.3$ & N\/A            & N\/A           & 2.8  & -0.7$\\pm0.2$ &N\/A\\\\\n166Dis & -7.5$\\pm6.9$  & 48.9$\\pm4.6$ & N\/A            & N\/A           & 0.5 & -0.8$\\pm$0.2 & N\/A\\\\\n106Dis & 11.6$\\pm$8.2  & 47.4$\\pm5.2$ & -85.0$\\pm8.2$  & 52.5$\\pm4.1$  & 3.5 & -0.9$\\pm$0.3 &  -1.0$\\pm0.3$\\\\\n105Dis & -16.4$\\pm7.2$ & 32.0$\\pm4.1$ & -54.8$\\pm8.7$  & 51.0$\\pm5.2$  & 4.0 & -0.7$\\pm$0.2 & -1.0$\\pm$0.2\\\\\n224Dis & -34.4$\\pm6.2$ & 24.3$\\pm2.5$ & -63.8$\\pm10.2$ & 55.0$\\pm12.5$ & 5.1 & -0.9$\\pm$0.1 & -1.1$\\pm$0.2\\\\\n232DiS & -37.2$\\pm3.0$ & 35.0$\\pm3.8$ & N\/A            & N\/A           & 2.0 & -0.8$\\pm$0.1 & N\/A\\\\\n104Dis & -37.9$\\pm6.5$ & 30.0$\\pm4.4$ & -114.3$\\pm9.5$ & 53.3$\\pm7.2$  & 9.0 & -0.7$\\pm$0.2 & -0.9$\\pm$0.1\\\\\n220Dis & -12.5$\\pm5.0$ & 39.6$\\pm2.9$ & -64.8$\\pm12.1$ & 54.7$\\pm10.9$ & 26.7 & -1.1$\\pm$0.3 & -1.6$\\pm$0.3\\\\\n213Dis & -10.9$\\pm3.2$ & 31.9$\\pm6.3$ & -80.5$\\pm10.0$ & 55.0$\\pm8.8$  & 39.2 & -0.8$\\pm$0.3 & -1.5$\\pm$0.3\\\\\n102Dis & -18.8$\\pm7.1$ & 22.2$\\pm5.2$ & -61.5$\\pm11.6$ & 48.1$\\pm5.6$  & 47.1 & -0.9$\\pm$0.2 & -1.0$\\pm$0.2\\\\\n231Dis &   4.7$\\pm15.2$& 22.9$\\pm4.2$ & -35.9$\\pm7.8$  & 53.4$\\pm7.2$  & 43.4 & -0.9$\\pm$0.1 & -1.1$\\pm$0.2\\\\\n223Dis &  -8.1$\\pm5.9$ & 20.1$\\pm4.9$ & -55.7$\\pm9.9$  & 51.1$\\pm6.8$  & 14.7 & -0.7$\\pm$0.2 & -1.0$\\pm$0.3\\\\\n101Dis & -16.2$\\pm3.9$ & 35.8$\\pm5.8$ & -57.3$\\pm8.6$  & 44.9$\\pm6.2$  & 3.1 & -1.1$\\pm$0.2 & -1.3$\\pm$0.3\\\\\n222Dis & -25.9$\\pm$4.8 & 33.6$\\pm2.9$ & N\/A            & N\/A           & 0.8 & -0.7$\\pm$0.1 & N\/A\\\\\n221Dis & -27.5$\\pm5.5$ & 35.0$\\pm3.6$ & -121.2$\\pm7.4$ & 52.2$\\pm6.3$  & 27.1 & -1.1$\\pm$0.2 & -1.4$\\pm$0.2\\\\\n50Disk & -16.9$\\pm8.1$ & 34.2$\\pm5.9$ & -149.2$\\pm13.2$& 42.3$\\pm7.2$  & 12.6 & -1.0$\\pm$0.2 & -1.2$\\pm$0.2\\\\\n107Ext & -55.2$\\pm6.9$ & 24.8$\\pm4.4$ & N\/A            & N\/A           & 2.4 & -1.0$\\pm$0.3 & N\/A\\\\\nw11old & -55.1$\\pm10.2$& 28.8$\\pm5.6$ & N\/A            & N\/A           & 1.4 & -1.0$\\pm$0.3 & N\/A\\\\\n167Hal & -72.6$\\pm8.3$ & 22.9$\\pm5.4$ & -120.3$\\pm12.2$& 35.1$\\pm7.8$  & 23.5 & -0.9$\\pm$0.3 & -1.0$\\pm0.3$\\\\\n148Ext & -17.2$\\pm7.0$ & 41.3$\\pm6.6$ & -154.5$\\pm11.0$& 25.7$\\pm5.1$  & 28.2 & -0.9$\\pm$0.3 & -1.0$\\pm$0.2\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{table}\n\\begin{center}\n\\caption{Average properties of thin and thick disc components derived in this work}\n\\label{avprops}\n\\begin{tabular}{lcccc}\n\\hline\nComponent & $\\sigma_v ({\\rm\\,km\\,s^{-1}})$ & $h_r$ (kpc) & z$_0$ (kpc)& [Fe\/H]$_{spec}$\\\\\n\\hline\nThin disc  & 35.7$\\pm1.0$ & 7.3$\\pm1.1$ & 1.1$\\pm0.2$ & -0.7$\\pm0.05$ \\\\\nThick disc & 50.8$\\pm1.9$ & 8.0$\\pm1.2$ & 2.8$\\pm0.6$ & -1.0$\\pm0.1$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Inferring the scale heights of the thin and thick discs}   \n\nOwing to the inclination of M31, we are unable to measure the height\nof either the thin or thick disc components directly. No photometric\nexcess above a typical bulge or extended disc profile is observed when\nperforming minor axis star counts \\citep{irwin05}, suggesting that\nthese components dominates the surface profile out to large radii. In\norder to infer probable scale heights for both components, we make use\nof the properties of the 34 edge-on galaxies measured by YD06. As the\nscale lengths and heights of both thin and thick discs in each of\nthese galaxies were derived, it is possible for us to search for a\nrelation between the scale length, $h_r$, and scale height, $z_0$ of\neach component. In the central panel of Fig.~\\ref{height}, we plot\n$h_r$ vs. $z_0$ for the YD06 sample as well as for the MW\n\\citep{ivezic08}, and fit it with a linear relation, on which we force\nan intercept of (0,0). We find that the data are well fit with a\ngradient for this relation of 0.18$\\pm0.04$, though there is\nsignificant scatter beyond $\\sim9$~kpc. From this, we deduce\n$z_0$=1.1$\\pm0.2$~kpc for the M31 extended disc (using $h_r$=7.3\nkpc). We repeat this for the thick disc (shown in the right panel of\nFig.~\\ref{height}) and find that these values are well fit with a\nlinear relation of gradient 0.35$\\pm0.06$, giving us\n$z_0$=2.8$\\pm0.6$~kpc for the M31 thick disc. If these values are\ncorrect, then not only are the discs of M31 more radially extended\nthan those of the MW by a factor of $\\sim2-3$, they are also\nsignificantly thicker.\n\n\n\\subsubsection{Contrast of the thin and thick discs}\n\nIn the previous sections, we have derived the density in each field of\nboth our components as a means to determine the scale lengths. We now use\nthese densities to work out how much of the total (disc related)\nstellar population is contained within either component. There are\nseveral caveats to such a comparison that should be\nmentioned. Firstly, our sampling of the field is likely to have\nan effect on our field-to-field estimates of the stellar density\n(which we discuss further in \\S~5). Secondly, as the disc is not\nobserved edge-on, we are measuring a 2D projection of the\ndensities which is difficult to interpret. We also note that the\nmeasurement errors associated with the densities of each field (shown\nin table~\\ref{densities}) are significant (of order $\\sim50$\\%). \n\nWith this in mind, we find that, on average, the thick disc component\naccounts for $35$\\% of the total stellar density, with an\ninter-quartile range of $\\pm$10\\%. In the Milky Way, we know that the\nthick disc contributes to $\\sim10$\\% of the stellar density in the\nsolar neighbourhood, and accounts for $\\sim1\/3$ of the {\\it total}\ndisc mass \\citep{juric08,schonrich09a}, comparable to what we derive here. \n\nFrom our calculated contrasts and individual density profiles for the\nthin and thick discs, we can estimate the mass contained within the\nthick disc component using values for the mass of the thin disc from\nthe literature. From our analysis above, we have determined that the\nthick disc contributes 35$\\pm10$\\% of the {\\it total} stellar density,\nmeaning the thick:thin disc density ratio is of order 55$\\pm15$\\%. We\ncan also estimate this fraction by integrating our stellar density\nprofiles (Fig.~\\ref{scale}) over the limits of our data, and from this\nwe calculate a thick:thin disc density ratio of $\\sim$65\\%, which is\nin good agreement with our contrast estimate. If we assume that both\ndiscs are composed of similar stellar populations, we can set the mass\nratio between the disc to be equivalent to the density ratio. In\n\\citet{yin09}, they quote a total mass for the thin stellar disc of\n$M_{*,thin}=5.9\\times10^{10}{\\rm\\,M_\\odot}$, calculated from the mass models of\n\\citet{widrow03} and \\citet{geehan06}. From this we estimate that the\ntotal mass of the M31 thick disc lies in the range\n2.4$\\times10^{10}{\\rm\\,M_\\odot}<M_{*,thick}<4.1\\times10^{10}{\\rm\\,M_\\odot}$. As we are\nunable to determine the full radial and underlying luminosity profile\nfor the thick disc, these values are obviously prone to large errors\nintroduced by our simplifying assumptions, and the mass of the thick\ndisc may be lower than our quoted range. For example, if we just\ncompare the number of stars we detect in the thin disc throughout our\nentire sample with the number we detect in the thick disc by\nintegrating the fitted Gaussians in Fig.~\\ref{fields} we find a\nthick:thin disc ratio of 20\\%. If we assume this value for the ratio\nof the masses between the components, our lower limit on the thick\ndisc is reduced to $M_{*,thin}=1.2\\times10^{10}{\\rm\\,M_\\odot}$. A future study\nof the thick disc which includes fields from the entirety of our\nsurvey will help us to better constrain both the radial profile and\nmass of this component.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Field by field densities of thin and thick disc stars}\n\\label{densities}\n\\begin{tabular}{lcccccc}\n\\hline\nField & $\\rho_{* thin}$ (*\/arcmin) & $\\rho_{* thick}$ (*\/arcmin) \\\\\n\\hline\n228Dis & 57.6$\\pm$22.3 & N\/A\\\\\n227Dis & 56.9$\\pm$22.3 & N\/A\\\\\n166Dis & 20.4$\\pm$9.4  & N\/A\\\\\n106Dis & 36.2$\\pm$15.3 & 23.6$\\pm$10.5\\\\\n105Dis & 45.3$\\pm$18.6 & 41.0$\\pm$16.3\\\\\n224Dis & 20.1$\\pm$8.4  & 17.1 $\\pm$7.1\\\\\n232DiS & 35.9$\\pm$15.7 & N\/A \\\\\n104Dis & 35.7$\\pm$14.6 & 6.6$\\pm$4.5\\\\\n220Dis & 19.9$\\pm$8.9  & 13.5$\\pm$6.4\\\\\n213Dis & 23.0$\\pm$12.2 & 9.8$\\pm$3.0\\\\\n102Dis & 24.0$\\pm$12.8 & 8.9$\\pm$4.5\\\\\n231Dis & 13.0$\\pm$6.6  & 16.6$\\pm$8.1\\\\\n223Dis & 15.1$\\pm$6.7  & 12.2$\\pm$5.6\\\\\n101Dis & 26.5$\\pm$12.9 & 11.1$\\pm$4.8\\\\\n222Dis & 4.9$\\pm$ 2.1  & N\/A \\\\\n221Dis & 14.4$\\pm$6.4  & 4.8$\\pm$3.3\\\\\n50Disk & 8.1$\\pm$3.9   & 2.0$\\pm$0.6\\\\\n107Ext & 9.6$\\pm$7.8   & N\/A\\\\\nw11old & 9.0$\\pm$6.5   & N\/A\\\\\n167Hal & 1.3$\\pm$0.8   & 0.8$\\pm0.5$\\\\\n148Ext & 1.1$\\pm$0.9   & 0.7$\\pm$0.5\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Spectroscopic metallicities}\n\n\nIn this section we present the spectroscopic values of [Fe\/H], both\nfor individual stars, and for the composite spectrum of each\ncomponent. Measuring individual metallicities from the Ca II triplet\nfor the stars in our survey, with S:N typically between\n5--15$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, is quite problematic. In \\citet{battaglia08}, they\nshow that the `best case' errors in measuring the equivalent widths\nof the Ca II lines, $\\Delta EW$, scale with S:N as:\n\n\\begin{equation}\n\\Delta EW=\\frac{\\sqrt{1.5\\times FWHM}}{S:N}\n\\end{equation}\n\n\\noindent assuming no contamination from residual sky lines and no covariance noise, where\nFWHM is the full width at half-maximum of the CaT lines which is\ntypically 2--3$\\; \\buildrel \\circ \\over {\\mathrm A}$. Using this equation, we can determine the average\nerrors in [Fe\/H] for stars in our sample at different S:N, and we find\nthat for spectra with S:N of (5, 8, 10)$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, the errors in their\ncalculated metallicity are of order (0.6, 0.4, 0.3) dex. To\ndemonstrate the effects of these large errors on the metallicity\ndistribution function (MDF) of our sample, we present histograms of\nthe individual spectroscopic metallicities for three quality cuts, one\nat S:N$\\ge10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ (left panel), one with S:N$\\ge8$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$\n(middle panel) and one with S:N$\\ge5$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ (right panel) in\nFig.