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+{"text":"\\section{Introduction}\n\n\nThe separability problem in Quantum Information Theory asks for a  criterion to detect  entanglement of quantum states. Denote by  $M_k$ and $P_k$  the set of complex matrices of order $k$ and the set of positive semidefinite Hermitian matrices of order $k$, respectively.  \n\nThis problem is known to be a hard problem \\cite{gurvits2003,gurvits2004} even for bipartite mixed states, which \n can be briefly described as: \\\\\n\n\\begin{quote}\nDetermine whether a given $A\\in M_k\\otimes M_m$ is separable or entangled ($A$ is separable if $A=\\sum_{i=1}^n C_i\\otimes D_i$, where $C_i\\in P_k, D_i\\in P_m$, for every $i$, and entangled otherwise).\\\\\n\\end{quote}\n\n\n\n\nThis problem was only solved  for $km\\leq 6$. Its original solution requires the classification of positive maps $T:M_k\\rightarrow M_m$, $km\\leq 6$, which is unknown for $km>6$.\n \nLet us identify $M_k\\otimes M_m\\simeq M_{km}$ via Kronecker product. We shall say that $A=\\sum_{i=1}^nA_i\\otimes B_i \\in M_k\\otimes M_m\\simeq M_{km}$ is positive under partial transposition, or simply PPT, if  $A$ and its partial transposition $A^{t_2}=\\sum_{i=1}^nA_i\\otimes B_i^t$ are positive semidefinite Hermitian matrices.\n\nNow, the separable matrices in  $M_k\\otimes M_m (km\\leq 6)$ are just the PPT matrices \\cite{peres, horodeckifamily}.\n\n\nAn alternative proof of this result (not based on positive maps)  was obtained  in $M_2\\otimes M_2$ \\cite{leinaas}.  Given a positive definite matrix $B\\in M_2\\otimes M_2\\simeq M_4$, there are invertible matrices $R,S\\in M_2$ such that \n\\begin{equation}\\label{eq=filter}\n(R\\otimes S)B(R\\otimes S)^*=\\lambda_1\\gamma_1\\otimes\\gamma_1 +\\lambda_2\\gamma_2\\otimes\\gamma_2+\n\\lambda_3\\gamma_3\\otimes\\gamma_3+\\lambda_4\\gamma_4\\otimes\\gamma_4,\n\\end{equation}\n\n\\vspace{0.2 cm}\\noindent\nwhere $\\gamma_1=\\frac{1}{\\sqrt{2}}Id$, $\\gamma_2,\\gamma_3,\\gamma_4$ are the normalized Pauli matrices and $\\lambda_i\\in\\mathbb{R}$. \nThe separability of this $B$ is equivalent  to satisfying the following inequality (see \\cite{leinaas}): \n\n\\begin{equation}\\label{eq=inequality}\n\\lambda_1\\geq |\\lambda_2|+|\\lambda_3|+|\\lambda_4|.\n\\end{equation}\n\n\n\\vspace{0.5 cm}\n\n\nThe canonical form presented in  (\\ref{eq=filter}) is the so-called filter normal form. \nIn general, we say that $A\\in M_k\\otimes M_m$ can be put in the filter normal form if there are invertible matrices $R\\in M_k, S\\in M_m$ such that \n$(R\\otimes S)A(R\\otimes S)^*=\\sum_{i=1}^sC_i\\otimes D_i$, where $C_1=\\frac{1}{\\sqrt{k}}Id$, $D_1=\\frac{1}{\\sqrt{m}}Id$ and $tr(C_iC_j)=tr(D_iD_j)=0$, for every $i\\neq j$ \\cite{filternormalform, leinaas}. \n\nNow, the inequality (\\ref{eq=inequality}) is a special case of a more general type of inequality (\\cite[Theorems 61-62]{carielloQIC}), which provides a sufficient condition for separability. \nNote that  (\\ref{eq=inequality}) is also necessary for separability in $M_2\\otimes M_2$ and has been obtained from the matrices in the filter normal form. Thus, matrices in this  form\nseem to provide sharper inequalities.\n\nMoreover, there are several criteria for detecting entanglement or separability \\cite{guhnesurvey}. The filter normal form has also been used to prove the equivalence of some of them \\cite{Git}.  Therefore, it plays an important role in Quantum Information Theory.\n\nIn this work, we tackle the problem of finding  a necessary and sufficient computable condition for the filter normal form of PPT states.  Our main result is based on a connection  between this form and the Sinkhorn-Knopp theorem for positive maps.\n\nLet $V\\in M_k$ be an orthogonal projection and  $VM_kV=\\{VXV,\\ X\\in M_k\\}$.\nA positive map $T:VM_kV\\rightarrow WM_mW$ is a linear transformation such that $T(VM_kV\\cap P_k)\\subset WM_mW\\cap P_m$.\nA positive map $T:M_k\\rightarrow M_m$ is called doubly stochastic \\cite{Landau} if\\\\ \\begin{center}\n$T(\\frac{Id}{\\sqrt{k}})=\\frac{Id}{\\sqrt{m}}$ and $T^*(\\frac{Id}{\\sqrt{m}})=\\frac{Id}{\\sqrt{k}}$,\n\\end{center} \n\\vspace{0.5 cm}\\noindent\nwhere $T^*$ is the adjoint of $T$ with respect to the trace inner product: $\\langle A,B\\rangle=tr(AB^*)$ . \n\n\n Two positive maps $L_1:M_k\\rightarrow M_m$ and $L_2:M_k\\rightarrow M_m$ are said to be  equivalent,  if there are invertible matrices $R_1\\in M_k$\nand $S_1\\in M_m$ such that $L_2(X)=S_1L_1(R_1XR_1^*)S_1^*$.\n\n\nNow, given a positive semidefinite Hermitian matrix $A=\\sum_{i=1}^nA_i\\otimes B_i \\in M_k\\otimes M_m$, consider the positive maps $G_A:M_k\\rightarrow M_m$ and  $F_A:M_m\\rightarrow M_k$ defined as \\\\\n\\begin{center}\n$G_A(X)=\\sum_{i=1}^n tr(A_iX)B_i$ and $F_A(X)=\\sum_{i=1}^n tr(B_iX)A_i$.\n\\end{center}\n\\vspace{0.5 cm}\n\nIt has been noticed that $A\\in M_k\\otimes M_m$ can be put in the filter normal form if and only if $G_A:M_k\\rightarrow M_m$ (or $G_A((\\cdot)^t):M_k\\rightarrow M_m$) is equivalent to a doubly stochastic map (e.g., \\cite{cariellosink}). \n\nThe next two conditions are known to be necessary and sufficient for the equivalence of a positive map $T:M_k\\rightarrow M_m$ with a doubly stochastic one:\\\\\n\\begin{itemize}\n\\item For square maps ($k=m$), the capacity of  $T:M_k\\rightarrow M_k$ is positive and achievable \\cite{gurvits2004}.\n\\item For rectangular maps (any $k,m$), $T:M_k\\rightarrow M_m$  is equivalent to a positive map with total support \\cite[Theorem 3.7]{cariellosink}. \\\\\n\\end{itemize}\n\nIn the classical Sinkhorn-Knopp theorem \\cite{Bapat,Brualdi,Sinkhorn,Sinkhorn2}, the concept of total support plays the key role. The second characterization above adapts this  concept to positive maps.\n\nSome easy properties on $A\\in M_k\\otimes M_m\\simeq M_{km}$ that grant the equivalence of $G_A$ with a doubly stochastic map were obtained in \\cite{cariellosink}. \nFor example, if $A$ is a positive semidefinite Hermitian matrix such that  either\\\\\n\n\\begin{itemize}\n\\item $\\dim(\\ker(A))<\\min\\{k,m\\}\\ ($if $k\\neq m)$ and  $\\dim(\\ker(A))<k-1\\ ($if $k=m)$ or\n\\item $G_A(Id)$ and $F_A(Id)$ are invertible and $\\dim(\\ker(A))<\\frac{\\max\\{k,m\\}}{\\min\\{k,m\\}}$.\\\\\n\\end{itemize}\n\nMoreover, if $k$ and $m$ are coprime then $G_A:M_k\\rightarrow M_m$ is equivalent  to a positive map with total support if and only if it has support \\cite[Corollary 3.8]{cariellosink}. \n\nNow, the positive map $G_A:M_k\\rightarrow M_m$ has support if and only if it is fractional-rank non-decreasing \\cite[Lemma 2.3]{cariellosink}. A polynomial-time algorithm that checks whether $G_A:M_k\\rightarrow M_m$ is fractional-rank non-decreasing was provided in \\cite{garg, garg2}. So the equivalence can be efficiently checked when $k$ and $m$ are coprime.\n\nUnfortunately, an algorithm that checks whether the capacity is achievable or the existence of an  equivalent map with total support is unknown (in general).\nOur main result is  an algorithm to solve this problem for positive maps $G_A((\\cdot)^t):M_k\\rightarrow M_m$, where $A\\in M_k\\otimes M_m$ is a PPT matrix with one extra property  (Algorithm 3). \n\nFirstly, we consider a PPT matrix $B\\in M_k\\otimes M_k$ such that there is  $v\\in\\Im(B)\\subset \\mathbb{C}^k\\otimes\\mathbb{C}^k$ with tensor rank $k$. \nThere is an invertible matrix $P\\in M_k$ such that $\\Im(T(X))\\supset\\Im(X)$, for every $X\\in P_k$, where $T(X)=G_C(X^t)$ and $C=(P\\otimes Id)B(P\\otimes Id)^*$ (Lemma \\ref{lemmaP}).\\\\\n\nA remarkable property owned by every PPT matrix was used in \\cite{carielloIEEE} to reduce the separability problem to the weakly irreducible PPT case.  Here, we use the same property (Proposition \\ref{propkey}) to show that this positive map $T=G_C(X^t):M_k\\rightarrow M_k$ is equivalent to a doubly stochastic map if and only if the following condition holds (Theorem \\ref{theoremprincipal}): \\\\\n\n\\begin{quote}\nFor every orthogonal projection $V\\in M_k$ such that $T(VM_kV)\\subset VM_kV$  and $T|_{VM_kV}$ is irreducible with spectral radius $\\lambda$ (see Definition \\ref{definitionIrredPerron}), there is a \\textit{\\textbf{unique}} orthogonal projection $W\\in M_k$ such that $T^*(WM_kW)\\subset WM_kW$, $T^*|_{WM_kW}$ is irreducible with spectral radius $\\lambda$, $rank(W)= rank(V)$ and\n $\\ker(W)\\cap\\Im(V)=\\{\\vec{0}\\}$.\\\\\n\\end{quote}\n\nOur algorithm 3 searches all possible pairs $(V,W)$. This is possible since there are at most $k$ pairs when $B\\in M_k\\otimes M_k$ is PPT (see Lemma \\ref{lemmakey}).\n\n \nNext, in order to find such $V$ (algorithm 1), we must look for Perron eigenvectors (see Definition \\ref{definitionIrredPerron}). \nGiven $V$,  the corresponding $W$ is related to the unique zero of  the function $f:M_{k-s\\times s}(\\mathbb{C})\\rightarrow \\mathbb{R}^+\\cup\\{0\\}$ defined as $$f(X)=trace\\left(T_1^*\\left(\\begin{pmatrix}Id& X^* \\\\ \nX& XX^*\\end{pmatrix}\\right)\\begin{pmatrix}X^*X & -X^* \\\\ \n-X & Id\\end{pmatrix}\\right),$$\nwhere $T_1(x)=\\frac{1}{\\lambda}QT(Q^{-1}X(Q^{-1})^*)Q^*$, for a suitable invertible matrix $Q\\in M_k$ (Lemma \\ref{lemmaQ}).\n\nNote that $f(X)$ seems to be a quartic function. It turns out that $f(X)$ is quadratic, since $T$ is completely positive (check the formula for $f(X)$ in Lemma \\ref{solutionW}).  