~\\ref{spechist}. In both plots, the blue hatched histogram\nrepresents our thin disc sample and the filled red histogram\nrepresents our thick disc population. For our higher quality spectra,\nwe calculate a median metallicity for the thin disc of [Fe\/H]=-0.9\nwith a dispersion of 0.4 dex (from the inter quartile range, IQR). For\nour thick disc population we calculate a median of [Fe\/H]=-1.0 also\nwith a dispersion of 0.4 dex. We note that the distributions of both\npopulations deviate from a Gaussian distribution, with a kurtosis of\n-0.6 and -0.3 for thin and thick discs respectively, implying a broad\npeaked distribution, with narrow tails. Both distributions are skewed\ntowards lower metallicity with skewness $\\alpha=-0.4$ and\n$\\alpha=-0.3$ for thin and thick disc. For our lowest S:N cut,\nhowever, much of this information is lost. While the median [Fe\/H]\nremains very similar with [Fe\/H]=-1.0 for the thin disc and\n[Fe\/H]=-1.1 for the thick, the distributions begin to broaden, with\ndispersions of 0.5 dex for both populations, and present almost no\nskew ($\\alpha=-0.1$ and $\\alpha=-0.2$ for thin and thick disc). This\nshows that by including data with larger measurement errors, we wash\nout our MDF considerably, and lose any meaningful information. As a\nsanity check, we compare the MDF for all stars with\nS:N$\\ge10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ with one for stars with S:N$\\ge15$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, and\nfind both the median values of [Fe\/H] and general distributions to be\ncomparable. We note that by requiring such a high S:N cut on our\nindividual measurements of [Fe\/H], we bias our sample towards more\nmetal-rich stars as these will have intrinsically stronger Ca II\nlines.\n\nOwing to the large errors associated with these measurements, this\nanalysis provides a crude indication of the metallicities of both\ndiscs, and so to get a more accurate estimate of the average\nmetallicities of both populations we construct composite spectra for\nboth components (using both the 2$\\sigma$ and Gaussian velocity cuts)\nby co-adding the individual spectra of all stars with\nS:N$>3$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$, weighted by their S:N values. The resulting S:N of\nthe composite is much greater than the individual spectra\n(S:N$\\sim60-100$ cf. S:N$\\sim3-25$), allowing a better fit to the CaII\nlines. We use a cut of S:N$>3$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$ as below this the velocity\nuncertainties of our stars begin to significantly increase (as\ndiscussed in \\S~3). As we shift all spectra to the rest frame before\nco-adding, including spectra where the velocity is uncertain could\nsmear out the Ca II lines, resulting in an over-estimate of [Fe\/H] for\nthe composite. We note that the results from our composites are only\nindicative of an average metallicity for each component, and can tell\nus nothing about the metallicity dispersion for the discs. We display\nthe resulting composites in Fig.~\\ref{spectra}. The top 2 panels show\nthe thin and thick spectra for the 2$\\sigma$ velocity cuts, while the\nbottom 2 panels show the same, but for the Gaussian velocity cuts. In\nthe case of the 2$\\sigma$ cuts, our thin composite comprises 511 stars\nthat match our kinematic and quality criteria, while our thick\ncomposite is constructed from 78 stars. For our Gaussian cuts, these\nnumbers fall to 380 and 52 stars respectively. We find an offset of\norder 0.2 dex between the thick and thin components for the 2$\\sigma$\ncuts, with the thick disc being more metal poor at [Fe\/H]$=-0.9\\pm0.1$\ncompared with [Fe\/H]$=-0.7\\pm0.05$ for the thin, inconsistent within\ntheir respective 1$\\sigma$ errors. For our Gaussian cuts, we find the\nthick disc to be more metal poor, giving us a larger difference in\nmetallicity between the two components of 0.3 dex (with\n[Fe\/H]$=-1.0\\pm0.1$ for the thick disc compared with\n[Fe\/H]$=-0.7\\pm0.05$ for the thin), although the two results for the\nthick disc are consistent within their 1$\\sigma$ errors. We also note\nthat we are liable to experience non-negligible thin disc\ncontamination of our thick disc component, which could cause us to\nover estimate the average [Fe\/H], so the true difference could be\nlarger still. We note that these results are consistent with\nperforming the same analysis on composites constructed from spectra\nwith S:N$>10$$\\; \\buildrel \\circ \\over {\\mathrm A}$$^{-1}$.\n\nFinally, the continuum fit to the third line in our composite spectra,\nparticularly for our thick disc selection, gives us some cause for\nconcern. Could this metallicity difference we derive be driven by poor\ncontinuum fitting in this region of the spectrum? To investigate this,\nwe analyse the [Fe\/H] for the thin and thick discs again, using solely\nthe first two lines (CaII$_{8498}$ and CaII$_{8542}$. In the case of\nour simple $2\\sigma$ cut, this narrows our difference in metallicity\nslightly from 0.2 dex to 0.15 dex, with [Fe\/H]$=-0.85\\pm0.1$ compared\nwith [Fe\/H]$=-0.7\\pm0.05$ for the thick and thin discs\nrespectively. However, in the case of our Gaussian cuts, which are\narguably less affected by cross contamination between the components,\nthe metallicity difference of 0.3 dex persists.\n\n\n\n\nWe also perform this composite analysis on a field-by-field basis. The\nresults of this analysis, shown in Table~\\ref{kprops} are again, less\naccurate than our overall composite, but they suggest a similar offset in\nmetallicity exists in the thin and thick components in each field. We\nplot this result as a function of radius in the lower panel of\nFig.~\\ref{summary}. We find no evidence for any evolution of\nmetallicity with radius.\n\nA slight concern in ascertaining the metallicity of a population from\na composite spectrum arises from inaccuracies in the estimate that\ncome from combining spectra with different effective\ntemperatures and $V$-band magnitudes, as the derived metallicities are\nweakly dependent on the apparent $V$-band colours of the stars. The\nrms dispersion in the $V$-band magnitudes within our sample are small\n($<0.5$ mag for both thin and thick discs) as we are sampling only a\nsmall region of the tip of the RGB, so the error introduced by this\neffect will be very small. However, to further assess this, we\nseparate our thin and thick disc spectra into bins of 0.2 mags in the\n$V$-band and create composite spectra for each bin, measuring the\nmetallicity of each. We show a sample of these spectra in\nFig.~\\ref{binned}, labelled with the metallicity and average $V$-band\nmagnitude. The typical errors in metallicities determined for these\ncomposites ranges from 0.1--0.3 dex. What we see is that the composite\nthick disc spectrum in each bin is more metal-poor than the\ncorresponding thin disc composite. We also find that the average\nmetallicities for both thin and thick discs agree with those that we\nderived from the composites for the entire sample.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{bin_sumspec_new.eps}\n\\caption{Composite spectra of our thin (left panel) and thick (right panel) disc samples, binned in $V$-band magnitude. Each bin spans 0.2 mags. We see that in each case, the thick disc is more metal-poor than the thin disc by $\\sim0.2$ dex. The errors in the values of [Fe\/H] for these composites ranges from 0.1--0.3 dex.}\n\\label{binned}\n\\end{center}\n\\end{figure*}\n   \n\n\\subsection{Photometric Metallicities}\n\nWe inspected the photometric metallicities of our sample using the\nDartmouth isochrone models \\citep{dart08}.  We select an age of 8 Gyrs\nas the work of \\citet{brown06} suggests that the age of the disc in\nthese outer regions varies between 4 and 8 Gyrs. We use an\n$\\alpha$-abundance of [$\\alpha$\/Fe]=+0.2 as it has been shown in\nvarious works (e.g. \\citealt{reddy06,abrito10}) that the\n$\\alpha$-enhancement of thin or extended stellar disc populations\ntypically ranges between [$\\alpha$\/Fe]=+0.0 and [$\\alpha$\/Fe]=+0.2. We\nthen interpolate between these isochrone models for every star within\nour sample to determine its metallicity.  We can then compare the\nMDFs for our thin and thick disc sample, selected by both the\n2$\\sigma$ and Gaussian cuts discussed above. The results of this are\nshown in the left panel of Fig.~\\ref{mdf}. This figure shows us that\nwhen using this set of isochrones, the MDFs of both populations trace\neach other remarkably well. We calculate a median metallicity for each\ncomponent and find [Fe\/H]$_{thin}=-0.79$ and [Fe\/H]$_{thick}=-0.80$,\nboth with IQRs of 0.2 dex. Neither population has a Gaussian\ndistribution, with positive kurtosis of +2.2 for both MDFs (i.e. more\npeaked, with broader tails), and both populations are skewed towards\nlower [Fe\/H] with $\\alpha\\sim-1.2$ for both discs. From this analysis,\none might conclude that the two discs are chemically\nindistinguishable. This is in contrast to our findings from the\ncombined spectra in \\S4.2 where we find an offset in the average\nmetallicities of thin and thick components of 0.2 dex. As our\nphotometric data are not deep enough to detect the MSTO of these\nfields, we are exposed to the age-metallicity-[$\\alpha$\/Fe] degeneracy\nproblem. If we analyse our data with isochrones of different ages and\nabundances, we find that the individual metallicities we measure\nchange. Increasing the age by 2 Gyrs has the effect of decreasing\n[Fe\/H] of a star by $\\sim0.05$ dex on average (shown in the centre\npanel of Fig~\\ref{mdf}) and increasing the abundance from\n[$\\alpha$\/Fe]=+0.2 to [$\\alpha$\/Fe]=+0.4 reduces [Fe\/H] by\n$\\sim0.1$~dex, (right hand panel, Fig.~\\ref{mdf}). The dispersions,\nkurtosis and skew remain largely unchanged by these variations. These\nfindings demonstrate that it may be difficult to discern slight\ndifferences in metallicity (such as the 0.2 dex measured in \\S4.2)\nusing photometric isochrones without knowing the ages and\/or\n$\\alpha$-abundances of the thick and thin disc. Studies of the thin\nand thick discs in the MW have shown that the thick disc is both older\nand more $\\alpha$-enriched than the thin disc\n\\citep{reddy06,abrito10}, and many of the formation scenarios of thick\ndiscs suggest this could be true for thick discs in general, including\nM31. Such differences would certainly affect our derived values of\n[Fe\/H] for both discs.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.3\\hsize]{SWcut_phot.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{SWcut_phot_age.eps}\n\\includegraphics[angle=0,width=0.3\\hsize]{SWcut_phot_alpha.eps}\n\\caption{Photometric MDFs derived from Dartmouth isochrones of varying\n  age and [$\\alpha$\/Fe] \\citep{dart08} for the thin and thick\n  components (shown as blue hatched and red filled histograms\n  respectively) as defined by our 2$\\sigma$ cuts. {\\bf Left panel: }\n  Analysis of [Fe\/H] for thin and disc using [$\\alpha$\/Fe]=+0.2 and an\n  age of 8 Gyrs. We detect no significant differences between the two\n  populations, calculating median [Fe\/H] and\n  [Fe\/H]$_{thick}=-0.8\\pm0.2$. {\\bf Centre panel: } Increasing the age\n  of isochrones used to calculate metallicity for the thick disc from\n  8 Gyrs to 10 Gyrs. An offset of $\\sim0.05$ dex in the average [Fe\/H]\n  of the two components is observed, with median\n  [Fe\/H]$_{thick}=-0.85$. The dispersion remains the same as\n  before. {\\bf Right panel: } Increasing [$\\alpha$\/Fe] for the thick\n  disc from +0.2 to +0.4. An offset of $\\sim0.1$ dex between the\n  median metallicities of the populations is now observed, with\n  [Fe\/H]$_{thick}=-0.93$.}\n\\label{mdf}\n\\end{center}\n\\end{figure*}\n\n\\section{Discussion}\n\nIn this section we discuss our findings, and comment on the\nmorphology of this thick component. First we compare our findings with\nan expected thin+thick disc population inclined to us along the line\nof sight by 77$^\\circ$, by creating a model of a galaxy with a\nthin\/extended disc with similar properties to those of M31 that has an\nadditional thick disc component and analysing it in the same way as\nour data. We then compare the M31 thick disc to the MW and the\n\\citet{yoachim06} sample of thick discs. Finally we comment on the\npossible formation mechanisms for this component.\n\n\\subsection{Comparison with thin + thick disc model}\n\nTo lend confidence to our defining the lagging component we isolate in the above\nanalysis as a thick disc, we create a simple kinematic model\nof a galaxy with properties similar to those of a MW-type galaxy,\nwhich has both a thin and thick stellar disc, and analyse this in the\nsame manner as our data. This is done as follows; first, we create a\nthin stellar disc of 9$\\times10^{6}$ stars, randomly generating radii\nfor each assuming the stars are distributed in an exponential disc\nwith a scale length equal to that of M31's (6.6 kpc,\nI05). We assign each particle with a velocity randomly\ndrawn from a Gaussian population centred on 0${\\rm\\,km\\,s^{-1}}$ with a velocity\ndispersion of 25${\\rm\\,km\\,s^{-1}}$ in the disc frame. We repeat this for our thick\ndisc component, assuming a thin:thick disc density ratio in M31 that\nis equal to that measured in the solar neighbourhood of 9:1\n\\citep{just10}, giving 1$\\times10^6$ stars, and we use a thick disc\nscale length of 8.0 kpc, as determined above. For the velocities, we\nassume the thick disc lags behind the thin by 50${\\rm\\,km\\,s^{-1}}$ and has the\nsame velocity dispersion as the MW thick disc, $\\sigma=40{\\rm\\,km\\,s^{-1}}$\n\\citep{ivezic08}. We also generate vertical heights within the discs\nfor both thin and thick populations, assuming MW scale heights for the\ndiscs (300 pc and 1000 pc for thin and thick disc,\n\\citealt{ivezic08}), $I$-band magnitudes between $25.0\\ge I\\ge20.3$\n(0.1 mags brighter than the tip of the RGB in M31) and angular\npositions within the disc. We then convert our disc frame velocities\ninto heliocentric velocities using the HI rotation curve of\n\\citet{chemin09}. Finally, we interpret this model in the same way as\nour data, by rotating it into the coordinate system of M31 as observed\nfrom the MW (with inclination and PA as discussed in \\S~3), and\nsubtracting off the assumed disc velocity at that position by\ninterpolating each of our model stars into our average disc velocity\nmap (Fig.~\\ref{velmap}).\n\nFrom this model data set, we select stars as we would select targets\nto observe when designing DEIMOS masks, requiring them to have\n$I$-band magnitudes between $22.0\\ge I\\ge20.5$. We then randomly\nselect the same number of stars as are observed at each field\nlocation, and make velocity histograms in the disc-lag frame for each\nfield. We then analyse these distributions with the same GMM technique\ndescribed in \\S~3.1, using a LRT to determine whether the distribution\nof each model field is best fit by a single thin disc component, or a\ndouble thin+thick component. This procedure is repeated 100 times,\nallowing us to compute the average velocities and dispersions for each\ncomponent, plus sampling errors which we tabulate in\nTable~\\ref{mprops}. In our final 100 samples, the thick disc is\ndetected in 15 of the 21 fields on average. The fact that we do not\nsee the thick disc component in all our model fields implies that the\nnon-detections in our data are an effect of our sampling of the DEIMOS\nfields rather than the component being absent in these fields. We show\nthe histograms and best fit Gaussians for three of these realizations\ncompared to our data in Fig.~\\ref{models}.\n\n\\begin{table*}\n\\begin{center}\n\\caption{Average kinematic properties of model fields from 100 MC realisations}\n\\label{mprops}\n\\begin{tabular}{lccccc}\n\\hline\nField & v$_{thin}$ (disc frame, ${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thin} ({\\rm\\,km\\,s^{-1}})$ & v$_{thick}$ (disc frame, ${\\rm\\,km\\,s^{-1}}$) & $\\sigma_{thick} ({\\rm\\,km\\,s^{-1}})$ \\\\\n\\hline\n228Mod & 11.4$\\pm5.0$ & 19.2$\\pm8.0$ & -50.0$\\pm8.9$  & 38.3$\\pm8.1$   \\\\\n227Mod & -1.7$\\pm0.6$ & 25.5$\\pm10.8$& -37.1$\\pm15.6$ & 46.3$\\pm12.4$  \\\\\n166Mod & -3.4$\\pm2.1$ & 25.9$\\pm7.3$ & -46.3$\\pm9.1$  & 43.0$\\pm13.4$  \\\\\n106Mod & -4.4$\\pm3.2$ & 25.9$\\pm8.2$ & -54.3$\\pm14.1$ & 42.8$\\pm9.3$   \\\\\n105Mod & -2.1$\\pm1.2$ & 22.0$\\pm9.4$ & -54.6$\\pm13.8$ & 39.4$\\pm12.8$  \\\\\n224Mod & -2.7$\\pm1.9$ & 23.1$\\pm9.9$ & -40.2$\\pm9.2$  & 42.3$\\pm10.3$  \\\\\n232Mod & -3.1$\\pm2.7$ & 21.8$\\pm8.2$ & -45.9$\\pm12.2$ & 41.2$\\pm13.2$  \\\\\n104Mod & -2.5$\\pm1.7$ & 22.8$\\pm10.2$& -57.6$\\pm14.4$ & 43.0$\\pm11.0$  \\\\\n220Mod & -7.0$\\pm3.8$ & 21.1$\\pm10.3$& -62.9$\\pm8.7$  & 36.2$\\pm10.4$  \\\\\n213Mod & -8.1$\\pm4.3$ & 19.5$\\pm9.9$ & -55.1$\\pm13.2$ & 32.2$\\pm13.9$  \\\\\n102Mod & -3.5$\\pm2.7$ & 20.6$\\pm8.9$ & -49.1$\\pm7.9$  & 34.9$\\pm12.4$  \\\\\n231Mod & -3.7$\\pm2.2$ & 19.4$\\pm7.4$ & -49.6$\\pm12.3$ & 33.2$\\pm9.3$   \\\\\n223Mod & -1.8$\\pm1.1$ & 20.8$\\pm7.2$ & -56.1$\\pm11.7$ & 36.7$\\pm8.8$   \\\\\n101Mod &  1.9$\\pm1.3$ & 22.3$\\pm7.0$ & -64.1$\\pm11.2$ & 38.4$\\pm10.8$  \\\\\n222Mod & -0.1$\\pm1.2$ & 19.9$\\pm6.6$ & -51.3$\\pm12.6$ & 34.8$\\pm11.0$  \\\\\n221Mod &  3.9$\\pm2.3$ & 19.9$\\pm8.6$ & -46.2$\\pm7.0$  & 38.1$\\pm11.1$  \\\\\n50Mod  &  4.2$\\pm2.5$ & 17.9$\\pm8.2$ & -42.2$\\pm15.5$ & 33.5$\\pm12.6$  \\\\\n107Mod &  5.1$\\pm2.7$ & 17.5$\\pm7.7$ & -57.1$\\pm11.6$ & 40.4$\\pm12.9$  \\\\\nw11Mod &  4.7$\\pm1.1$ & 19.5$\\pm6.8$ & -63.1$\\pm10.4$ & 43.4$\\pm10.5$  \\\\\n167Mod &  2.0$\\pm0.7$ & 22.5$\\pm5.7$ & -52.2$\\pm8.6$ &  41.6$\\pm9.7$  \\\\\n148Mod & -1.2$\\pm1.5$ & 25.2$\\pm6.8$ & -53.2$\\pm9.7$  & 36.5$\\pm10.3$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=0,width=0.4\\hsize]{velcuts_new.eps}\n\\includegraphics[angle=0,width=0.4\\hsize]{model_gauss.ps}\n\\includegraphics[angle=0,width=0.4\\hsize]{model_gauss2.ps}\n\\includegraphics[angle=0,width=0.4\\hsize]{model_gauss3.ps}\n\\caption{A comparison of the data (top left panel) with 3 realisations\n  of parsing our thin + thick disc model through the same analysis as\n  our data, selecting stars from the same regions as the data. It can\n  be seen that the model data resembles the actual data very closely,\n  and that non-detections are likely an effect of sampling.  }\n\\label{models}\n\\end{center}\n\\end{figure*} \n\n\nWe now assess how both the lag between components and the dispersions\nof each component evolve with radius for our model data set, and how\naccurately we can recover these values from our model. In\nFig.~\\ref{modsummary}, we plot these values for our model (black\ncircles) alongside the values we obtained from our data (red squares),\nand fit the evolution of the model results with linear functions as\nbefore. In the top panel, we show the measurement of $\\Delta v$ for\nour model fields as a function of projected radius. The model results\nshow no evolution of $\\Delta v$ with radius, recovering an average lag across\nall fields of 48.9$\\pm6.7{\\rm\\,km\\,s^{-1}}$, which is similar to the constant lag\nof 50~${\\rm\\,km\\,s^{-1}}$ implemented in our model.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.9\\hsize]{model_summary.eps}\n\\caption{This figure compares the results from our data with results\n  from our model analysed in the same way. In all cases, data is\n  represented by filled red squares and dot-dashed lines, and the\n  model results are shown as filled black circles and solid\n  lines. {\\bf Top panel:} The difference in velocity, $\\Delta v$,\n  between the thin disc and thick component of data and model as a\n  function of projected radius. The model lag is consistent with no\n  evolution with radius, and shows an average lag of 48.9~${\\rm\\,km\\,s^{-1}}$, very\n  close to our input lag of 50~${\\rm\\,km\\,s^{-1}}$. {\\bf Middle panel: }Dispersion,\n  $\\sigma_{thin}$, of the thin disc is plotted for both data and model\n  as a function of projected radius. The model thin disc is best fit\n  with an average dispersion of 21.5~${\\rm\\,km\\,s^{-1}}$, very close to the input of\n  25.0${\\rm\\,km\\,s^{-1}}$. {\\bf Lower panel} Results for both data and model for\n  the dispersion of the thick disc ($\\sigma_{thick}$) as a function of\n  radius. For our model, the thick disc is consistent with no\n  evolution with radius, unlike our data, and has an average\n  dispersion of $\\sigma_v=38.8{\\rm\\,km\\,s^{-1}}$, which recovers our input\n  dispersion of 40~${\\rm\\,km\\,s^{-1}}$ relatively well. }\n\\label{modsummary}\n\\end{center}\n\\end{figure}\n \n\\begin{table}\n\\begin{center}\n\\caption{Average densities for thin and thick discs in model fields from 100 MC realisations}\n\\label{moddens}\n\\begin{tabular}{lccccc}\n\\hline\nField & $\\rho_{* thin}$ (*\/arcmin) & $\\rho_{* thick}$ (*\/arcmin) \\\\\n\\hline\n228Mod   &73.0$\\pm$35.2  &    31.0$\\pm$    19.2 \\\\\n227Mod   &15.7   $\\pm$    7.1   &    18.4   $\\pm$    8.1 \\\\\n166Mod   &14.2   $\\pm$    7.7   &    14.1   $\\pm$    7.7 \\\\\n106Mod   &11.9   $\\pm$    5.7   &    11.8   $\\pm$    5.7 \\\\\n105Mod   &13.5   $\\pm$    6.3   &    10.8   $\\pm$    5.3 \\\\\n224Mod   &10.6   $\\pm$    5.0   &    10.1   $\\pm$    4.8 \\\\\n232Mod   &10.9   $\\pm$    6.1   &    7.6    $\\pm$    4.5 \\\\\n104Mod   &10.7   $\\pm$    4.9   &    7.9    $\\pm$    3.8 \\\\\n220Mod   &4.7    $\\pm$    1.8   &    4.2    $\\pm$    1.6 \\\\\n213Mod   &4.6    $\\pm$    2.6   &    4.4    $\\pm$    2.4 \\\\\n102Mod   &3.7    $\\pm$    1.5   &    3.8    $\\pm$    1.5 \\\\\n231Mod   &3.8    $\\pm$    1.9   &    2.8    $\\pm$    1.3 \\\\\n223Mod   &2.7    $\\pm$    1.0   &    2.7    $\\pm$    0.9 \\\\\n101Mod   &2.9    $\\pm$    1.1   &    2.8    $\\pm$    1.0 \\\\\n222Mod   &2.1    $\\pm$    0.7   &    2.1    $\\pm$    0.6 \\\\\n221Mod   &1.7    $\\pm$    0.5   &    1.9    $\\pm$    0.5 \\\\\n50Mod    &1.3    $\\pm$    0.3   &    1.2    $\\pm$    0.2 \\\\\n107Mod   &1.4    $\\pm$    0.3   &    1.5    $\\pm$    0.3 \\\\\nw11Mod   &1.4    $\\pm$    0.4   &    1.5    $\\pm$    0.3 \\\\\n167Mod   &0.7    $\\pm$    0.4   &    0.5    $\\pm$    0.4 \\\\\n148Mod   &0.6    $\\pm$    0.4   &    0.5    $\\pm$    0.3 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=0,width=0.99\\hsize]{mod_slength_thin.eps}\n\\includegraphics[angle=0,width=0.99\\hsize]{mod_slength_thick.eps}\n\\caption{Results from MC recovery of the scale lengths in our\n  thin+thick disc model for our input thin (top panel) and thick\n  (bottom panel) discs. The error bars on individual points represent\n  the dispersion of calculated densities in the MC analysis, while the\n  shaded regions represent the 1$\\sigma$ uncertainties from the\n  weighted least-squares fit. We recover a scale length for the thin\n  disc of $h_r$=6.2$\\pm0.8{\\rm\\,kpc}$ and h$_r=7.8\\pm0.9$ for the thick\n  disc, which are consistent with our input values.}\n\\label{modscale}\n\\end{center}\n\\end{figure}\n\nNext, we compare the evolution of the disc dispersions for our model\nwith the data, shown in the central and lower panels of\nFig.~\\ref{modsummary}. The model thin disc is best fit with a constant\nrelation, giving an average lag of $\\sigma_{thin}=21.5\\pm1.7{\\rm\\,km\\,s^{-1}}$,\nvery close to the input of 25.0${\\rm\\,km\\,s^{-1}}$. For the thick disc dispersion,\n$\\sigma_{thick}$ the measured dispersion in the model\nscatters about a mean dispersion of $\\sigma_{thick}=38.8\\pm2.4{\\rm\\,km\\,s^{-1}}$,\nwith no evidence of evolution with radius. \n\nFinally, we can use our model to get a handle on how accurate our\nestimates of the scale length of the M31 discs might be. We use the\nsame Monte Carlo (MC) technique above to calculate the density of stars in each\ncomponent in our model fields 100 times, then we compute the average\ndensity from these results. These results are shown in\nTable~\\ref{moddens}, and the errors represent the dispersion of the\ndensities computed in each field. We then plot the densities as a\nfunction of radius for the thin and thick discs (shown in\nFig.~\\ref{modscale}), and fit the result with an exponential profile\nto determine the scale length. For the thin disc, we calculate\nh$_r=6.2\\pm0.8{\\rm\\,kpc}$, which is consistent with our input of 6.6 kpc,\nand for the thick disc we compute h$_r=7.8\\pm0.9{\\rm\\,kpc}$, consistent with\nour input of 8.0 kpc. The shaded regions indicate the 1$\\sigma$\nuncertainties from the fit. These results suggest that our\nobservationally derived scale lengths for the thin and thick discs are\na good indicator of their true scale lengths.\n\n\\subsection{Comparison to the MW and `edge-on' thick discs}\n\nNow that we have characterised the radial profile, kinematics and\nmetallicity of the thick disc in M31, we are able to compare it to the\nproperties of other thick discs that have been observed in the\nuniverse. We shall begin with the most well studied thick disc\ncurrently known -- that of our own Galaxy. Given that these two\ngalaxies are relatively close to one another (separated by 785 kpc),\nand have similar morphologies (both large spiral galaxies),\ncomparisons between the MW and M31 are often made. But for all their\napparent similarities, these two galaxies are quite different from one\nanother. Work by \\citet{hammer07} has shown that the MW is quite\ndifferent in terms of its structure and evolutionary history from the\nmajority of local spiral galaxies, whereas M31 is actually quite\n``typical'', so these differences are perhaps unsurprising. In this\nwork, we have demonstrated that the scale lengths of the M31 discs are\nlarger than those of the MW by a factor of $\\sim2$, as shown in\nFig.~\\ref{height}. Given that we derive the scale heights of the M31\ndiscs from these scale lengths, this results in scale heights in M31\nthat are of order $\\sim3$ times as thick as those of the MW. However,\nwe note that as we calculate scale heights for the M31 disc based on a\nrelation determined from disc galaxies that are quite different in\nterms of their mass to both the MW and M31, our values may be an\noverestimate. The M31 discs are also seemingly hotter than the MW\ndiscs, with $\\sigma_{thin,M31}=32.0{\\rm\\,km\\,s^{-1}}$ cf.\n$\\sigma_{thin,MW}=20.0{\\rm\\,km\\,s^{-1}}$ \\citep{ivezic08}, and\n$\\sigma_{thick,M31}=45.7{\\rm\\,km\\,s^{-1}}$ cf. $\\sigma_{thick,MW}=40.0{\\rm\\,km\\,s^{-1}}$\n\\citep{ivezic08}. This could tell us something about the merger\nhistory of M31. If the thick discs in both galaxies are formed as a\nresult of heating by mergers, the hotter discs of M31 could imply that\nthis galaxy has undergone a more active merger history than the MW.