If we regard $M_{k-s\\times s}(\\mathbb{C})$ as a real vector space then the search for a zero of $f(X)$ is an unconstrained quadratic minimization problem (see Lemma \\ref{solutionW} and Remark \\ref{remarkquadratic}). \n\nSince $W$ must be unique then we shall solve this unconstrained quadratic minimization problem only when the solution is known to be unique (Remark \\ref{remarkquadratic}). Therefore, the only difficult  part of our algorithm is finding Perron eigenvectors of completely positive maps. \n\n There are some algorithms that can be used to find positive definite Hermitian matrices within linear subspaces \\cite{HUHTANEN, Zaidi}, however these Perron eigenvectors might be positive semidefinite.\\\\\n\nSecondly, we extend the result to PPT matrices in $M_k\\otimes M_m$ using  \\cite[Corollary 3.5]{cariellosink}. Recall the identification $M_m\\otimes M_k \\simeq M_{mk}$. Let $B=\\sum_{i=1}^nC_i\\otimes D_i\\in M_k\\otimes M_m$ be a PPT matrix and define $\\widetilde{B}\\in M_{mk}\\otimes M_{mk}$ as $$\\widetilde{B}=\\sum_{i=1}^n (Id_{m}\\otimes C_i)\\otimes (D_i\\otimes Id_k).$$ \n\n We show that $G_B((\\cdot)^t): M_k\\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $G_{\\widetilde{B}}((\\cdot)^t): M_{mk}\\rightarrow M_{mk}$ is equivalent to a doubly stochastic map. Thus, our algorithm 3 can be applied on $\\widetilde{B}$ to determine whether $B$ can be put in the filter normal form.\\\\\n\n The existence of $v$ with full tensor rank within the range of $A\\in M_k\\otimes M_k$ is essential for our algorithm to work. There are several possible properties to  impose on a subspace  of $\\mathbb{C}^k\\otimes \\mathbb{C}^k$ in order to guarantee the existence of such $v$   within this subspace \\cite{lovasz, meshulam1985, meshulam1989, meshulam2017, pazzis, flanders, dieudonne}. The Edmonds-Rado property \\cite{gurvits2003, gurvits2004, ivanyos} is the most relevant to our problem.\n\nNote that, if $D=\\sum_{i=1}^nv_i\\overline{v_i}^t\\in M_k\\otimes M_k\\simeq M_{k^2}$, where $v_i\\in \\mathbb{C}^k\\otimes\\mathbb{C}^k\\simeq \\mathbb{C}^{k^2}$, then $$G_D(X^t)=\\sum_{i=1}^nR_iXR_i^*,$$ where $R_i=F(v_i)^t$ and $F:\\mathbb{C}^k\\otimes\\mathbb{C}^k\\rightarrow M_k$ is defined by $F(\\sum_{l=1}^t a_i\\otimes b_i)=\\sum_{l=1}^t a_ib_i^t$. In other words, $D$ is the Choi matrix associated to the completely positive map $G_D(X^t)$ \\cite{Choi}. Actually, the map $D\\rightarrow F_D((\\cdot)^t)$ is the inverse of Jamio{\\l}kowski isomorphism \\cite{Jamio}. \\\\\n\nA well known necessary condition for the equivalence of $G_D(X^t)$ with a doubly stochastic map is to be rank non-decreasing, i.e.,  $\\text{rank}(G_D(X^t))\\geq \\text{rank}(X)$ for every $X\\in P_k$ \\cite{gurvits2003}.\nWe say that the $\\Im(D)=\\text{span}\\{v_1,\\ldots,v_n\\}$ has the Edmonds-Rado property, if the existence of $v\\in\\Im(D)$ with tensor rank $k$ is equivalent to  $G_D(X^t)$ being rank non-decreasing.\\\\\n\nNow, if $\\Im(D)$ is generated by rank 1 tensors then it has the Edmonds-Rado property  \\cite{lovasz, gurvits2003}. Therefore the range of every separable matrix has this property \\cite{pawel}. Thus, if $D$ is separable with no vector $v$ with full tensor rank in its range then $D$ can not be put in the filter normal form.\n\nWe do not know whether $\\Im(D)$ has the Edmonds-Rado property when $D$ is only PPT. If the range of any PPT matrix had this property then our algorithm would work for any PPT matrix, since the existence of $v$ would be granted whenever $G_D(X^t)$ is rank non-decreasing. \n\n\n\nFinally, there are polynomial-time algorithms that check whether there is $v\\in\\Im(D)$ with tensor rank $k$  \\cite{gurvits2003} and whether $G_D(X^t)$ is rank non-decreasing \\cite{garg}.\\\\\n\nThis paper is organized as follows. In Section 2, we present some very important preliminary results. It is worth noticing that Proposition \\ref{propkey} is the key that makes our main algorithm (algorithm 3) to work.\nThis proposition describes the complete reducibility property owned by every PPT matrix \\cite{carielloIEEE}. \nIn Section 3, the reader can find our main result (Theorem \\ref{theoremprincipal}) and one problem derived from this main result (Problem \\ref{problem}). This problem can be solved as a unconstrained quadratic minimization problem (Lemma \\ref{solutionW}, Remark \\ref{remarkquadratic}). In Section 4, we bring all the results together in our algorithms. \nAlgorithm 3 provides a way to determine whether a completely positive map originated from a PPT matrix in $M_k\\otimes M_k$ (with a rank $k$ tensor within its image) is equivalent to a doubly stochastic map or not. This algorithm can be used to compute the filter normal form of PPT matrices.\nIn Section 5, we extend our results to PPT matrices in $M_k\\otimes M_m$, $k\\neq m$. \\\\\n\n\\textbf{Notation:} Let $V\\in M_k$ be an orthogonal projection.\nDenote by  $VM_k+M_kV=\\{VX+YV,\\ X,Y\\in M_k\\}$ and $V^{\\perp}=Id-V$, where $V\\in M_k$ is an orthogonal projection. Let $\\Im(A)$ be the image of the matrix $A$ and $tr(A)$ be its trace. We shall use the trace inner product in $M_k$: $\\langle A,B\\rangle=tr(AB^*)$. \n\n\n\n\n\\newpage\n\n\n\n\\section{Preliminary Results}\n\nIn this section, we present some preliminary results. It is worth noticing that Proposition \\ref{propkey} is the key that makes our main algorithm (algorithm 3) to work.\nThis proposition describes the complete reducibility property  \\cite{carielloIEEE}.  \\\\\n\n\\begin{definition}\\label{definitionIrredPerron} Let $V,V'\\in M_k$ be orthogonal projections. A non-null positive map $T:VM_kV\\rightarrow VM_kV$ is called irreducible if $T(V'M_kV')\\subset V'M_kV'$ implies that $V=V'$ or $V'=0$. The spectral radius of $T:VM_kV\\rightarrow VM_kV$ is the largest absolute value of an eigenvalue of $T:VM_kV\\rightarrow VM_kV$. A eigenvector  $\\gamma\\in P_k$  associated to the spectral radius of $T:VM_kV\\rightarrow VM_kV$ is called a Perron eigenvector of $T$.\n\\end{definition}\n\\begin{remark} The existence of a Perron eigenvector of an arbitrary positive map $T:VM_kV\\rightarrow VM_kV$ is granted by Perron-Frobenius Theory \\cite[Theorem 2.3]{evans}.\n\\end{remark}\n\n\\begin{definition}\\label{defcompletepositive} A positive map $T:M_k\\rightarrow M_k$ is completely positive if $T(X)=\\sum_{i=1}^nA_iXA_i^*$, where $A_i\\in M_k$ for every $i$. \n\\end{definition}\n\n\nThe next three lemmas are well known. We present their proof for the convenience of the reader in Appendix.\n\n\\begin{lemma}\\label{lemma1} Let $T:M_k\\rightarrow M_k$ be a completely positive map. If $V\\in M_k$ is an orthogonal projection  such that $T(VM_kV)\\subset VM_kV$ then $T(VM_k+M_kV)\\subset VM_k+M_kV$.\n\\end{lemma}\n\n\\vspace{0.3 cm}\n\n\\begin{lemma}\\label{lemma2} Let $T:M_k\\rightarrow M_k$ be a positive map. Let $V_1,V\\in M_k$ be orthogonal projections such that $\\Im(V_1)\\subset\\Im(V)$ and $T(VM_kV)\\subset VM_kV$.\n\\begin{itemize}\n\\item[a)] If $T(V_1M_kV_1)\\subset V_1M_kV_1$ then $VT^*((V-V_1)M_k(V-V_1)))V\\subset (V-V_1)M_k(V-V_1)$.\n\\item[b)] $T:VM_kV\\rightarrow VM_kV$ is irreducible if and only if $VT^*(\\cdot)V:VM_kV\\rightarrow VM_kV$ is irreducible.\n\\item[c)] If $\\delta\\in P_k\\cap VM_kV$ is such that $\\Im(\\delta)=\\Im(V)$ and $T(\\delta)=\\lambda\\delta$, $\\lambda>0$, then $\\lambda$ is the spectral radius of $T|_{VM_kV}$.\n\\end{itemize}\n\\end{lemma}\n\n\\vspace{0.3 cm}\n\n\\begin{lemma}\\label{lemma3} Let $T:M_k\\rightarrow M_k$ be a completely positive map. Let us assume that $T(VM_kV)\\subset VM_kV$, where $V\\in M_k$ is an orthogonal projection. Let $\\lambda$ be the spectral radius of $T|_{VM_kV}$. Therefore,  $T|_{VM_kV}$ is irreducible if and only if the following conditions hold.\n\\begin{enumerate}\n\\item There  are $\\gamma,\\delta \\in VM_kV\\cap P_k$ such that $T(\\gamma)=\\lambda\\gamma$, $VT^*(\\delta)V=\\lambda\\delta$ and $\\Im(\\gamma)=\\Im(\\delta)=\\Im(V)$.\n\\item The geometric multiplicity of $\\lambda$ for $T|_{VM_kV}$ and $VT^*(\\cdot)V|_{VM_kV}$ is 1.\\\\\n\\end{enumerate}\n\\end{lemma}\n\nThe next proposition is the key that makes our algorithm to  work for PPT matrices. This proposition describes a remarkable property owned by these matrices. The author of \\cite{carielloIEEE} used this complete reducibility property to reduce the separability problem in Quantum Information Theory to the weakly irreducible PPT case. \\\\\n\n\\begin{proposition}\\label{propkey} $($\\textbf{The complete reducibility property of PPT states \\cite{carielloIEEE}}$)\\\\ $Let $B\\in M_k\\otimes M_k$ be  a PPT matrix. Let  $\\{W_1,\\ldots, W_s\\}\\subset  M_k$ be orthogonal projections such that\n\\begin{itemize}\n\\item[a)] $W_iW_j=0$, for $i\\neq j$,\n\\item[b)] $\\sum_{i=1}^sW_i=Id$,\n\\item[c)] $W_iM_kW_i$ is left invariant by $G_B((\\cdot)^t):M_k\\rightarrow M_k$ for every $i$.\n\\end{itemize}\nThen $B=\\sum_{i=1}^s(W_i^t\\otimes W_i)B(W_i^t\\otimes W_i)$.\n\\end{proposition}\n\\begin{proof}\nNote that $tr(B(W_j^t\\otimes W_m))=tr(G_B(W_j^t)W_m)=0$, $ j\\neq m$, by item $a)$ and $c)$.\\\\\n\nSince $B$ and $W_j^t\\otimes W_m$ are positive semidefinite Hermitian matrices then \\begin{equation}\\label{eq=0}\nB(W_j^t\\otimes W_m)=(W_j^t\\otimes W_m)B=0,\\ \\   j\\neq m.\\\\\n\\end{equation}\n\nThus, for $j\\neq m$,  $(B(W_j^t\\otimes W_m))^{t_2}=(Id\\otimes W_m^t)B^{t_2}(W_j^t\\otimes Id)=0$ \nand \\begin{center}\n$tr((Id\\otimes W_m^t)B^{t_2}(W_j^t\\otimes Id))=tr(B^{t_2}(W_j^t\\otimes W_m^{t}))=0$\n\\end{center}\n\nSince $B$ is PPT then $B^{t_2}$ is positive semidefinite. Thus, \\begin{equation}\\label{eq=1}\nB^{t_2}(W_j^t\\otimes W_m^t)=(W_j^t\\otimes W_m^t)B^{t_2}=0,\\ \\   j\\neq m.\\\\\n\\end{equation}\n\nNow, by item $b)$ and equation \\ref{eq=0}, we have $$B=\\sum_{i,l,j,m=1}^s(W_i^t\\otimes W_l)B(W_j^t\\otimes W_m)=\\sum_{i,j=1}^s(W_i^t\\otimes W_i)B(W_j^t\\otimes W_j).$$\n\nBy equation \\ref{eq=1}, $B^{t_2}=\\sum_{i,j=1}^s(W_i^t\\otimes W_j^t)B^{t_2}(W_j^t\\otimes W_i^t)=\\sum_{i=1}^s(W_i^t\\otimes W_i^t)B^{t_2}(W_i^t\\otimes W_i^t).$\\\\\n\nFinally, $B=\\sum_{i=1}^s(W_i^t\\otimes W_i)B(W_i^t\\otimes W_i).$\n\\end{proof}\n\\vspace{0.5 cm}\n\nThe next lemma shall be used  in our main theorem to prove the uniqueness of $W$ (item 3 of \\ref{theoremprincipal}). Its proof is another consequence of the complete reducibility property. \n\n\\vspace{0.5 cm}\n\n\\begin{lemma}\\label{lemmakey} Let $B\\in M_k\\otimes M_k$ be  a PPT matrix and $\\{W_1,\\ldots,W_s\\}\\subset M_k$  be orthogonal projections as in Proposition \\ref{propkey}. Suppose that $WM_kW$ is left invariant by $G_B((\\cdot)^t):M_k\\rightarrow M_k$ and $G_B((\\cdot)^t)|_{WM_kW}$ is irreducible. If there is $t\\leq s$ such that $G_B((\\cdot)^t)|_{W_iM_kW_i}$ is irreducible for every $1\\leq i\\leq t$ then either\n$$W=W_j\\ (\\text{for some } 1\\leq j\\leq t)\\ \\ \\ \\ or \\ \\ \\ \\ \\Im(W)\\subset \\Im(Id-W_1-\\ldots-W_t).$$\nThis result is also valid if $G_B((\\cdot)^t):M_k\\rightarrow M_k$ is replaced by its adjoint $F_B(\\cdot)^t:M_k\\rightarrow M_k$.\n\\end{lemma}\n\\begin{proof}\nBy Proposition \\ref{propkey}, $G_B(W^t)=\\sum_{i=1}^sW_iG_B(W_i^tW^tW_i^t)W_i.$ \nSo, $W_iG_B(W^t)=G_B(W^t)W_i$, $\\forall i$.\\\\\n\nSince $W_i$ and $G_B(W^t)$ commute, if  $W_iG_B(W^t)\\neq 0$, for some $i\\leq t$,   then \n\\begin{center}\n$\\Im(W_i)\\cap\\Im(G_B(W^t))\\neq\\{\\vec{0}\\}$.\n\\end{center}\n\n Let $\\widetilde{W}$ be the orthogonal projection onto $\\Im(W_i)\\cap\\Im(G_B(W^t))$. \n\n\nNext,  $\\widetilde{W}M_k\\widetilde{W}\\subset W_iM_kW_i\\cap WM_kW$ is left invariant by $G_B((\\cdot)^t)$ and \\begin{center}\n$G_B((\\cdot)^t)|_{W_iM_kW_i}$, $G_B((\\cdot)^t)|_{WM_kW}$ are irreducible.\n\\end{center}  \n\nTherefore,  $\\widetilde{W}= W_i= W$, for some $i\\leq t$.\n\nNow, if $W_iG_B(W^t)= 0$, for every $1\\leq i\\leq t$, then $\\Im(G_B(W^t))\\subset\\Im(Id-W_1-\\ldots-W_t)$. \n\nFinally, since  $G_B((\\cdot)^t)|_{WM_kW}$ is irreducible then $\\Im(W)=\\Im(G_B(W^t))$.\n\\end{proof}\n\n\\vspace{0.5 cm}\n\n\\begin{lemma}\\label{lemmapropertyowned} Let $T:M_k\\rightarrow M_k$ be a positive map and $Q\\in M_k$ an invertible matrix. Let us assume that for every orthogonal projection $V\\in M_k$ such that $T(VM_kV)\\subset VM_kV$, $T|_{VM_kV}$ is irreducible with spectral radius $\\lambda$, there is a unique orthogonal projection $W\\in M_k$ such that $T^*(WM_kW)\\subset WM_kW$, $T^*|_{WM_kW}$ is irreducible with spectral radius $\\lambda$, rank$(V)=$rank$(W)$ and $\\ker(W)\\cap\\Im(V)=\\{\\vec{0}\\}$. Then the same property is owned by $S:M_k\\rightarrow M_k$, $S(X)=QT(Q^{-1}X(Q^{-1})^*)Q^*$.\n\\end{lemma}\n\\begin{proof} Let $V'$ be an orthogonal projection such that $S(V'M_kV')\\subset V'M_kV'$, $S|_{V'M_kV'}$ is irreducible with spectral radius $\\lambda$. \n\nSince $S,T$ are similar maps then $T(VM_kV)\\subset VM_kV$, $T|_{VM_kV}$ is irreducible with spectral radius $\\lambda$, where $V$ is the orthogonal projection onto $\\Im(Q^{-1}V'(Q^{-1})^*)$.\\\\\n\nLet $W$ be the orthogonal projection described in the statement of this theorem and $W'$ the orthogonal projection  \nonto $\\Im((Q^{-1})^*WQ^{-1})$. Note that $S^*(X)=(Q^{-1})^*T^*(Q^*XQ)Q^{-1}$. \\\\\n\n Therefore,\n\n\\begin{itemize}\n\\item $S^*(W'M_kW')\\subset W'M_kW',$\n\\item $S^*|_{W'M_kW'}$ is irreducible with spectral radius $\\lambda$,\n\\item rank$(W')=$ rank$(W)=$ rank$(V)=$ rank$(V')$.\\\\\n\\end{itemize}\n\nFurthermore, since $\\ker(W)\\cap\\Im(V)=\\ker(Q^*W'Q)\\cap \\Im(Q^{-1}V'(Q^{-1})^*)=\\{\\vec{0}\\}$ then $$\\ker(W')\\cap \\Im(V')=\\{\\vec{0}\\}.$$\n\nNext, let $W''$ be another orthogonal projection satisfying the same properties of $W'$. \n\nSince $T^*(X)=Q^*S^*((Q^{-1})^*XQ^{-1})Q$ then the orthogonal projection onto $\\Im(Q^*W''Q)$ satisfies the same properties of $W$ (by the argument above). \\\\\n\nBy the uniqueness of $W$, we have $\\Im(W)=\\Im(Q^*W''Q)$. Hence, $$\\Im(W'')=\\Im((Q^{-1})^*W(Q^{-1}))=\\Im(W').$$\n\n So the uniqueness of $W'$ follows.\n\\end{proof}\n\n\\vspace{0.5 cm}\n\n\\begin{lemma}\\label{lemmaP} Let $B\\in M_k\\otimes M_k\\simeq M_{k^2}$ be a positive semidefinite Hermitian matrix. Suppose there is $v\\in\\Im(B)\\subset \\mathbb{C}^k\\otimes\\mathbb{C}^k$ with tensor rank $k$. There is an invertible matrix $P\\in M_k$ such that $\\Im(G_A(X^t))\\supset\\Im(X)$ for every $X\\in P_k$, where $A=(P\\otimes Id)B(P\\otimes Id)^*$.\n\\end{lemma}\n\\begin{proof}\nLet $P\\in M_k$ be an invertible matrix such that $(P\\otimes Id)v=u$, where $u=\\sum_{i=1}^k e_i\\otimes e_i$ and $\\{e_1,\\ldots, e_k\\}$ is the canonical basis of $\\mathbb{C}^k$. Thus, $u\\in\\Im(A)$, where $A=(P\\otimes Id)B(P\\otimes Id)^*$.\n\nThere is $\\epsilon>0$ such that $A-\\epsilon (uu^t)$ is a positive semidefinite Hermitian matrix. Thus, $G_{A-\\epsilon uu^t}(X^t)$ is a positive map.\nNote that\\begin{center}\n $G_A(X^t)=G_{A-\\epsilon uu^t}(X^t)+\\epsilon\\ G_{uu^t}(X^t)$ and $G_{uu^t}(X^t)=X$.\n\\end{center} \n\nFinally, $\\Im(G_A(X^t))\\supset\\Im(X)$ for every $X\\in P_k$.\n\\end{proof}\n\n\\section{Main Results}\n\nIn this section we present our main result (Theorem \\ref{theoremprincipal}) and one problem derived from this main result (Problem \\ref{problem}). This problem can be solved as a unconstrained quadratic minimization problem (Lemma \\ref{solutionW}, Remark \\ref{remarkquadratic}).\n\n\\begin{theorem}\\label{theoremprincipal} Let $A\\in M_k\\otimes M_k$ be  a PPT matrix and $T:M_k\\rightarrow M_k$ be the completely positive map $G_A((\\cdot)^t):M_k\\rightarrow M_k$. Suppose that $\\Im(T(X))\\supset \\Im(X)$ for every $X\\in P_k$. The following statements are equivalent:\n\\begin{enumerate}\n\\item $T:M_k\\rightarrow M_k$ is equivalent to a doubly stochastic map.\\\\\n\n\\item There are orthogonal projections $\\{V_1,\\ldots,V_s\\}\\subset M_k$ such that  \\ \\  $\\mathbb{C}^k=\\bigoplus_{i=1}^s\\Im(V_i)$,\\\\ $T(V_iM_kV_i)\\subset V_iM_kV_i$, $T|_{V_iM_kV_i}$ is irreducible for every $i$.\\\\\n\n\\item For every orthogonal projection $V\\in M_k$ such that $T(VM_kV)\\subset VM_kV$, $T|_{VM_kV}$ is irreducible with spectral radius $\\lambda$, there is a unique orthogonal projection $W\\in M_k$ such that $T^*(WM_kW)\\subset WM_kW$, $T^*|_{WM_kW}$ is irreducible with spectral radius $\\lambda$, $rank(W)= rank(V)$ and\n $\\ker(W)\\cap\\Im(V)=\\{\\vec{0}\\}$.\n\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n$(1\\Leftrightarrow 2)\\ $The existence of the orthogonal projections described in item 2 and the condition $\\Im(T(X))\\supset \\Im(X)$ for every $X\\in P_k$ imply that $T:M_k\\rightarrow M_k$ has a positive achievable capacity. Then it is equivalent to a doubly stochastic map (See \\cite{gurvits2004}). Now, if a positive map $T:M_k\\rightarrow M_k$ is equivalent to a doubly stochastic map and $\\Im(T(X))\\supset \\Im(X)$ for every $X\\in P_k$ then we can easily find orthogonal projections as described in  item 2. So the first two conditions are equivalent. Check also \\cite[Theorem 3.4]{cariellosink} for a different approach  based on Sinkhorn and Knopp original proof.\\\\\\\\\n$(2\\Rightarrow 3)\\ $\nThere is an invertible matrix  $P\\in M_k$ such that $\\Im(PV_iP^*)\\perp\\Im(PV_jP^*)\\ (i\\neq j),$ since $\\mathbb{C}^k=\\bigoplus_{i=1}^s\\Im(V_i)$.\n\nLet $W_i$ be the orthogonal projection onto $\\Im(PV_iP^*)$. Hence, $\\sum_{i=1}^sW_i=Id$, $W_iW_j=0$ $(i\\neq j)$.\\\\\n\nDefine $B=((P^{-1})^t\\otimes P)A((P^{-1})^t\\otimes P)^*$. Note that $B$ is PPT and $$G_B(X^t)=P(T(P^{-1}X(P^{-1})^*))P^*\\text{ and } (F_B(X))^t=(G_B(X^t))^*=(P^{-1})^*T^*(P^*XP)P^{-1}$$ \n\nThus, the hypotheses of Lemma \\ref{lemmakey} hold for this $B$. Moreover, it follows from the hypothesis of item 2 that\n\\begin{center}\n$G_B((\\cdot)^t)|_{W_iM_kW_i}$ is irreducible, $1\\leq i\\leq s$, and $\\Im(G_B(X^t))\\supset\\Im(X)$ for every $X\\in P_k$.\n\n\\end{center}\n\nNext, if $T(VM_kV)\\subset VM_kV$ and $T|_{VM_kV}$ is irreducible then $G_B((\\widetilde{V}M_k\\widetilde{V})^t)\\subset \\widetilde{V}M_k\\widetilde{V}$ and $G_B((\\cdot)^t)|_{\\widetilde{V}M_k\\widetilde{V}}$ is irreducible, where $\\widetilde{V}$ is the orthogonal projection onto $\\Im(PVP^*)$. \n\nBy Lemma \\ref{lemmakey}, $\\widetilde{V}=W_i$ for some $i$. Hence, \\begin{center}\n$\\Im(PVP^*)=\\Im(PV_iP^*)$ and $V=V_i$.\n\\end{center}\n\nAnalogously, if $T^*(WM_kW)\\subset WM_kW$ and $T^*|_{WM_kW}$ is irreducible then, by Lemma \\ref{lemmakey}, \\begin{center}\n$\\Im((P^{-1})^*WP^{-1})=\\Im(W_j)$ for some $j$.