\n\nThe MW thick disc is more metal poor, enriched in $\\alpha$ metals and\nolder than the thin disc. While we are unable to measure the age and\n$\\alpha$ abundances of the M31 discs, we have shown that there exists\nan offset in the average metallicities of the two components of\n$\\sim0.2$ dex when measured spectroscopically. While we do not see\nthis offset photometrically, this could be due to our analysis\ntechnique as we use isochrones of the same $\\alpha$ abundance\n([$\\alpha$\/Fe]=+0.2) and age (8 Gyrs) for both components. If we\nmodify the $\\alpha$-abundance and age of these isochrones to\n[$\\alpha\/$Fe]=+0.4 and 10 Gyrs for our thick disc sample, we see an\noffset of $\\sim0.2$ dex. We also note that both the thin and thick\ndiscs in M31 appear to be more metal-poor than the MW discs, which\nhave average metallicities of [Fe\/H]$\\sim-0.3$ and [Fe\/H]$\\sim-0.6$\n\\citep{gilmore02,abadi03,carollo10} respectively, although there is a\nsignificant metallicity spread in both discs. \\citet{carollo10} also\ndemonstrated evidence of a secondary, more metal poor thick disc in\nthe MW, whose metallicities span the range $-1.8\\le[Fe\/H]\\le-0.8$,\npeaking at [Fe\/H]=-1.3. This component also appears to be hotter than\nthe traditional MW thick disc component, with $\\sigma_z$=44$\\pm3{\\rm\\,km\\,s^{-1}}$,\nvery similar to what we observe in M31.\n\nIn \\S4.1.3, we inferred scale heights for the thin and thick discs of\nM31 of $h_z=1.1\\pm0.2$~kpc and $h_z=2.8\\pm0.6$~kpc respectively, using\na sample of 34 galaxies with thick discs measured by YD06 to determine\na relationship between scale length and scale height of a stellar\ndisc. As we noted in \\S4.1.3, a comparison with the YD06 sample might\nnot be desirable, as these galaxies are typically much less massive\nthan M31, and selected to be bulgeless. A more appropriate comparison\nwould be the MW analogue, NGC 891, an edge-on galaxy that was recently\nthe subject of a structural analysis by \\citet{ibata09} using HST\/ACS\nimaging. They detected the presence of a thick disc component in the\ngalaxy and were able to measure both a scale length and height for\nthis component of $h_r=4.8\\pm0.1{\\rm\\,kpc}$ and $z_0=1.44\\pm0.03{\\rm\\,kpc}$,\ncompared with $h_r=4.2\\pm0.01{\\rm\\,kpc}$ and $z_0=0.57\\pm0.01{\\rm\\,kpc}$ for the\nthin disc component in this galaxy. This gives a ratio of $\\sim1.1$\nbetween the scale lengths and $\\sim2.5$ for the scale heights of these\ncomponents, which is identical to what we observe in M31. To\nillustrate this, we overplot these values for NGC 891 in\nFig.~\\ref{scale} as a green circle.\n\nIn \\citet{yoachim08a} the authors present kinematics of the thin and\nthick discs of 9 of their initial sample of 34 galaxies, obtained\nusing the GMOS spectrograph on Gemini. To measure velocities and\ndispersions in both thin and thick components, they placed slits in\npositions corresponding to the midplane of the galaxy to measure the\nthin disc properties, and above the midplane where the contribution\nfrom the thin disc was thought to be negligible. As their typical\nvelocity resolution was 60${\\rm\\,km\\,s^{-1}}$, they were unable to draw robust\nconclusions on the velocity dispersions of these components, but they\nwere able to measure velocity rotation curves for each component, and\nfound a wide variety of behaviour amongst their thick disc components,\nwith discs which lagged behind the thin disc by only $\\sim5{\\rm\\,km\\,s^{-1}}$,\ndiscs that show no evidence of rotation and one case where the thick\ndisc is counter-rotating with respect to the thin disc. The average\nlag between the thin and thick components of $\\Delta v=46.0{\\rm\\,km\\,s^{-1}}$ we\nsee in the M31 system is larger than the majority that they\nobserve. We note that the galaxies in their sample were typically of\nmuch lower mass than M31 (V$_{circ}<150{\\rm\\,km\\,s^{-1}}$ cf\nV$_{circ}\\sim230{\\rm\\,km\\,s^{-1}}$). In the most massive of their sample (which are\nstill less massive than M31), they do not detect a lag in the thick\ndisc kinematics at all, and they attribute this to contrast\nissues. Their sample were also selected to be ``bulgeless'', unlike\nM31 which has a significant bulge, and so a direct comparison may not\nbe advisable. Owing to the wide range of kinematic behaviour exhibited\nin their sample, they conclude that the dominant formation process of\nthick discs is via minor mergers and accretions of satellites. In\n\\citet{yoachim08b}, they use Lick indices to measure ages and\nmetallicities in 9 low mass galaxies with thick disc components. While\nwe measure an offset of 0.2 dex in the metallicities of the M31 thick\nand thin discs, they were unable to measure any such offset in their\nsample, though this could be a result of the insensitivity of Lick\nindices to such differences at low metallicity. They do find that the\nthick discs are host to older stellar populations than the thin disc,\nhowever with our current data set, we are unable to comment on the\nages of stars in the M31 discs.\n\n\\subsection{Possible formation scenarios}\n\nIn this section, we discuss the various formation scenarios mentioned\nin \\S~1. Owing to our inability to measure ages and vertical dispersions in\nM31, we are not able to confirm or reject any of these formation\nmechanisms at present, so we discuss additional constraints for these\nmodels that could help to rule out or confirm each scenario with\nfurther data and analysis.\n\n\\subsubsection{Heating by minor mergers}\n\nNumerous studies have identified that impacts and mergers of\nsatellites with masses less than a third of their hosts can\nkinematically heat the thin stellar disc, puffing it out into a\nsubstantially thicker disc\n(e.g. \\citealt{quinn93,robin96,walker96,velazquez99,chen01,sales09,villalobos09}). M31\nis known to have recently undergone at least one significant minor\nmerger event, resulting in the GSS tidal stream. In recent work by\n\\citet{purcell10}, the authors model the heating of the stellar discs\nby minor mergers and trace disc stars ejected into the stellar halo by\nthese simulated events. In addition to the stars ejected into the\nhalo, they observe a concomitant increase in the number of stars\nlocated in the kinematic regime of the thick disc, contributing\n$\\sim~10-20$\\% of the total stellar density along the major axis,\nsimilar to what we observe in M31. They also find that their simulated\nplanar infall produces two-component systems with scale heights\n(z$_{thin}\\sim 1~{\\rm\\,kpc}$ and z$_{thick}\\sim3~{\\rm\\,kpc}$), consistent with our\nmeasurements for M31. The similarities between our findings and those\nof \\citet{purcell10} could suggest that thin disc stars heated by the\nmerging event that created the GSS may contribute some non-trivial\nfraction of stars to the thick disc. \n\n According to the simulations of \\citet{kazantzidis09}, thick discs\n produced in this vein imprint a number of dynamical signatures on\n both the kinematic and structural properties of the galaxy. These\n include considerable thickening and heating at all radii, prominent\n flaring, particularly in the outskirts of the disc (beyond 3 scale\n lengths), surface density excesses at large radii, radial\n anisotropies and substantial tilting of the disc. As M31 is not edge\n on, we are unable to comment on the evolution of the height of the\n thick disc with radius, and so we cannot use this as a measure of\n flaring in the outer regions of the disc. However, one might expect\n that if there was a substantial flaring beyond 3 disc scale lengths\n ($\\sim24$~kpc), that this may be reflected by an increase in the\n velocity dispersions of both thin and thick disc components. Our\n results for evolution in the thin and thick disc dispersions remain\n inconclusive, and so it is possible that such flaring may exist. At\n present, we possess few fields between R$\\sim32$ and 39.6~kpc, so\n populating this region with kinematics, as well as additional fields\n further out, may further enlighten us to any potential\n flaring. Another test of this formation scenario would be to include\n fields from both the minor axis and NE portion of M31 to test for any\n radial anisotropy, assuming one can reliably disentangle\n contamination from foreground and substructure from the signatures of\n the discs. The work of \\citet{sales09} also tells us that thick discs\n that are produced as a result of heating present structures with low\n orbital eccentricity.\n\n\\subsubsection{Accretion of satellite on a coplanar orbit}\n\nNumerical simulations by \\citet{abadi03} and \\citet{penarrubia06} show\nthat an old, thick disc of stars could form via the accretion of stars\nfrom satellite galaxies on an approximately coplanar orbit with its\nhost. Such discs are similar in radial extent and contain older\nstellar populations when compared to the thin disc. The thick disc we\nfind in M31 is consistent with this model in so far as the radial\nextents of both discs (5.9--7.3~kpc and 8.0~kpc) are comparable with\none another. They also argue that the mass and luminosity of the\nprogenitor satellite can be inferred from the metallicity of the\ncomponent. We deduce [Fe\/H]$_{thick}=-1.0\\pm0.1$ for M31, which would\ncorrespond to a satellite of $M_V\\sim-15$ ($L_v\\sim9\\times10^7{\\rm\\,L_\\odot}$),\nsimilar to the M31 dwarf elliptical, NGC 147 ($M_V=-15.1$,\n\\citealt{vandenbergh99}). However, given the mass we calculate for the\nthick disc in \\S4.1.4 of 2--4$\\times10^{10}{\\rm\\,M_\\odot}$, it seems very\nunlikely that the thick disc of M31 could have been formed from such a\nsatellite.