\n\\end{center} \n\nNote that $\\ker(W)\\cap\\Im(V)=\\{\\vec{0}\\}$ if and only if $\\ker(W_j)\\cap\\Im(PV_iP^*)=\\{\\vec{0}\\}$. \\\\\n\nSince $\\Im(PV_iP^*)=\\Im(W_i)$ and $W_lW_m=0$ $(l\\neq m)$ then $\\ker(W)\\cap\\Im(V_i)=\\{\\vec{0}\\}$ if and only if  $W$ is the orthogonal projection onto $\\Im(P^*W_iP)$ $($i.e. $j=i)$. Thus, there is a unique $W$ such that \\begin{center}\n$T^*(WM_kW)\\subset WM_kW$, $T^*|_{WM_kW}$ is irreducible and $\\ker(W)\\cap\\Im(V)=\\{\\vec{0}\\}$.\n\\end{center}\n\nRecall that, $\\Im(W)=\\Im(P^*W_iP)=\\Im(P^*PVP^*P)$. So rank$(W)=$ rank$(V)$.\\\\\n\nNext, $T|_{VM_kV}$ and $G_B((\\cdot)^t)|_{W_iM_kW_i}$ have the same spectral radius, since they are similar and \nthe same is valid for $T^*|_{WM_kW}$ and $(F_B(\\cdot))^t|_{W_iM_kW_i}$. \\\\\n\nSince  $W_lW_m=0$ for every $l\\neq m$ then \\begin{center}\n$(G_B((\\cdot)^t)|_{W_iM_kW_i})^*=(F_B(\\cdot))^t|_{W_iM_kW_i}$.\n\\end{center} \n\nThus, $T|_{VM_kV}$ and $T^*|_{WM_kW}$ have also the same spectral radius.\\\\\\\\\n$(3\\Rightarrow 2)\\ $ Let $V_1\\in M_k$ be an orthogonal projection such that\\begin{center}\n $T(V_1M_kV_1)\\subset V_1M_kV_1$, $T|_{V_1M_kV_1}$ is irreducible. \n\\end{center}\n\nIf $V_1=Id$ then the proof is complete. \n\nIf $V_1\\neq Id$ then, by hypothesis of item 3, there is an orthogonal projection $W_1$ such that \n\n\\begin{itemize}\n\\item $T^*(W_1M_kW_1)\\subset W_1M_kW_1$,\n\\item $T^*|_{W_1M_kW_1}$ is irreducible,\n\\item rank$(W_1)=\\ $rank$(V_1)$ and $\\ker(W_1)\\cap\\Im(V_1)=\\{\\vec{0}\\}$.\n\\end{itemize}\n\n \nThus, $\\Im(V_1)\\oplus\\Im(Id-W_1)=\\mathbb{C}^k$ and  \\begin{center}\n$T((Id-W_1)M_k(Id-W_1))\\subset (Id-W_1)M_k(Id-W_1)$,\n\\end{center} by item $a)$  of Lemma \\ref{lemma2}.\n\nLet $Q_1$ be an invertible matrix such that \\begin{center}\n$Q_1V_1=V_1$ and $\\Im(Q_1(Id-W_1))=\\Im(Id-V_1)$.\n\\end{center}\n\nDefine \\begin{center}\n$A_1=((Q_1^{-1})^t\\otimes Q_1)A((Q_1^{-1})^t\\otimes Q_1)^*$ and $T_1(X)=G_{A_1}(X^t)=Q_1T(Q_1^{-1}X(Q_1^{-1})^*)Q_1^*$.\n\\end{center}\n\nNote that \n$T_1(V_1M_kV_1)\\subset V_1M_kV_1$, $T_1|_{V_1M_kV_1}$ is irreducible, \\begin{center}\n$T_1((Id-V_1)M_k(Id-V_1))\\subset (Id-V_1)M_k(Id-V_1)$\n\\end{center}\n and the same occurs if we replace $T_1$ by $T_1^*$.\n \n Now, let $V_2$ be an orthogonal projection such that \\begin{center}\n$T_1(V_2M_kV_2)\\subset V_2M_kV_2 \\subset  (Id-V_1)M_k(Id-V_1)$ and $T_1|_{V_2M_kV_2}$ is irreducible.\n\\end{center}\n\nIf $V_2=Id-V_1$ then $T_1$ satisfies the conditions of item 2. Hence, the same conditions hold for $T$ and the proof is complete. \n\nNext, assume that $V_2\\neq Id-V_1$. By Lemma \\ref{lemmapropertyowned}, the property owned by $T$ described in item 3 of the statement of this theorem is also owned by $T_1$. Hence, there is an orthogonal projection $W_2$ such that \\begin{itemize}\n\\item $T_1^*(W_2M_kW_2)\\subset W_2M_kW_2$, \\item $T_1^*|_{W_2M_kW_2}$ is irreducible, \\item rank$(W_2)=$ rank$(V_2)$ and $\\ker(W_2)\\cap\\Im(V_2)=\\{\\vec{0}\\}$.\n\\end{itemize}\n\nMoreover, since $A_1$ satisfies the hypotheses of Lemma \\ref{lemmakey}  with $t=1$ and $s=2$ then \\begin{center}\n$W_2=V_1$ or $\\Im(W_2)\\subset \\Im(Id-V_1)$. \n\\end{center}\n\nHowever,  $W_2=V_1$ is not possible, since $\\Im(V_2)=\\ker(V_1)\\cap \\Im(V_2)$ and $\\ker(W_2)\\cap\\Im(V_2)=\\{\\vec{0}\\}$. \n\nTherefore, $\\Im(W_2)\\subset \\Im(Id-V_1)$. \n\nLet $Q_2$ be an invertible matrix such that \\begin{center}\n$Q_2(V_1+V_2)=V_1+V_2$ and $\\Im(Q_2(Id-V_1-W_2))=\\Im(Id-V_1-V_2)$.\n\\end{center}\n\nDefine \\begin{center}\n$A_2=((Q_2^{-1})^t\\otimes Q_2)A_1((Q_2^{-1})^t\\otimes Q_2)^*$ and $T_2(X)=G_{A_2}(X^t)=Q_2T_1(Q_2^{-1}X(Q_2^{-1})^*)Q_2^*$.\n\\end{center}\n\nNote that $T_2(V_iM_kV_i)\\subset V_iM_kV_i$, $T_2|_{V_iM_kV_i}$ is irreducible for $1\\leq i\\leq 2$\n and  \n \n \\begin{center}\n $T_2((Id-V_1-V_2)M_k(Id-V_1-V_2))\\subset (Id-V_1-V_2)M_k(Id-V_1-V_2)$.\n\\end{center}\n\nRepeating this argument $s$ times $(s\\leq k)$, we obtain \n $T_s(V_iM_kV_i)\\subset V_iM_kV_i$, \n $T_s|_{V_iM_kV_i}$ is irreducible\n for every $1\\leq i\\leq s$, $V_iV_j=0$ for $i\\neq j$ and $\\sum_{i=1}^sV_i=Id$, where $T_s(X)=RT(R^{-1}X(R^{-1})^*)R^*$ for some invertible $R\\in M_k$. \n\\end{proof}\n\n\\vspace{0.5 cm}\n\n\nThis last theorem gives rise to the following problem. We present a solution for this problem in the next two lemmas (\\ref{lemmaQ}, \\ref{solutionW}). \n\n\\vspace{0.5 cm}\n\n\n\\begin{problem}\\label{problem} Given a completely positive map $T:M_k\\rightarrow M_k$ and an orthogonal projection $V\\in M_k$ such that \n$T(VM_kV)\\subset VM_kV$  and $T|_{VM_kV}$ is irreducible,\nfind\n an orthogonal projection $W$ such that \n\\begin{itemize}\n\\item $T^*(WM_kW)\\subset WM_kW$,\n\\item $T^*|_{WM_kW}$ is irreducible,\n\\item $rank(W)= rank(V)$ and \n\\item $\\ker(W)\\cap\\Im(V)=\\{\\vec{0}\\}$.\n\\end{itemize}\n\nWe shall call such $W$ a solution to the Problem \\ref{problem} subjected to $T$ and $V$.\n\\end{problem}\n\n\\vspace{0.5 cm}\n\nThe next lemma simplifies our search for $W$ of problem \\ref{problem} and relates  it  to a solution of a  unconstrained quadratic minimization problem (lemma \\ref{solutionW}). Recall that item 3 of  theorem \\ref{theoremprincipal} requires the uniqueness of  this solution. Thus, we shall solve this minimization problem only when the uniqueness of the solution is granted (see remark \\ref{remarkquadratic}). \n\\vspace{0.5 cm}\n\n\n\\begin{lemma}\\label{lemmaQ} Let $T:M_k\\rightarrow M_k$ and $V\\in M_k$ be as in Problem \\ref{problem}.\nLet $\\lambda$ be the spectral radius  of $T|_{VM_kV}$. There is an invertible matrix $Q\\in M_k$ such that a solution to the Problem \\ref{problem} subjected to $T$ and $V$ is the orthogonal projection onto $\\Im(Q^*W_1Q)$, where $W_1$ is a solution to the Problem \\ref{problem} subjected to\\\\\n\\begin{enumerate}\n\\item $T_1:M_k\\rightarrow M_k,\\ T_1(x)=\\frac{1}{\\lambda}QT(Q^{-1}X(Q^{-1})^*)Q^*$,\n\\item $V_1=\\begin{pmatrix}Id_{s\\times s} & 0_{s\\times k-s} \\\\ \n0_{k-s\\times s} & 0_{k-s\\times k-s}\\end{pmatrix}$, where $s= rank(V)$, \\\\\n\\item $V_1T_1^*(V_1)V_1=V_1$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} By Lemma \\ref{lemma2}, item b), $VT^*(\\cdot)V:VM_kV\\rightarrow VM_kV$ is irreducible. \\\\\n\nBy Lemma \\ref{lemma3}, there is \n$\\delta_1\\in P_k\\cap VM_kV$ such that $VT^*(\\delta_1)V=\\lambda\\delta_1$ and $\\Im(\\delta_1)=\\Im(V)$.\\\\\n\n\nDefine $R=\\delta_1^{\\frac{1}{2}}+V^{\\perp}$, where $V^{\\perp}=Id-V$. Note that \\begin{center}\n$R^*=R$, $RVR=\\delta_1$ and $VR^{-1}=R^{-1}V$.\n\\end{center}\n\nNow, consider $S:M_k\\rightarrow M_k$ defined by $S(X)=\\frac{1}{\\lambda}RT(R^{-1}X(R^{-1})^*)R^*.$\\\\\n\nSince $S^*(X)=\\frac{1}{\\lambda}(R^{-1})^*T^*(R^*XR)R^{-1}$ then $VS^*(V)V=$ \n\n$$=\\frac{1}{\\lambda}V(R^{-1})T^*(RVR)R^{-1}V=\\frac{1}{\\lambda}R^{-1}VT^*(\\delta_1)VR^{-1}=\\frac{\\lambda}{\\lambda}R^{-1}\\delta_1R^{-1}=V.$$\n\nNote that  $S(VM_kV)\\subset VM_kV$ and $S|_{VM_kV}$ is irreducible, since $R^{-1}V=VR^{-1}$ and $T|_{VM_kV}$ is irreducible. \nTherefore, by Lemma \\ref{lemma2},\n\n\\begin{center}\n$S^*(V^{\\perp}M_kV^{\\perp})\\subset V^{\\perp}M_kV^{\\perp}$ and $VS^*(\\cdot)V|_{VM_kV}$ is irreducible.\n\\end{center}\n\nNext, let $U$ be a unitary matrix such that $U^*VU=\\begin{pmatrix}Id_{s\\times s}& 0 \\\\ \n0& 0\\end{pmatrix}$. \n\nLet $V_1=U^*VU$ and define $T_1:M_k\\rightarrow M_k$ as $T_1(X)=U^*S(UXU^*)U$. \\\\\n\nSince $T_1^*(X)=U^*S^*(UXU^*)U$ and $UV_1=VU$ then $V_1T^*_1(V_1)V_1=$\n\n\\begin{center}\n$=V_1U^*S^*(UV_1U^*)UV_1=U^*VS^*(UV_1U^*)VU=U^*VS^*(V)VU=U^*VU=V_1.$\n\\end{center}\n\nThus, $V_1T_1^*(V_1M_kV_1)V_1\\subset V_1M_kV_1$. Moreover,   $V_1T_1^*(\\cdot)V_1:V_1M_kV_1\\rightarrow V_1M_kV_1$ is irreducible, since $VS^*(\\cdot)V|_{VM_kV}$ is irreducible and $V_1U^*=U^*V$.  \\\\\n\nNext,\n $T_1^*(V_1^{\\perp}M_kV_1^{\\perp})\\subset U^*S^*(UV_1^{\\perp}M_kV_1^{\\perp}U^*)U=$\\begin{center}\n$U^*S^*(V^{\\perp}M_kV^{\\perp})U\\subset U^*V^{\\perp}M_kV^{\\perp}U= V_1^{\\perp}M_kV_1^{\\perp}$.\n\\end{center}\n\nHence, $T_1(V_1M_kV_1)\\subset V_1M_kV_1$, by Lemma \\ref{lemma2}. Since  $V_1T_1^*(\\cdot)V_1|_{V_1M_kV_1}$ is irreducible then $T_1|_{V_1M_kV_1}$ is irreducible, by Lemma \\ref{lemma2}.\\\\\n\nRecall that $V_1T_1^*(V_1)V_1=V_1$ then, by Lemma \\ref{lemma2},  the spectral radius of $V_1T_1^*(\\cdot)V_1|_{V_1M_kV_1}$ is $1$. Thus, the spectral radius of its adjoint $T_1|_{V_1M_kV_1}$ is also $1$.\\\\\n\nNow, let $W_1$ be a solution to the Problem \\ref{problem} subjected to $T_1:M_k\\rightarrow M_k$ and $V_1$. Thus,\n\\begin{itemize}\n\\item  rank$(W_1)=$ rank$(V_1)$,\n\\item  $\\Im(V_1)\\cap\\ker(W_1)=\\{\\vec{0}\\}$\n\\item $T^*_1(W_1M_kW_1)\\subset W_1M_kW_1$\n\\item $T_1^*|_{W_1M_kW_1}$ is irreducible.\n\\end{itemize}\n\n\n\nNext, define $Q=U^*R$. Note that $T_1(X)=\\frac{1}{\\lambda}QT(Q^{-1}X(Q^{-1})^*)Q^*$, \\begin{center}\n$T(X)=\\lambda Q^{-1}T_1^*(QXQ^*)(Q^{-1})^*$ and $T^*(X)=\\lambda Q^*T_1^*((Q^{-1})^*XQ^{-1})Q$.\n\\end{center}\n\n\nLet $W$ be the orthogonal projection onto $\\Im(Q^*W_1Q)$. Note that $T^*(WM_kW)\\subset WM_kW$.\\\\\n\nNow, since \n$WM_kW=Q^*W_1M_kW_1Q$ and $T_1^*|_{W_1M_kW_1}$ is irreducible then \\begin{center}\n$T^*(WM_kW)\\subset WM_kW$, $T^*|_{WM_kW}$ is irreducible.\n\\end{center}\n\nSince $\\Im(V_1)=\\Im(U^*\\delta_1U)=\\Im(U^*RVR^*U)=Im(QVQ^*)$ then $\\Im(V)=\\Im(Q^{-1}V_1(Q^{-1})^*)$.\\\\\n\nMoreover, $\\ker(W)=\\ker(Q^*W_1Q)$.\nThus, $\\ker(W)\\cap \\Im(V)=\\{\\vec{0}\\}$, since $\\ker(W_1)\\cap \\Im(V_1)=\\{\\vec{0}\\}$.\\\\\n\nFinally, rank$(W)=$ rank$(W_1)=$ rank$(V_1)=$ rank$(V).$\n\\end{proof}\n\n\n\\vspace{0.5 cm}\n\n\n\\begin{lemma}\\label{solutionW}  Let $T_1:M_k\\rightarrow M_k$ be a completely positive map such that $T_1(V_1M_kV_1)\\subset V_1M_kV_1$, $T_1|_{V_1M_kV_1}$ is irreducible and $V_1T_1^*(V_1)V_1=V_1$, where $V_1=\\begin{pmatrix}Id_{s\\times s} & 0_{s\\times k-s} \\\\ \n0_{k-s\\times s} & 0_{k-s\\times k-s}\\end{pmatrix}$. The following statements are equivalent:\n\\begin{itemize}\n\\item[a)] $W$ is a solution to the Problem \\ref{problem} subjected to these $T_1$ and $V_1$.\n\\item[b)] $W$ is an orthogonal projection onto $\\Im\\begin{pmatrix}Id_{s\\times s}\\\\ \nS_{k-s\\times s} \\end{pmatrix}$, where $S\\in M_{k-s\\times s}(\\mathbb{C})$\\\\ is a zero of\n$f:M_{k-s\\times s}(\\mathbb{C})\\rightarrow \\mathbb{R}^+\\cup\\{0\\}$, defined by \\\\\n\n$f(X)=tr\\left(T_1^*\\left(\\begin{pmatrix}Id& 0 \\\\ \n0& 0\\end{pmatrix}\\right)\\begin{pmatrix}0 & 0 \\\\ \n0 & Id\\end{pmatrix}\\right)+tr\\left(T_1^*\\left(\\begin{pmatrix}Id& 0 \\\\ \n0& 0\\end{pmatrix}\\right)\\begin{pmatrix}X^*X & -X^* \\\\ \n-X & 0\\end{pmatrix}\\right)\\\\ $\n\n$ +tr\\left(T_1^*\\left(\\begin{pmatrix}0& X^* \\\\\nX& XX^*\\end{pmatrix}\\right)\\begin{pmatrix}0 & 0 \\\\ \n0 & Id\\end{pmatrix}\\right)+tr\\left(T_1^*\\left(\\begin{pmatrix}0& X^* \\\\\nX& 0\\end{pmatrix}\\right)\\begin{pmatrix}0 & -X^* \\\\ \n-X & 0\\end{pmatrix}\\right).\\\\\n$\n\n\n\n\n\\end{itemize}\n\\end{lemma}\n\n\n\n\\begin{proof} First, define $f:M_{k-s\\times s}(\\mathbb{C})\\rightarrow \\mathbb{R}^+\\cup\\{0\\}$ as $$f(X)=tr\\left(T_1^*\\left(\\begin{pmatrix}Id& X^* \\\\ \nX& XX^*\\end{pmatrix}\\right)\\begin{pmatrix}X^*X & -X^* \\\\ \n-X & Id\\end{pmatrix}\\right).$$\n\nNow, by Lemmas \\ref{lemma1} and \\ref{lemma2},  since $T_1(V_1M_kV_1)\\subset V_1M_kV_1$\nthen \n\\begin{center}\n$T_1^*(V_1^{\\perp}M_kV_1^{\\perp})\\subset V_1^{\\perp}M_kV_1^{\\perp}$ and $T_1^*(V_1^{\\perp}M_k+M_kV_1^{\\perp})\\subset V_1^{\\perp}M_k+M_kV_1^{\\perp}.$\n\n\\end{center}\n\n\nTherefore,\n\\begin{center}\n$T_1^*\\left(\\begin{pmatrix}0 & 0 \\\\ \n0& XX^*\\end{pmatrix}\\right)=\\begin{pmatrix}0 & 0 \\\\ \n0& B\\end{pmatrix}$ and $T_1^*\\left(\\begin{pmatrix}0 & X^* \\\\ \nX& 0\\end{pmatrix}\\right)=\\begin{pmatrix}0 & Y^* \\\\ \nY& D\\end{pmatrix}$.\n\n\\end{center}\n\nHence,\n $$tr\\left(T_1^*\\left(\\begin{pmatrix}0& X^* \\\\\nX& XX^*\\end{pmatrix}\\right)\\begin{pmatrix}X^*X & -X^* \\\\ \n-X & 0\\end{pmatrix}\\right)=tr\\left(\\begin{pmatrix}0& Y^* \\\\\nY& D+B\\end{pmatrix}\\begin{pmatrix}X^*X & -X^* \\\\ \n-X & 0\\end{pmatrix}\\right)=$$\n\n$$=tr\\left(\\begin{pmatrix}0& Y^* \\\\\nY& D\\end{pmatrix}\\begin{pmatrix}0 & -X^* \\\\ \n-X & 0\\end{pmatrix}\\right)=tr\\left(T_1^*\\left(\\begin{pmatrix}0& X^* \\\\\nX& 0\\end{pmatrix}\\right)\\begin{pmatrix}0 & -X \\\\ \n-X^* & 0\\end{pmatrix}\\right).$$\n\nThis  equality yields the formula for $f(X)$ in the statement of this lemma. \n \n Next, let us show the equivalence between $a)$ and $b)$.\\\\\n \n\\textit{Proof of }$a)\\Rightarrow b): $  Let $W$ be a solution of Problem \\ref{problem} subjected to  $T_1$ and $V_1$.\n\nSince $T_{1}^*|_{WM_kW}$ is irreducible, there is $\\delta\\in P_k$ such that \\begin{center}\n$\\Im(\\delta)=\\Im(W)$ and \n$T_1^*(\\delta)=\\lambda\\delta$, \n\\end{center} \n\nwhere $\\lambda$ is the spectral radius of $T_{1}^*|_{WM_kW}$ by Perron-Frobenius theory \\cite[Theorem 2.3]{evans}.\n\nNow, $T_1^*(V_1^{\\perp}M_k+M_kV_1^{\\perp})\\subset V_1^{\\perp}M_k+M_kV_1^{\\perp}$ implies that\n\\begin{center}\n$V_1T_1^*(V_1^{\\perp}\\delta +V_1\\delta V_1^{\\perp})V_1=0$. \n\\end{center}\n\nThus, \n$\\lambda V_1\\delta V_1=V_1T_1^*(\\delta)V_1=V_1T_1^*(V_1^{\\perp}\\delta +V_1\\delta V_1^{\\perp})V_1+V_1T_1^*(V_1\\delta V_1)V_1=V_1T_1^*(V_1\\delta V_1)V_1$.\\\\\n\n\nSince $\\ker(\\delta)\\cap\\Im(V_1)=\\ker(W)\\cap\\Im(V_1)=\\{\\vec{0}\\}$ then  \n$V_1\\delta V_1\\in (P_k\\cap  V_1M_kV_1)\\setminus \\{0\\}$.\n\nBy Lemma \\ref{lemma2}, $V_1T_1^*(\\cdot)V_1|_{V_1M_kV_1}$ is irreducible. Thus,\n$\\Im(V_1\\delta V_1)=\\Im(V_1)$. \\\\\n\nNext, since\n$V_1T_1^*(V_1\\delta V_1)V_1=\\lambda V_1\\delta V_1$, $\\Im(V_1\\delta V_1)=\\Im(V_1)$ and  \n$V_1T_1^*(V_1)V_1= V_1$\nthen $\\lambda=1$,\n\n by item $c)$ of Lemma \\ref{lemma2}.\nMoreover, the geometric multiplicity of $\\lambda$   is $1$ (Lemma \\ref{lemma3}). \\\\\n\nTherefore,\n$V_1\\delta V_1=\\mu V_1$, for some $\\mu>0$.\nThus, \n$\\frac{\\delta}{\\mu}=\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}$, for some $S\\in M_{k-s\\times s}(\\mathbb{C})$.\n\n\nSo $T_1^*\\left(\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}\\right)=\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}$ and $W$ is the orthogonal projection onto $\\Im\\left(\\begin{pmatrix}Id \\\\\nS \\end{pmatrix}\\right)$. \\\\\n\nFinally, note that $S$ is a zero of $f(X)$. The proof that $a)\\Rightarrow b)$ is complete.\\\\\n\n\\textit{Proof of }$b)\\Rightarrow a):$ Let $S\\in M_{k-s\\times s}(\\mathbb{C})$ be a zero of $f(X)$.\\\\\n\nSince $V_1T_1^*(V_1)V_1=V_1$ and $T_1^*(V_1^{\\perp}M_k+M_kV_1^{\\perp})\\subset V_1^{\\perp}M_k+M_kV_1^{\\perp}$ then \n\n\\begin{center}\n$T_1^*\\left(\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}\\right)=\\begin{pmatrix}Id& Z^* \\\\\nZ& R\\end{pmatrix}\\in P_k.$\n\n\\end{center}\n\nSince $f(S)=0$ then  $$\\Im\\left(\\begin{pmatrix}S^*S & -S^* \\\\ \n-S & Id\\end{pmatrix}\\right)=\\Im\\left(\\begin{pmatrix}S^* \\\\ \n-Id\\end{pmatrix}\\right)\\subset\\ker\\left(\\begin{pmatrix}Id& Z^* \\\\\nZ& R\\end{pmatrix}\\right),$$ \n\nwhich is equivalent to $S^*=Z^*$ and $ZS^*=R$.\nThus, \\begin{center}\n$T_1^*\\left(\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}\\right)=\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}.$\n\n\\end{center}\n\n\n\nLet $W_1$ be the orthogonal projection onto $\\Im\\left(\\begin{pmatrix}Id& S^* \\\\\nS& SS^*\\end{pmatrix}\\right)$.  \n\nNote that\n\\begin{itemize}\n\\item $T_1^*(W_1M_kW_1)\\subset W_1M_kW_1$,\n\\item $T_1^*|_{W_1M_kW_1}$ has spectral radius equals to 1 (by item $c)$ of Lemma \\ref{lemma2}),\n\\item $\\Im(W_1)\\cap\\ker(V_1)=\\{\\vec{0}\\}$,\n\\item $\\ker(W_1)\\cap\\Im(V_1)=\\{\\vec{0}\\}$.\n\\end{itemize}\n\nIn order to complete this proof, we must show that $T_1^*|_{W_1M_kW_1}$ is irreducible. \nIf this is not the case, then \nthere is $\\delta_1\\in P_k\\cap  W_1M_kW_1$ such that $T_1^*(\\delta_1)=\\alpha\\delta_1$, $\\alpha>0$ and $0<$ rank$(\\delta_1)<$ rank$(W_1)$.\\\\\n\nNext, if $V_1\\delta_1V_1=0$ then $\\Im(\\delta_1)\\subset\\ker(V_1)$. Since $\\Im(\\delta_1)\\subset \\Im(W_1)$ then $\\Im(W_1)\\cap\\ker(V_1)\\neq\\{\\vec{0}\\}$, which is a contradiction.\nSo $V_1\\delta_1V_1\\neq 0$.\\\\\n\nRepeating the same argument used previously in $[a)\\Rightarrow b)]$, we have \n$V_1T_1^*(V_1\\delta_1 V_1)V_1=\\alpha V_1\\delta_1 V_1.$\\\\\n\n \nSince rank$(V_1\\delta_1V_1)\\leq$ rank$(\\delta_1)<$ rank$(W_1)=$ rank$(V_1)$ then $V_1T_1^*(\\cdot)V_1|_{V_1M_kV_1}$ is not irreducible, which is a contradiction with Lemma \\ref{lemma2}. \\\\\n\nFinally, $T_1^*|_{W_1M_kW_1}$ is irreducible, $\\ker(W_1)\\cap\\Im(V_1)=\\{\\vec{0}\\}$ and rank$(W_1)=$ rank$(V_1)$.\n\\end{proof}\n\n\n\\vspace{0.5 cm}\n\n\\begin{remark} \\label{remarkquadratic}\n\n\nFinding a zero for $f(X)$ is an unconstrained quadratic minimization problem, if $M_{k-s\\times s}(\\mathbb{C})$ is regarded as a real vector space. We are only interested in this zero when its uniqueness is granted. The uniqueness occurs only when the real symmetric matrix associated to the bilinear form $g: M_{k-s\\times s}(\\mathbb{C})\\times M_{k-s\\times s}(\\mathbb{C})\\rightarrow\\mathbb{R}$  is positive definite, where  $g(X,Y)\\text{ is the real part of }$ \\\\\n$$tr\\left(T_1^*\\left(\\begin{pmatrix}Id& 0 \\\\ \n0& 0\\end{pmatrix}\\right)\\begin{pmatrix}X^*Y & 0 \\\\ \n0 & 0\\end{pmatrix}+T_1^*\\left(\\begin{pmatrix}0& 0 \\\\\n0 & XY^*\\end{pmatrix}\\right)\\begin{pmatrix}0 & 0 \\\\ \n0 & Id\\end{pmatrix}+T_1^*\\left(\\begin{pmatrix}0& X^* \\\\\nX& 0\\end{pmatrix}\\right)\\begin{pmatrix}0 & -Y^* \\\\ \n-Y & 0\\end{pmatrix}\\right).\\\\\n$$\n\\end{remark} \n\n\n\\section{Algorithms}\nIn this section, we bring all the results together in our algorithms. Our main algorithm  (Algorithm 3) checks whether a PPT state $B\\in M_k\\otimes M_k$ with a vector  $v\\in\\Im(B)$ with tensor rank $k$ can be put in the filter normal form or not. It searches for all pairs of orthogonal projections $(V,W)$ as described in problem \\ref{problem} for a positive map $T:M_k\\rightarrow M_k$ equivalent to $G_B((\\cdot)^t):M_k\\rightarrow M_k$ . This  procedure reproduces the proof of the theorem \\ref{theoremprincipal}, particularly the part $(3\\Rightarrow 2)$ .\n\n\\vspace{0.5 cm}\n\\noindent\\textbf{Algorithm 1:} Given a completely positive map $T:VM_kV\\rightarrow VM_kV$, this algorithm finds $V_1M_kV_1\\subset VM_kV$ left invariant by $T:VM_kV\\rightarrow VM_kV$ such that $T|_{V_1M_kV_1}$ is irreducible. Note that every time $V$ is redefined its rank decreases. So the process shall stop.  \\\\\n\n\\begin{itemize}\n\\item[Step 1:] Compute rank$(V)$.\\\\\\\\\n$\\bullet$ If rank$(V)=1$ then define $V_1=V$.\\\\\n$\\bullet$ If rank$(V)\\neq 1$ then do Step 2.