\n\nResults of the simulations of \\citet{sales09} show that stars accreted\ninto a thick disc from satellites on coplanar orbits exhibit high\neccentricity orbits. Our present data set does not allow us to probe\nthe eccentricity of the orbits within the thick disc at this\ntime. With a larger data set, we could perhaps see the effects of\norbital differences in the form of structural asymmetries.\n\n\\subsubsection{Radial migration and internal heating}\n\nThe scattering of stars by spiral structure and molecular clouds has\nlong been proposed as a method of heating the stellar disc, moving\nstars out onto more eccentric and inclined orbits\n\\citep{sellwood02,haywood08,roskar08,schonrich09a}, and it has been\nargued in \\citet{schonrich09a,schonrich09b} that these processes\nnaturally produce an old, $\\alpha$-enhanced thick disc, whose\nproperties are consistent with those observed in the MW. These models\nalso show wide MDFs and an increase in the scatter of the\nAge-Metallicity relation. This is also demonstrated in\n\\citet{quillen09}, where they investigate radial mixing induced by an\norbiting subhalo. Again they find evidence of wide MDFs in both the\nthin and thick discs. With deeper photometry that allowed us to reach\nthe MSTOs of the two discs we could derive the average ages of these\ncomponents, and high resolution spectroscopy (R$\\sim15,000$) of M31\nthick disc stars that would allow us to determine accurate abundances\nfrom unblended Fe lines for individual stars, we could comment more\nrobustly on the likelihood of such a formation scenario.\n\n\\subsubsection{Thick disc forms thick}\n\n\\citet{kroupa02} posited that thick discs could be formed as a result of \nvigorous star formation in massive star clusters ($\\sim10^5-10^6{\\rm\\,M_\\odot}$) \nduring the period of assembly of the stellar disc. If this is true, \na number of these massive clusters may have survived to the present day, \nand would possess large vertical velocity dispersions. \\citet{kroupa02} \nsuggests that these clusters could be the metal-rich globular cluster system \nin the MW. Once again, owing to the inclination of M31, we are unable to \nmeasure the vertical dispersions of its metal-rich globular cluster system, \nand can therefore neither confirm nor reject this formation model.\n\n\\section{Conclusions}\n\nUsing the DEIMOS spectrograph on the Keck II telescope, we have\nidentified a statistically significant population of stars in M31 that\nlags behind the thin and extended discs by 46.0$\\pm3.9{\\rm\\,km\\,s^{-1}}$. Comparing\nthis with a model of a thin+thick disc system with the same distance\nand inclination as M31 shows this component to be consistent with a\nthick disc component. Analysing its kinematics, we find it to be\nhotter than the thin disc, with average dispersion\n$\\sigma_{thick}=50.8\\pm1.9{\\rm\\,km\\,s^{-1}}$ cf. $\\sigma_{thin}=35.7\\pm1.0{\\rm\\,km\\,s^{-1}}$,\nlarger than the dispersions observed in the MW discs. From composite\nspectra for each component, constructed from highly probable thin and\nthick disc stars (selected using stringent Gaussian cuts) we measure a\nmetallicity offset of $\\sim0.3$ dex between the two disc, with the\nthick disc being metal-poor than the thin disc\n(${\\rm[Fe\/H]}_{thick}=-1.0\\pm0.1$ cf. ${\\rm[Fe\/H]}_{thin}=-0.7\\pm0.05$). The fact\nthat this metallicity offset is not observed when analysing the thin\nand thick disc RGB stars with isochrones of identical age and\n$\\alpha$-abundance suggests that the two populations differ in these\nproperties, with the thick disc likely being older and more enriched\nin $\\alpha$ elements.\n\nWe measure scale lengths for both thin and thick discs, finding\n$h_r=8.0\\pm1.2{\\rm\\,kpc}$ for the thick disc, and $h_r=7.3\\pm1.1{\\rm\\,kpc}$ for\nthe thin disc, comparable to previous estimates. Using the data of\nYD06 we infer scale heights for both discs at $z_0=2.8\\pm0.6{\\rm\\,kpc}$ and\n$z_0=1.1\\pm0.2{\\rm\\,kpc}$ for thick and thin discs respectively. These\nvalues are of order 2--3 times larger than those measured in the MW,\nperhaps suggesting that M31 has undergone more heating than our\nGalaxy. \n\nBy measuring the ratio of the densities of both discs, we are able to\nestimate a mass range for the thick disc component of\n2.4$\\times10^{10}{\\rm\\,M_\\odot}<M_{*,thick}<4.1\\times10^{10}{\\rm\\,M_\\odot}$. This value\nprovides a useful constraint on possible formation mechanisms, as any\nproposed method for forming a thick disc must be able to heat (or\ndeposit) at least this amount of material.\n\nOwing to current limitations within our data set, we are not able to\ndistinguish between the different thick disc formation\nmechanisms. However, with further analysis of this component using our\ncomplete kinematic sample (including regions from the minor axis and\nNE of M31) and spectroscopic follow up of fields where this component\nis strongly observed, we will be able to better understand the\nchemistry of this component and distinguish between various formation\nmechanisms.\n\n\\section{Acknowledgements}\n\nMLMC would like to acknowledge the support of an STFC studentship. GFL\nacknowledges the award of the Cambridge Raymond \\& Beverly Sackler\ndistinguished visitor fellowship. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAstrophysical \\water masers, like other maser species such as OH\nand SiO, are excellent probes of magnetic fields in a variety of\ninteresting regions. The magnetic field strength is derived from\nobservations of maser polarization caused by Zeeman\nsplitting. Polarization observations of the 22~GHz \\water masers in the\ncircumstellar envelopes (CSEs) around evolved stars have made important\ncontributions to the understanding of the magnetic fields of evolved\nstars \\citep[][hereafter V02 and V05a]{V02,V05a}. \\water maser observations have also provided\ninformation on the strength and structure of the magnetic field in\nstar-forming regions \\citep{FG89, S01, S02} and have provided upper limits for the field in the megamaser galaxy NGC 4258 \\citep{M05}.\n\nHowever, the exact relationship between the observed maser\npolarization and the magnetic field strength is not straightforward,\nleading to uncertainties in the determined field strength. It was\nshown in \\citet[][hereafter NW92]{NW92} that the influence of the\ndifferent hyperfine components of the \\water ($6_{16}-5_{23}$)\nrotational transition as well as of the degree of maser saturation is\nsignificant. The effect of maser saturation on the magnetic field\nstrength determination can be as large as a factor of 2 (NW92).\nAdditionally, velocity and magnetic field gradients might play an\nimportant role. While typical \\water maser spot sizes are between\n$\\sim$10$^{12}$ and $10^{13}$~cm \\citep{RM81, I97}, the actual size of\nthe maser region will be several times larger due to beaming\n\\citep[e.g.][]{GK72}. As a result the maser path length could be as\nlong as $10^{14}-10^{15}$~cm. Across such a path length, velocity and\nmagnetic field gradients likely influence the observed\npolarization. In \\citet[][hereafter V05b]{V05b} it was shown that\nvelocity gradients alter the shape of the total intensity line\nprofile. From this analysis and from earlier observations\n\\citep[e.g.][]{R99} it is apparent that in CSEs velocity shifts of\n$\\sim$1~\\kms are common. In this paper the effects of velocity and\nmagnetic field gradients along the maser path on the magnetic field\nstrength determined from circular polarization measurements and on the\nobserved fractional linear polarization are examined.\n\n\\begin{figure*}[ht!]\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig1.eps}}\n   \\hfill\n\\caption[]{Normalized total intensity (I) and circular polarization (V) spectra for $v_{\\rm th}=0.6, 1.0$ and $1.5$~\\kms in the presence of a velocity gradient. From thick to thin the solid lines denote velocity shifts of $\\Delta V_m = 0, 0.5, 1.0$ and $1.5$~\\kms respectively except for $v_{\\rm th}=0.6$~\\kms where $\\Delta V_m = 1.5$~\\kms is omitted.}\n\\label{Fig:ex}\n\\end{figure*}\n\n\\section{Background}\n\\label{back}\n\nSimilar to the analysis in V02 and V05b, the equations of state for\nthe populations of the upper ($6_{\\rm 16}$) and lower ($5_{\\rm 23}$)\nrotational levels of the three strongest hyperfine components\n($F-F'=7-6, 6-5$ and $5-4$) of the 22~GHz \\water maser have been\nsolved using the method described in NW92. The equation of state\nfor the number density $n(F,a,\\nu)$ of the upper energy levels ($F$)\nis\n\n\\begin{eqnarray}\n0 & = & \\lambda_F(v_s) - (\\Gamma+\\Gamma_v)n(F,a,v) \\nonumber \\\\\n& & + {R(F,F',a,b,v)}(n(F',b,v) - n(F,a,v)) \\nonumber \\\\ \n& & + \\phi(v_s)({\\Gamma_v \\over \\sum g_{F}})\\int dv\\sum g_{F} n(F,a,v).\n\\label{eq1}\n\\end{eqnarray}\n\nThe equation for the number density $n(F',b,\\nu)$ of the lower levels\n$(F')$ is similar but reversed. Here $a$ and $b$ denote the different\nmagnetic sub-states of each hyperfine component. The population levels\nare solved as a function of molecular velocity $v$ and at different\npositions along the maser propagation direction. The statistical\nweights are designated by $g_F$ and $g_F'$, and are from\n\\citet{K69}. The pump rate $\\lambda_F(v_s)$ is assumed to be the same\nfor the different hyperfine components and has a Maxwellian\ndistribution.  