\\\\\n\n\\item[Step 2:] Find the spectral radius $\\lambda$ of $T:VM_kV\\rightarrow VM_kV$, compute $\\dim(\\ker(T-\\lambda Id|_{VM_kV})))$ and find a Perron eigenvector $\\gamma\\in VM_kV$ associated to $\\lambda$.\\\\\\\\\n$\\bullet$ If $\\dim(\\ker(T-\\lambda Id|_{VM_kV})))=1$ and $\\Im(\\gamma)=\\Im(V)$ then do Step 3.\\\\\n$\\bullet$ If $\\dim(\\ker(T-\\lambda Id|_{VM_kV})))=1$  and $\\Im(\\gamma)\\neq\\Im(V)$ then redefine V as the orthogonal projection onto $\\Im(\\gamma)$ and do Step 3.\\\\\n$\\bullet$ If $\\dim(\\ker(T-\\lambda Id|_{VM_kV})))\\neq 1$ then find an Hermitian matrix $\\gamma'\\in VM_kV$  and $\\epsilon>0$ such that $T(\\gamma')=\\lambda\\gamma'$, $\\gamma-\\epsilon\\gamma'\\in P_k\\setminus\\{0\\}$ and rank$(\\gamma-\\epsilon\\gamma')<$rank$(\\gamma)$. Redefine $V$ as the orthogonal projection onto $\\Im(\\gamma-\\epsilon\\gamma')$ and repeat Step 2.\\\\\n\n\n\\item[Step 3:] Find a Perron eigenvector $\\delta\\in VM_kV$ of $VT^*(\\cdot)V:VM_kV\\rightarrow VM_kV$ associated to $\\lambda$.\\\\\\\\\n$\\bullet$ If $\\Im(\\delta)=\\Im(\\gamma)$ then $T:VM_kV\\rightarrow VM_kV$ is irreducible. Define $V_1=V$.\\\\\n$\\bullet$ If $\\Im(\\delta)\\neq\\Im(\\gamma)$ then redefine $V$ as the orthogonal projection onto $\\Im(V)\\cap\\ker(\\delta)$ and return to Step 1.\\\\\n\n\\end{itemize}\n\n\n\\noindent\\textbf{Algorithm 2:} This algorithm finds the unique solution $W$ of Problem \\ref{problem} subjected to $T$ and $V$.\\\\\n\n\\begin{itemize}\n\\item[Step 1:] Find the spectral radius $\\lambda$ of $T:VM_kV\\rightarrow VM_kV$ and the invertible matrix $Q\\in M_k$ of Lemma \\ref{lemmaQ}.\\\\\n\n\\item[Step 2:] Define $T_1:VM_kV\\rightarrow VM_kV$ as $T_1(X)=\\frac{1}{\\lambda}QT(Q^{-1}X(Q^{-1})^*)Q^*$. Regard  $M_{k-s\\times s}(\\mathbb{C})$ as a real vector space and find the unique zero $S\\in M_{k-s\\times s}$ of the quadratic function $f:M_{k-s\\times s}(\\mathbb{C})\\rightarrow \\mathbb{R}^+\\cup\\{0\\}$ defined in Lemma \\ref{solutionW}.\\\\\n$\\bullet$ If the unique zero exists then define $W$ as the orthogonal projection onto $\\Im\\left(Q^*\\begin{pmatrix}Id_{s\\times s}\\\\ \nS_{k-s\\times s} \\end{pmatrix}\\right)$\\\\\n$\\bullet$ If the zero does not exist or it is not unique then there is no such $W$.\\\\\\\\\n\\end{itemize}\n\n\\noindent\\textbf{Algorithm 3:} Given a PPT matrix $B\\in M_k\\otimes M_k$ and a vector $v\\in\\Im(B)$ with tensor rank $k$, this algorithm checks whether $G_B((\\cdot)^t):M_k\\rightarrow M_k$ is equivalent to a doubly stochastic map.\\\\\n\nFind the invertible matrix $P\\in M_k$ of Lemma \\ref{lemmaP}. Let $A=(P\\otimes Id)B(P\\otimes Id)^*$ and $T:M_k\\rightarrow M_k$ be $T(X)=G_A((\\cdot)^t)$. In order to start the procedure set $V'=Id$.\\\\\n\n\\begin{itemize}\n\\item[Step 1:] Find $VM_kV\\subset V'M_kV'$ such that $T(VM_kV)\\subset VM_kV$ and $T|_{VM_kV}$ is irreducible via algorithm 1.\\\\\\\\\n$\\bullet$ If $V=V'$ then $T$ is equivalent to a doubly stochastic map and also $G_B((\\cdot)^t):M_k\\rightarrow M_k$.\\\\\n$\\bullet$ If $V\\neq V'$ then do Step 2.\\\\\n\n\\item[Step 2:] Search for the unique solution $W$ of Problem \\ref{problem} subjected to $T$ and $V$ via algorithm 2.\\\\\\\\\n$\\bullet$ If there is no such $W$ then $T$ is not equivalent to a doubly stochastic map and neither is $G_B((\\cdot)^t):M_k\\rightarrow M_k$.\\\\\n$\\bullet$ If there is such $W$ then find an invertible matrix $R\\in M_k$ such that\\begin{center}\n \n$R(Id-V'+V)=(Id-V'+V)$  and $\\Im(R(V'-W))=\\Im(V'-V)$.\n\\end{center} \n\nRedefine $T$ as $RT(R^{-1}X(R^{-1})^*)R^*$ and  $V'$ as $V'-V$ then repeat Step 1.\n\n\\end{itemize}\n\n\\vspace{0.5 cm}\n\n\\begin{remark}  If the algorithm finds out that $T:M_k\\rightarrow M_k$ is equivalent to a doubly stochastic map then \nthe $s$ values attained by $V'$ $($in this run$)$ are orthogonal projections $V_1,\\ldots V_s$ \nsuch that\n\\begin{tabbing}\n\\hspace{8 cm}\\=\\kill\n \\hspace{3 cm}$V_iV_j=0$ for $i\\neq j$,  \\> $\\sum_{i=1}^sV_i=Id$, \\\\ \n\\hspace{3 cm} $T'(V_iM_kV_i)\\subset V_iM_kV_i$, \\> $T'|_{V_iM_kV_i}$ is irreducible\n for every $1\\leq i\\leq s$,\n\\end{tabbing} \n where $T'(X)$ is the last value attained by $T$.  \\\\\n \n Recall that $T'(X)=QG_A((Q^{-1}X(Q^{-1})^*)^t)Q^*$ for some invertible $Q\\in M_k$. Thus, $T'(X)=G_C(X^t)$, where $C=((Q^{-1})^t\\otimes Q)A((Q^{-1})^t\\otimes Q)^*$. \nBy Proposition \\ref{propkey},\n$C=\\sum_{i=1}^s C_i$, where $C_i=(V_i^t\\otimes V_i)C(V_i^t\\otimes V_i)$.\\\\\n\n\n\nNote that $\\Im (T'(X))\\supset\\Im(X)$, for every $X\\in P_k$, and $T'|_{V_iM_kV_i}=G_{C_i}((\\cdot)^t)|_{V_iM_kV_i}$ is irreducible. Therefore, $G_{C_i}((\\cdot)^t)|_{V_iM_kV_i}$ is fully indecomposable. So the scaling algorithm \\cite{gurvits2003, gurvits2004, cariellosink} applied to $G_{C_i}((\\cdot)^t)|_{V_iM_kV_i}$ converges to  a doubly stochastic map $G_{D_i}((\\cdot)^t)|_{V_iM_kV_i}$. Note that $D_i$ is the filter normal form of $C_i$.\n\\end{remark}\n\n\n\n\\section{A simple extension to $M_k\\otimes M_m$}\n\nRecall the identification $M_m\\otimes M_k \\simeq M_{mk}$. Let $B=\\sum_{i=1}^nC_i\\otimes D_i\\in M_k\\otimes M_m$ be a PPT matrix and define $\\widetilde{B}\\in M_{mk}\\otimes M_{mk}$ as $$\\widetilde{B}=\\sum_{i=1}^n (Id_{m}\\otimes C_i)\\otimes (D_i\\otimes Id_k).$$ \nThe next result is the key to extend algorithm $3$ to PPT matrices in $M_k\\otimes M_m$.\n\\begin{lemma}\\label{lemmaextension} Let $B=\\sum_{i=1}^nC_i\\otimes D_i\\in M_k\\otimes M_m$ be a PPT matrix. Consider the PPT matrix $\\widetilde{B}\\in M_{mk}\\otimes M_{mk}$ as defined above. Then, $G_B((\\cdot)^t): M_k\\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $G_{\\widetilde{B}}((\\cdot)^t): M_{mk}\\rightarrow M_{mk}$ is equivalent to a doubly stochastic map.\n\\end{lemma}\n\\begin{proof}\nLet $e_1,\\ldots,e_m$ be the canonical basis of $\\mathbb{C}^m$. Note that for every $\\sum_{i,j=1}^m e_ie_j^t\\otimes B_{ij}\\in M_m\\otimes M_k$, $$G_{\\widetilde{B}}(\\sum_{i,j=1}^m e_je_i^t\\otimes B_{ij}^t)=G_B(\\sum_{i=1}^m B_{ii}^t)\\otimes Id_{ k}.$$\n\nTherefore, by \\cite[Corollary 3.5]{cariellosink}, $G_B((\\cdot)^t): M_k\\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $G_{\\widetilde{B}}((\\cdot)^t): M_{mk}\\rightarrow M_{mk}$ is equivalent to a doubly stochastic map.\n\\end{proof}\n\n\\begin{corollary} Let $B\\in M_k\\otimes M_m$ and $\\widetilde{B}\\in M_{mk}\\otimes M_{mk}$ be as in Lemma \\ref{lemmaextension}. Then, $B$ can be put in the filter normal form if and only if $\\widetilde{B}$ can be put in the filter normal form.\n\\end{corollary}\nThus, if there is a vector $v \\in \\Im(\\widetilde{B})$ with tensor rank $mk$ then we can run algorithm $3$ with $\\widetilde{B}$. If the algorithm finds out that $G_{\\widetilde{B}}((\\cdot)^t): M_{mk}\\rightarrow M_{mk}$ is equivalent to a doubly stochastic map then $B$ can be put in the filter normal form.\n\n\n\n\\section*{Summary and Conclusion}\n\nIn this work  we described a procedure to determine whether $G_A:M_k\\rightarrow M_k$, $G_A(X)=\\sum_{i=1}^n tr(A_iX)B_i$ is equivalent to a doubly stochastic map or not when $A=\\sum_{i=1}^nA_i\\otimes B_i \\in M_k\\otimes M_k\\simeq M_{k^2}$ is a PPT matrix and there is a full tensor rank vector within its image. The difficult  part of the algorithm is finding Perron eigenvectors of $G_A((\\cdot)^t):M_k\\rightarrow M_k$. \n\nThis procedure can be used to determine whether a PPT matrix can be put in the filter normal form, which is a very useful tool to study entanglement of quantum mixed states. 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Mathematical Phys.},\n   volume={3},\n   year={1972},\n   pages={275-278},\n}\n\n\n\\bib{pawel}{article}{\n   title={Separability criterion and inseparable mixed states with positive partial transposition},\n  author={Horodecki, Pawel},\n  journal={Physics Letters A},\n  volume={232},\n  number={5},\n  pages={333--339},\n  year={1997},\n  publisher={Elsevier}\n}\n\n\\bib{evans}{article}{\n  title={Spectral properties of positive maps on C*-algebras},\n  author={Evans, David E.}\n  author={H{\\o}egh-Krohn, Raphael},\n  journal={Journal of the London Mathematical Society},\n  volume={2},\n  number={2},\n  pages={345--355},\n  year={1978},\n  publisher={Oxford University Press}\n}\n\n\n\\bib{Bhatia1}{book}{\n  title={Positive definite matrices},\n  author={Bhatia, Rajendra},\n  year={2009},\n  publisher={Princeton (NJ): Princeton university press}\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{biblist}\n\\end{bibdiv}\n\n\\section*{Appendix}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemma1}]\n\nBy Definition \\ref{defcompletepositive}, $T(X)=\\sum_{i=1}^mA_iXA_i^*.$ \n\nNow, if $T(V)=\\sum_{i=1}^mA_iVA_i^*\\in VM_kV$ then \n$A_iVA_i^*\\in VM_kV$, for every $i$. \n\n\nSince $A_iVA_i^*=A_iV(A_iV)^*$ then $A_iV\\in VM_k$. Thus,  $VA_i^*\\in M_kV$, for every $i$.\n\nHence, \n$A_i(VX+YV)A_i^*\\in VM_k+M_kV$, for every $X,Y\\in M_k$.  \n\\end{proof}\n\\vspace{0,5cm}\n\n\\begin{proof}[Proof of Lemma \\ref{lemma2}] \na) Note that\n$$0=tr(T(V_1)(V-V_1))=tr(V_1T^*(V-V_1))=tr(VV_1VT^*(V-V_1))=tr(V_1VT^*(V-V_1)V).$$\n\nThus, $\\Im(VT^*(V-V_1)V)\\subset\\Im(V-V_1)$ and $$VT^*((V-V_1)M_k(V-V_1))V\\subset (V-V_1)M_k(V-V_1).$$\n\n\\noindent\nb) If $T|_{VM_kV}$ is not irreducible, neither is $VT^*(\\cdot)V|_{VM_kV}$ by item a).\\\\\n\nNow, if $VT^*(\\cdot)V:VM_kV\\rightarrow VM_kV$ is not irreducible then there is a proper subalgebra $V_1M_kV_1\\subset VM_kV$ such that \n$VT^*(V_1M_kV_1)V\\subset V_1M_kV_1$ and $V_1\\neq 0$.\\\\\n\nHence, $0=tr(VT^*(V_1)V(V-V_1))=tr(T^*(V_1)V(V-V_1)V)=tr(V_1T(V-V_1))$. Thus, $$T((V-V_1)M_k(V-V_1))\\subset (V-V_1)M_k(V-V_1).$$\n\n\\noindent\nc) Let $\\gamma\\in P_k\\cap VM_kV$ be such that $\\gamma^2=\\delta$. Denote by $\\gamma^{+}$ the  Hermitian pseudo-inverse of $\\gamma$. So $\\gamma^{+}\\gamma=\\gamma\\gamma^{+}=V$. \n\nNote that $\\frac{1}{\\lambda}\\gamma^{+}T(\\gamma V\\gamma)\\gamma^{+}=V$. Therefore, the operator norm of \\begin{center}\n$\\frac{1}{\\lambda}\\gamma^{+}T(\\gamma (\\cdot) \\gamma)\\gamma^{+}:VM_kV\\rightarrow VM_kV$\n\\end{center} induced by the spectral norm on $M_k$ is 1 \\cite[Theorem 2.3.7]{Bhatia1}. \\\\\n\nLet $Y\\in VM_kV$ be an eigenvector of $T:VM_kV\\rightarrow VM_kV$ associated to $\\alpha$ such that the spectral norm of $\\gamma^{+}Y\\gamma^{+}$ is 1. Then, the spectral norm of  $\\frac{1}{\\lambda}\\gamma^{+}T(\\gamma \\gamma^{+}Y\\gamma^{+} \\gamma)\\gamma^{+}=\\frac{\\alpha}{\\lambda}\\gamma^{+}Y\\gamma^{+}$ is smaller or equal to the operator norm of $\\frac{1}{\\lambda}\\gamma^{+}T(\\gamma (\\cdot) \\gamma)\\gamma^{+}:VM_kV\\rightarrow VM_kV$. Thus, $\\frac{|\\alpha|}{\\lambda}\\leq 1$. \n\\end{proof}\n\\hspace{1cm}\n\\begin{proof}[Proof of Lemma \\ref{lemma3}]\nSince $VT^*(\\cdot)V|_{VM_kV}$ and $T|_{VM_kV}$ are adjoint with respect to the trace inner product and $\\lambda\\in \\mathbb{R}$ then $\\text{rank}(VT^*(\\cdot)V-\\lambda Id|_{VM_kV})=       \n\\text{rank}(T-\\lambda Id|_{VM_kV})$. So the geometric multiplicity of $\\lambda$ is the same for both maps.\\\\\n\nMoreover, $T|_{VM_kV}$ is irreducible if and only if $VT^*(\\cdot)V|_{VM_kV}$ is irreducible by Lemma \\ref{lemma2}. item $b)$. So conditions $1)$ and $2)$ are necessary for irreducibility by Perron-Frobenius theory (\\cite[Theorem 2.3]{evans}).\\\\\n\nNext, let us assume by contradiction that conditions $1)$ and $2)$ hold and $T|_{VM_kV}$ is not irreducible.\n\nThus, there is an orthogonal projection $V_1\\in M_k$ such that $T(V_1M_kV_1)\\subset V_1M_kV_1$, $V_1V=VV_1=V_1$ and $V-V_1\\neq 0$.\\\\\n\nBy item $a)$ of Lemma \\ref{lemma2}, $VT^*((V-V_1)M_k(V-V_1))V\\subset (V-V_1)M_k(V-V_1)$. Since $T^*:M_k\\rightarrow M_k$ is completely positive then $VT^*(\\cdot)V:M_k\\rightarrow M_k$ is too. \\\\\n\nHence, by Lemma \\ref{lemma1},\n\\begin{equation}\\label{eq=2}\nVT^*((V-V_1)M_k+M_k(V-V_1))V\\subset (V-V_1)M_k+M_k(V-V_1).\n\\end{equation}\n\nNow, since $V_1V=V_1$ and $\\lambda\\delta=VT^*(\\delta)V=VT^*((V-V_1)\\delta+V_1\\delta (V-V_1))V+VT^*(V_1\\delta V_1)V$ then, by Equation \\ref{eq=2},\n$\\lambda V_1\\delta V_1=V_1T^*(V_1\\delta V_1)V_1.$  \n\nNote that $V_1\\delta V_1\\neq 0$, since $\\Im(V_1)\\subset \\Im(V)=\\Im(\\delta)$. Hence, $\\lambda$ is an eigenvalue of $V_1T^*(\\cdot)V_1: V_1M_kV_1\\rightarrow V_1M_kV_1$. Therefore, it is also an eigenvalue of its adjoint $T:V_1M_kV_1\\rightarrow V_1M_kV_1$. \\\\\n\nActually, $\\lambda$ is the spectral radius of $T|_{V_1M_kV_1}$, since $V_1M_kV_1\\subset VM_kV$ and by hypothesis. So, by Perron-Frobenius theory, there is $\\gamma'\\in (V_1M_kV_1 \\cap P_k)\\setminus \\{0\\}$ such that $T(\\gamma')=\\lambda\\gamma'$.\\\\\n\nNote that $\\gamma$ and $\\gamma'$ are linearly independent, since $\\Im(\\gamma')\\subset\\Im(V_1)\\neq\\Im(V)=\\Im(\\gamma)$. Thus, the geometric multiplicity of $\\lambda$ is not 1. Absurd!\n\\end{proof}\n\n\\end{document} \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Nearly 100 years of nuclear astrophysics}\n\\label{intro}\nWith Eddington's brilliant conjecture in 1920 that the Sun's energy\nreservoir ``can scarcely be other than sub-atomic energy\", based on\nground-breaking experimental work by Aston and Rutherford, the field\nof nuclear astrophysics was born~\\cite{Eddi20}.\nWork accomplished in the following decades demonstrated that nuclear\nphysics was not only key to describing how stars lived and died, but\nalso how they made the elements we see (and are made of)\ntoday~\\cite{Burb57,Came57}.\nTo date, great strides have been made in a worldwide effort (See\nFig.~\\ref{GlobalRIB}.) toward answering fundamental questions about\nour universe, chief among them: \\emph{Where were the elements\nmade?}, \\emph{How does nuclear energy generation impact stars and\nstellar explosions?}, and \\emph{How does matter behave at high\ndensity and low temperature?}. This progress has relied on the study\nof atomic nuclei over nearly the entire nuclear landscape, the\nregion bounded by the so-called proton and neutron driplines, where\nprotons or neutrons `drip' out of the nucleus due to the extreme\nmismatch in their respective numbers.\n\nThe dramatic enhancements of\nexperimental capabilities offered by next generation\nradioactive ion beam facilities such as the Facility for Rare\nIsotope Beams (FRIB)~\\cite{FRIB} and the NuSTAR experiments at the Facility for Antiproton and\nIon Research (FAIR)~\\cite{FAIR}, coupled with advances in\nobservational and computational capabilities\n(e.g. Refs.~\\cite{NUSTAR,UNEDF}), are certain to deepen our\nunderstanding of nature and likely yield more than a few surprises.\nThe following sections will briefly touch on recent accomplishments in\nexperimental nuclear astrophysics and\noutstanding questions across the nuclear landscape. Due to\nspace limitations, many exciting works and research topics have been\nomitted. For more comprehensive reviews, see\nRefs.~\\cite{Jose11,Wies12,Scha16}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth,angle=0]{GlobalRIBFacilities.pdf}\n\\caption{\n Global census of radioactive ion beam facilities.\n\\label{GlobalRIB}}\n\\end{center}\n\\end{figure}\n\n\\section{Progress on the proton-rich side}\n\\label{prich}\nVarious astrophysical environments feature nuclear reactions on the\nneutron-deficient side of the valley of $\\beta$-stability that are\nresponsible for nuclear energy generation and element formation.\nThese proton-rich conditions, caused by an injection of large amounts\nof hydrogen from a stellar envelope or transmutation of neutrons\ninto protons via neutrino capture, enable sequences of proton-capture reactions\nto proceed at high-temperature. These reactions drive the rapid proton-capture\n($rp$)-process that largely powers type-I x-ray bursts~\\cite{Pari13} and\nclassical novae~\\cite{Bode12} and are a main player in the reaction\nnetwork leading to nucleosynthesis via the neutrino-p ($\\nu\np$)-process in core-collapse supernovae~\\cite{Froh06}.\n\nClassical novae are thermonuclear explosions on the surfaces of white\ndwarf stars that recur due to the reaccumulation of hydrogen fuel\nfrom a binary companion star~\\cite{Bode12}. These explosions, aside from\ngenerating astronomical displays occasionally visible to the naked\neye, contribute to the creation of light elements in the universe.\nEven though these objects have been a focus of intense study for\ndecades, including numerous observational, theoretical, and\nexperimental efforts, their exact contribution\nto the cosmic abundances is still unknown. Major advances\nhave been made recently by using the power of pure, exotic\nradioactive ion beams to investigate the origins of \npresolar dust grains that may have been produced by novae (e.g.\nRef.~\\cite{Benn16}).\nIn fact, studies such as the aforementioned work make novae one of\nthe few astrophysical phenomena for which most nuclear\nreaction rates are based on experimental data.\n\nType-I x-ray bursts are similar but much more powerful explosions\nrecurring on the surfaces of hydrogen and helium accreting neutron\nstars, where the larger surface gravity of the neutron star relative\nto the white dwarf ultimately leads to the enhanced energy\nrelease~\\cite{Pari13}. These observables are one of the main\ntools used for understanding neutron stars, unique astronomical\nlaboratories that provide insight into the behavior of dense matter at\nlow temperature. Recent work has focused on understanding reactions\nthat trigger the $rp$-process (e.g. Ref.~\\cite{Parp05}) and the\nlocations of the nuclear landscape where the $rp$-process reaction\nsequence is significantly stalled, termed waiting-point nuclides (e.g.\nRef.~\\cite{DelS14}). Though the strengths of all waiting-points are\nsoon to be well constrained, theoretical work has shown that a host\nof other nuclear uncertainties remain that must ultimately be\nremoved or reduced by future experiments~\\cite{Cybu16}.\n\n\\section{New and old directions for neutron-rich nuclides}\n\\label{nrich}\nOn the opposite side of the nuclear chart, an array of nuclear\nreaction sequences operate in astronomical environments to various\nends, such as the forging of new elements and the alteration of\ndense objects' thermal and compositional structures. The oldest and most well known\nof these is the rapid neutron-capture ($r$)-process that is\nresponsible for creating roughly half of the elements heavier than\niron in as-yet undetermined astrophysical sites~\\cite{Mump16}.