The results of the calculations depend only on the the\nratio $(\\lambda_F - \\lambda_F') \/ \\lambda_F$, which is of the order of\na few percent \\citep{AW93}. Generally $v_s=v$, but when velocity\ngradients are introduced later, the molecular velocity of the maser\npump $v_s$ changes along the maser path. The rate for stimulated\nemission $R(F,F',a,b,v)$ (hereafter $R$) is calculated using the local\nmaser intensity and the hyperfine interaction coefficients as\ndescribed in NW92. $\\Gamma$ is the decay rate for the molecular\nexcitations.  The cross-relaxation rate $\\Gamma_\\nu$ describes the\nreabsorbtion of previously emitted infrared pump photons trapped in\noptically thick transitions of the maser system, which generate a\nnewly excited molecule at random, within a Maxwellian velocity\ndistribution ($\\phi(v_s)$). \nElastic collisions between the maser molecules and intermixed H$_2$\nmolecules have not been included. The effects of these are described\nin detail in \\citet{E90} and are unimportant for the \\water masers\ndiscussed here.\n\n\\begin{figure*}[t!]\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig2.eps}}\n   \\hfill\n\\caption[]{The dependence of the magnetic field strength determined from circular polarization observation, represented by the $A_{F-F'}$ coefficient from Eq.~\\ref{eq2}, on the magnitude of the velocity gradient $\\Delta V_m$ for two values of intrinsic thermal line width $v_{\\rm th}$. $A_{F-F'}$ is normalized by $A^0_{F-F'}$, which is $A_{F-F'}$ for $\\Delta V_m=0$~\\kms. The solid lines represents masers with different emerging brightness temperatures. Note that the noise ($<1$\\%) on the curves is due to the velocity resolution of our model spectra. The results are independent of the magnetic field angle $\\theta$.}\n\\label{affvsv}\n\\end{figure*}\n\n\nThe maser intensity, which is solved using the radiative transfer\nequations from NW92, influences the level populations through\n$R$. However, as it is itself dependent on the level populations\ndetermined with Eq.~\\ref{eq1}, the maser total intensity (I), linear\npolarization (Q) and circular polarization (V) are solved iteratively\nalong the maser path. Assuming nearly one-dimensional maser\npropagation the beaming of the maser radiation is represented by the\nsolid angle $\\Delta\\Omega$. The emerging maser fluxes will thus be\nrepresented in $T_{\\rm b}\\Delta\\Omega$, with $T_{\\rm b}$ the\nbrightness temperature. All calculations were performed with a\nbrightness temperature for the radiation incident on the maser region\nof $T_{\\rm b}\\Delta\\Omega = 0.1$ K~sr. It was verified in NW92 and V02\nthat the results are insensitive to the chosen initial value. The\ncalculations are performed for different thermal line widths ($v_{\\rm\nth}$) of the Maxwellian particle velocity distribution, where $v_{\\rm th}\\!\\approx$0.5$ (T\/100)^{1\/2}$~\\kms with $T$ the temperature of the masing\nmolecules in $K$. \n\\citet{NW91} and NW92 have shown that when Eq.~\\ref{eq1} is\nsolved without including the contribution of the cross-relaxation rate\n$\\Gamma_\\nu$, the results for the line profiles, fractional\npolarizations and line width in the case of non-negligable\n$\\Gamma_\\nu$ can be obtained by scaling the results with\n$[\\Gamma+\\Gamma_\\nu]$. Following this result, the calculations in this\npaper are performed for $\\Gamma=1$~s$^{-1}$ and\n$\\Gamma_\\nu=0$~s$^{-1}$. Thus, in all the results of this\npaper, the emerging brightness temperature $T_{\\rm b}\\Delta\\Omega$ can\nbe replaced with $T_{\\rm b}\\Delta\\Omega \\times [\\Gamma+\\Gamma_\\nu]$ to\nobtain the results for different values of the decay and\ncross-relaxation rates.\n\nTo solve the equations of state it is assumed that the Zeeman\nfrequency shift $g\\Omega$ is much greater than $R$, $\\Gamma$ and\n$\\Gamma_\\nu$. Then the off-diagonal elements of the density matrix\ndescribing the molecular states are negligible, greatly simplifying\nthe calculations. For a magnetic field strength of $\\sim$1.0~G,\n$g\\Omega$ of the dominant hyperfine transitions is\n$\\sim$10$^4$~s$^{-1}$ while for $T_{\\rm b}\\Delta\\Omega < 10^{12}$~K~sr\nthe rate for stimulated emission $R \\leq 100$~s$^{-1}$ (V02). Most of\nthe calculations in this paper are performed for a magnetic field\nstrength of $B=1$~G. For the highest $T_{\\rm b}\\Delta\\Omega$, the\nresults will be valid for fields down to $\\sim$10~mG, when $g\\Omega$\nbecomes comparable to $R$. However, the low observed linear polarization of the 22~GHz \\water masers (e.g. V02, V05a) indicates that\n$T_{\\rm b}\\Delta\\Omega$ is at most $10^{11}$~K~sr and typically even lower. Thus, the results presented in this paper are valid for $B$ well below $1$~mG.\n\n22~GHz \\water masers begin to saturate when $R\/\\Gamma \\gtrsim\n  1$~s$^{-1}$ and approach full saturation when $R\/\\Gamma \\gtrsim\n  100$~s$^{-1}$. Thus, for our models with $\\Gamma=1$~s$^{-1}$,\nsaturation starts at $T_{\\rm b}\\Delta\\Omega \\gtrsim 10^{10}$~K~sr and\nfull saturation occurs for $T_{\\rm b}\\Delta\\Omega \\gtrsim\n10^{12}$~K~sr. The maser spectral line profile (through rebroadening)\nand the polarization properties of the maser changes at\n$R=(\\Gamma+\\Gamma_\\nu)$ (e.g. NW92). While typically for the 22~GHz\n\\water masers this is close to when saturation occurs, a large\ncross-relaxation rate means the changes will only occur when the maser\nhas already become partly saturated. As $\\Gamma_\\nu$ for the\n  astrophysical \\water masers is at most $\\approx 5$~s$^{-1}$ for\n  \\water masers at $T\\sim 1000~K$ and is less than that lower\n  temperatures \\citep{AW93}, $\\Gamma_\\nu\\approx\\Gamma$ is assumed in\n  this paper when discussing unsaturated and saturated masers. It is\n  worth noting that \\water maser observations indicate that in\n  actuality $\\Gamma$ is between one and two orders of magnitude less\n  than $\\Gamma_\\nu$ \\citep{AW93}. However, as our results scale\n  linearly with $(\\Gamma+\\Gamma_\\nu)$ this does not affect the\n  conclusions in this paper.\n\n\\begin{figure*}[t!]\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig3.eps}}\n   \\hfill\n\\caption[]{a) Derived magnetic field strength $B'$ over the average field strength along the maser $\\langle B\\rangle$ as a function of $T_{\\rm b}\\Delta\\Omega$. The solid and dashed lines correspond to positive and negative magnetic field gradients respectively. b) $A'_{F-F'}$ as a function of $\\Delta\\theta$ for different values of emerging brightness temperature $T_{\\rm b}\\Delta\\Omega$. Calculations were performed for $v_{\\rm th}=0.8$~\\kms.}\n\\label{bchange}\n\\end{figure*}\n\n\\section{Radiative Transfer Modeling Results}\n\n\\subsection{Circular Polarization}\n\nAs shown in NW92 and V02, the V-spectra produced from the radiative\ntransfer equations are not anti-symmetric. They are also not always\ndirectly proportional to the derivative of the I-spectrum due to\nhyperfine interaction as is often assumed. However, as was discussed\nin V02, it is impossible to directly observe the intrinsic shape of\nthe V-spectrum due to instrumental effects and necessary data\ncalibration steps. Still, the percentage of circular polarization\n$P_{\\rm V}$ can be related to the magnetic field strength using\n\\begin{eqnarray}\nP_{\\rm V} & = & (V_{\\rm max} - V_{\\rm min})\/I_{\\rm max} \\nonumber\\\\\n& = & 2\\cdot A_{F-F'}\\cdot B_{\\rm [Gauss]} \\rm{cos}\\theta\/\\Delta v_{\\rm L}[\\rm{km~s^{-1}}].\n\\label{eq2}\n\\end{eqnarray}\nHere $V_{\\rm max}$ and $V_{\\rm min}$ are the minimum and maximum of\nthe V-spectrum. $I_{\\rm max}$ and $\\Delta v_{\\rm L}$ are the peak flux\nand the full width half-maximum (FWHM) of the I-spectrum respectively.\n$B$ is the magnetic field strength at an angle $\\theta$ to the maser\npropagation direction. While in most analyses of maser circular\npolarization, the coefficient $A_{F-F'}$ is taken to be a fixed value\n\\citep[e.g][]{FG89}, it was shown in NW92 to depend on the level of\nmaser saturation and the intrinsic thermal velocity $v_{\\rm th}$ of\nthe maser. Figure 6 of V02 shows $A_{F-F'}$ for different values of\n$v_{\\rm th}$ as a function of emerging maser brightness temperature\n$T_{\\rm b}\\Delta\\Omega$. \nFor maser brightness temperatures $T_{\\rm b}\\Delta\\Omega >\n10^9$~K~sr it was shown in NW92 that the ${\\rm cos}\\theta$ dependence\nof Eq.~\\ref{eq2} breaks down effectively also introducing a dependence\non $\\theta$ to $A_{F-F'}$. This was later shown in more detail in\n\\citet{WW01} for masing involving angular momentum $J=$1--0 and\n$J=$2--1 transitions. In V02, figure 7 shows the derived magnetic\nfield strength dependence on $\\theta$ to $A_{F-F'}$ for the 22~GHz\n$J=$6--5 transition.\n\n\\subsubsection{velocity gradients}\n\n\\begin{figure*}\n   \\resizebox{0.9\\hsize}{!}{\\includegraphics{fig4.eps}}\n   \\hfill\n\\caption[Linpol]{a) Magnetic field angle gradient $\\Delta\\theta$ vs. the fractional linear polarization for three different stages of maser saturation and $\\theta_0 = 0^\\circ$. b) The angle $\\theta$ between the maser propagation direction and the magnetic field vs. the fractional linear polarization for different values of emerging maser brightness. The thick solid line denotes the theoretical limit from \\citet{GKK} for a completely saturated maser.}\n\\label{Fig:linpol}\n\\end{figure*}\n\nFirst, a velocity gradient is introduced by uniformly shifting, in\nEq.~\\ref{eq1}, the velocity $v_s$ of the pump rate $\\lambda_F$ for\neach integration step along the maser path. By varying the amount of\nshift, a velocity difference $\\Delta V_m$ is created between the start\nand end of the maser amplification path of up to $\\sim$1.5~\\kms. This\ncorresponds to a velocity gradient of $\\Delta V_m \/ S$~\\kms m$^{-1}$,\nwhere $S$ is the length of the maser path. As $S$ depends on the exact\nvalues of the pumping rate $\\lambda_F$, while our results, expressed\nin emerging brightness temperature are only dependent on the ratio of\npumping rates as addressed above, any mention of the velocity gradient\nwill henceforth be referring to the value of $\\Delta V_m$. In\n\\citet{NW88} it was shown that the maser splits into several\ndistinctly separate narrow features when $\\Delta V_m$ becomes a few\ntimes $v_{\\rm th}$, each of the features becoming an independent\nmaser. The calculations in this paper are therefore limited to values\nof $\\Delta V_m$ up to $\\sim$1.5~\\kms.\n\nFig.~\\ref{Fig:ex} shows the effect of a velocity gradient on the total\nintensity and circular polarization profiles for three different\nintrinsic thermal velocities $v_{\\rm th}$, produced for the same\nmagnetic field strength (1~G). While for low $\\Delta V_m$, the effect\nof the velocity gradient on the shape of the I-profile is small, the\neffect on the V-spectrum can still clearly be seen. To examine how a\nvelocity gradient influences the magnetic field determination from the\ncircular polarization measurements, the normalized $A_{F-F'}$\ncoefficient from Eq.~\\ref{eq2} is plotted in Fig.~\\ref{affvsv} as a\nfunction of velocity gradient for $v_{\\rm th}=0.6$ and $1.5$~\\kms and\nemerging brightness temperatures $T_{\\rm b}\\Delta\\Omega = 10^9,\n10^{10}$ and $10^{11}$~K~sr. Because of the normalization, the\nresults are independent of the magnetic field angle $\\theta$. For\n$v_{\\rm th}=0.6$~\\kms the effect of the hyperfine components is\nclear. For $T_{\\rm b}\\Delta\\Omega = 10^9$~K~sr none of the hyperfine\ncomponents are becoming saturated yet, thus $A_{F-F'}$ is mostly\nsymmetric around $\\Delta V_m = 0$~\\kms. The symmetry breaks down when\nthe hyperfine lines are slowly starting to saturate for $T_{\\rm\nb}\\Delta\\Omega \\gtrsim$10$^{10}$~K~sr, as maser growth is somewhat\ninhibited for velocity gradients in the direction of the weaker\nhyperfine components ($F-F'=6-5$ and $5-4$), while a gradient in the\nopposite direction allows for stronger maser amplification. As\nnoted above this effect can be postponed further into the saturated\nregime when $\\Gamma_\\nu$ increased. It can be seen in the figure, that\n$A_{F-F'}$ typically decreases for increasing $|\\Delta V_m|$, implying\nthat in the presence of velocity gradients the magnetic field strength\n$B'$ derived with Eq.~\\ref{eq2} underestimates the true $B$. For small\n$v_{\\rm th}$ the field strength can sometimes be underestimated by\nmore than a factor of $2$. However, for a narrow range in $\\Delta\nV_m$, $A_{F-F'}$ is actually enhanced. In those cases $B$ could\nactually be overestimated by up to $\\sim$20\\%. This effect is largest\nfor the more saturated masers.\n\n\\subsubsection{magnetic field gradients}\n\nThere are different ways a magnetic field gradient along the maser\namplification path can occur. The actual field strength $B$ can change\nor the angle $\\theta$ between the field and the maser propagation\ndirection can vary. Similar to $\\Delta V_m$, differences\n$\\Delta\\theta$ and $\\Delta B$ are introduced between the start and end\nof the maser path and their influence on the maser polarization is\nexamined. Fig.~\\ref{bchange}a shows the ratio between the magnetic\nfield $B'$ determined with Eq.~\\ref{eq2} and the average magnetic\nfield $\\langle B\\rangle$ along the maser as a function of $T_{\\rm\nb}\\Delta\\Omega$ when a gradient $\\Delta B$ is included. The shape of\nthis function does not depend on the actual value of $\\Delta B$ or on\nthe intrinsic thermal line width $v_{\\rm th}$. While the maser is\nunsaturated, or more precisely while $R<\\Gamma_\\nu$, $B'$ is\nequal to $\\langle B\\rangle$. However, when the maser\nsaturates, $B'$ will be dominated by the average magnetic field in the\nunsaturated maser region.\n\nFig.~\\ref{bchange}b shows the change in $A'_{F-F'}$ versus\n$\\Delta\\theta$ for different $T_{\\rm b}\\Delta\\Omega$. The angle\n$\\theta_0$, the value of $\\theta$ at the start of maser amplification\nis taken to be $0^\\circ$. Here $A'_{F-F'} = A_{F-F'}$cos$\\theta$, as\nthe actual angle $\\theta$ cannot be specified due to the inclusion of\nthe gradient $\\Delta\\theta$. Comparing the unsaturated maser ($T_{\\rm\nb}\\Delta\\Omega = 10^9$~K~sr) with the partly saturated maser ($T_{\\rm\nb}\\Delta\\Omega = 10^{11}$~K~sr) it is again apparent that, as seen in\nFig.~\\ref{bchange}a, the unsaturated maser regime provides the\ndominant contribution to the magnetic field strength determined with\nEq.~\\ref{eq2}. Comparing Fig.~\\ref{bchange}b with figure 7 from V02\nindicates that the angle average $\\langle\\theta\\rangle$ in the\nunsaturated maser region determines the observed circular\npolarization.\n\n\\subsection{Linear Polarization}\n\nThe linear polarization in the presence of a magnetic field was also\ndiscussed in NW92. The fractional linear polarization is not found to\nbe affected when including a velocity gradient $\\Delta V_m$ or\nmagnetic field gradient $\\Delta B$ in the calculations. Only when\nincluding a gradient $\\Delta\\theta$ does the linear polarization\nchange. In Fig.~\\ref{Fig:linpol}a the fractional linear polarization\n$|Q_0|\/I_0$ is shown as a function of $\\Delta\\theta$.  This can be\ncompared with $|Q_0|\/I_0$ as a function of the angle between the\nmagnetic field and the maser propagation direction $\\theta$ shown in\nFig.~\\ref{Fig:linpol}b. This relationship, in the limiting case of a\ncompletely saturated maser, was solved in \\citet{GKK}. In the case of\nan unsaturated maser ($T_{\\rm b}\\Delta\\Omega \\lesssim 10^{10}$~K~sr), the\nrelationship between $|Q_0|\/I_0$ and the magnetic field angle gradient\n$\\Delta\\theta$ is very similar to the relationship between $|Q_0|\/I_0$\nand $\\theta$, while when the saturation level increases the\nlinear polarization is quenched and shifted. This indicates that the\nobserved fractional linear polarization is determined by $\\theta$ at\nthe end of the unsaturated maser regime, where the largest\namplification occurs, and not by the average $\\langle\\theta\\rangle$\nover the unsaturated maser core.\n\n\\section{Conclusions}\n\\label{concl}\n\nUsing a maser radiative transfer code which includes the three\nstrongest hyperfine components of the 22~GHz \\water $6_{\\rm 16} -\n5_{\\rm 23}$ rotational transition and their magnetic sub-states, the\neffects of velocity and magnetic field gradients on the determination\nof magnetic fields from maser polarization observations have been\ncalculated.  It was shown that, due to velocity gradients, the\nmagnetic field strength determined on \\water masers in CSEs (V02,\nV05a), where $v_{\\rm th}\\!\\approx$0.8$-1.0$~\\kms and where velocity\ngradients of $\\sim$1.0~\\kms have been observed (V05b), can be\nunderestimated by up to a factor of 2. However, for maser features\nwith smaller velocity gradients there will be cases where $B$ is\nactually overestimated by $\\sim$10$-20\\%$. The magnetic fields\ndetermined on the \\water masers in star-formation regions \\citep{FG89,\nS01, S02}, which have higher thermal line widths ($v_{\\rm\nth}\\!\\approx$1.5~\\kms) can be underestimated by $\\sim$20$-40\\%$. These\nuncertainties can only be overcome by including a consistent model for\nthe velocity gradients throughout the \\water maser source when\nderiving the magnetic field strengths derived from circular\npolarization observations.\n\nWhen there is a magnetic field gradients along the maser amplification\npath, $B'$ determined from polarization observations of an unsaturated\nmaser (more formally when $R\\lesssim\\Gamma_\\nu$), is typically\nequal to the average field along the maser. When the maser is\nsaturated, the $B'$ is dominated by the average magnetic field\nstrength or the average magnetic field angle in the unsaturated maser\ncore. The fractional linear polarization on the other hand, is not\naffected by velocity or magnetic field gradients. Only a gradient in\nthe angle between the magnetic field and the maser propagation axis\nalters the linear polarization. In contrast to the circular\npolarization, the fractional linear polarization is mainly determined\nby the magnetic field angle in the part of the unsaturated maser where\nthe largest amplification occurs and not by the angle averaged over\nthe entire unsaturated maser region.\n\n\\begin{acknowledgements}\nThe author acknowledges the helpfull comments by the anonymous\nreferee.  This work has been supported by an EC Marie Curie Fellowship\nunder contract number MEIF-CT-2005-010393.  \n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}