\nHowever, other reaction sequences such as the\n$\\alpha$-process~\\cite{Arco11} and $i$-process~\\cite{Jone16} have\nrecently joined the club of potential mechanisms for nucleosynthesis\non the neutron-rich side of stability. Equally exciting are the new\ndevelopments that have shown individual nuclear properties are\ncritical to understanding the various observables from neutron stars\nthat provide unique windows into the behavior of high-density,\nlow-temperature matter~\\cite{Scha14}.\n\nThe $r$-process, the rapid neutron-capture sequence likely operating\nin core-collapse supernovae and\/or neutron star mergers, has long\nbeen out of reach of even the most advanced radioactive ion beam\nfacilities. To date, much of the relevant experimental work has focused on\nconstraining key nuclear quantities nearer to stability so that\nthese results can guide theoretical estimates for nuclides on the\n$r$-process path~\\cite{Mump16}. However, recent innovative\ntechniques have allowed some of the first experimental constraints\nto be made for nuclear reactions on the path itself (e.g.\nRef.~\\cite{Spyr14}).\nThis and other approaches will be particularly powerful when coupled\nwith the extended reach toward the neutron dripline anticipated for\nFRIB (See Fig.~\\ref{FRIBproduction}.) and FAIR and promise to dramatically advance our understanding\nof this long-studied problem.\n\nIn the past decade other avenues for nucleosynthesis have been\nidentified that likely operate nearer to stability on the\nneutron-rich side. The $\\alpha$-process, which may operate via a\nsequence of $\\alpha$-capture, neutron-emission\nreactions~\\cite{Peri14}, and\n$i$-process, a reaction sequence similar in spirit to the\n$r$-process but at lower neutron densities, have only just begun to\nbe investigated. Numerous experimental studies are anticipated in the near future\nthat will drastically expand our knowledge of these processes and provide\ncritical tests of their viability as mechanisms of element\nformation.\n\nLately it has been shown that neutron-rich nuclides are\njust as important as their proton-rich cousins with regards to their\nimpact on our understanding of dense matter. When the proton-rich\nashes of the $rp$-process are compressed on the neutron star surface\nby subsequent accretion from a binary companion, electrons are\nforced into nuclei, converting protons into neutrons and\nfundamentally altering the neutron star thermal and compositional\nouter structure~\\cite{Scha14}. It has been shown that it is\ncritical to determine the properties of individual neutron-rich\nnuclides in order to accurately describe the accreted neutron star\nocean and crust~\\cite{Estr11,Meis15,Meis16}. Efforts in the near\nfuture are planned to focus on nuclides which have been identified\nto have the greatest potential impact on astronomical observables~\\cite{Deib16}.\n\n\n\\section{Nuclear astrophysics near stability}\n\\label{stable}\nAlthough most nuclides in and near the valley of $\\beta$-stability\nhave been accessible in the laboratory for some time, several\noutstanding questions in nuclear astrophysics require their further\nstudy. These involve quiescent and explosive stellar environments and require\nexperimental approaches using both indirect and direct techniques to\nstudy the most important reactions~\\cite{Brun15}. Open questions\ninclude the extent to which photodisintegration reactions impact the\ncosmic elemental abundances~\\cite{Raus13}, the exact abundance yield\nfrom slow neutron capture in stellar envelopes~\\cite{Reif14}, and\nthe role of electron captures in high-density astrophysical\nenvironments~\\cite{Sull16}. Progress in these areas has been driven\nby several nuclear physics labs around the world, especially the\nmany stable-ion beam facilities which are far too numerous to\ninclude in Fig.~\\ref{GlobalRIB}.\n\nPrecisely describing the nuclear reaction sequences of stars has\nremained a challenge since Eddington first approached the\nsubject~\\cite{Eddi20}.\nPartially due to triumphs of astrophysical modeling and observations, such as\nasteroseismology and measurements of neutrinos from our sun, high\nprecision experimental studies are needed to advance our\nunderstanding of quiescent nuclear burning in stars~\\cite{Brun15}.\nSpecialized equipment such as recoil mass separators and underground laboratories (e.g.\nRefs.\\cite{Coud08} and \\cite{Robe16}, respectively) have\nplayed and will continue to play a major role in this effort. When\ndirect measurements via these and other methods are not possible, as is the\ncase for some branch-point nuclides in the slow neutron-capture\n($s$)-process reaction network, indirect techniques will be required\nto experimentally constrain important reactions~\\cite{Reif14}.\n\nIn spite of their association with the most exotic nuclides, models\nof stellar explosions require a thorough understanding of nearly the\nentire nuclear landscape, including nuclides along and near\nstability. A case-in-point is the photodisintegration-driven\n$p$-process operating in supernovae, which is currently the favored creation\nmechanism of the so-called $p$-nuclides whose\norigins cannot\nbe explained by the $s$ and $r$ processes~\\cite{Raus13}. Sustained\nefforts have reduced the nuclear physics\nuncertainties of this process, where the focus has generally been on\nconstraining the Wolfenstein-Hauser-Feshbach reaction theory that\nprovides essential input to astrophysics models in the absence of\nexperimental data (e.g. Refs.~\\cite{Quin15,Yalc15}). Additional\nmeasurements on and near stability have focused on reducing the\nuncertainties in nuclear weak rates that limit the ability to\ndescribe the mechanisms through which supernovae operate (e.g.\nRef.~\\cite{Noji14}). Here theory calculations have provided\nimportant guidance, identifying the most essential nuclear data and\nfilling in the large gaps left by insufficient experimental\ninformation~\\cite{Sull16}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=0.9\\columnwidth,angle=0]{FRIBproduction.pdf}\n\\caption{\n  Predicted FRIB production rates in particles per\n  second~\\cite{Boll11}. See Ref.~\\cite{Fran08} for a similar\n  prediction for FAIR.\n\\label{FRIBproduction}}\n\\end{center}\n\\end{figure}\n\n\n\\section{FRIB, FAIR, and the future}\n\\label{outlook}\nRoughly 100 years after its inception, nuclear astrophysics\nresearch continues to enhance our understanding of nature. At\npresent the field is poised to build upon our current body of\nknowledge by leaps and bounds, in no small part due to upcoming developments such\nas new recoil separators~\\cite{Coud08,Meis16b}, underground\nlaboratories~\\cite{Robe16}, and storage rings dedicated to nuclear\nphysics studies~\\cite{Fran08}. Frontier nuclear physics facilities\nsuch as FRIB and the NuSTAR experiments at FAIR will play a central role in this advancement\nby providing unprecedented access to ever more exotic nuclides (See\nFig.~\\ref{FRIBproduction}.). Meanwhile, stable beam facilities will\ncontinue to play a complementary role in answering astrophysical\nquestions both new and old. In the near future, together with advances in observation\nand theory, experimental nuclear astrophysics studies from dripline to dripline\npromise to offer profound insight into how our universe operates.\n\n\\section*{Acknowledgements}\nThis work was supported by the National Science Foundation Grants No. 1419765 and 1430152.\n\n\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nA wide array of astrophysical data point to us living in a universe comprised of\n$4\\%$ baryons, $\\sim 25\\%$ cold dark matter (CDM) and $\\sim 70\\%$ dark energy.\nIn fact, the cosmic abundance of CDM has been recently measured\nto high precision by the WMAP collaboration\\cite{wmap5}, which finds\n\\be\n\\Omega_{CDM}h^2=0.110\\pm 0.006 ,\n\\ee\nwhere $\\Omega=\\rho\/\\rho_c$ is the dark matter density relative to the \nclosure density, and $h$ is the scaled Hubble constant.\nNo particle present in the Standard Model (SM) of particle physics \nhas the correct properties to constitue the CDM, so some form of new physics \nis needed. It is compelling, however, that candidate CDM particles do emerge naturally \nfrom two theories which provide solutions to longstanding problems in particle physics. \n\nThe first problem-- known as the gauge hierarchy problem-- arises due\nto quadratic divergences in the scalar sector of the SM. \nThese divergences\nlead to scalar masses blowing up to the highest scale in the theory\n({\\it e.g.} in grand unified theories (GUTS), \nthe GUT scale $M_{GUT}\\simeq 2\\times 10^{16}$ GeV), unless\nan enormous fine-tuning of parameters is invoked. \nOne solution to the gauge hierarchy problem occurs by introducing supersymmetry (SUSY) into the\ntheory. The inclusion of softly broken SUSY leads to\na cancellation of quadratic divergences between fermion and boson loops, \nso that only log divergences remain. \nThe log divergence is soft enough that\nvastly different scales remain stable within a single effective theory.\nIn SUSY theories, the lightest neutralino emerges as an excellent\nWIMP CDM candidate. \nGravity-mediated SUSY breaking models (supergravity, or SUGRA) contain gravitinos with weak-scale masses. \nSUGRA models experience tension due to\npossible overproduction of gravitinos in the early universe, leading to an overabundance\nof CDM.\nIn addition, gravitinos usually decay during or after Big Bang nucleosynthesis (BBN),\nand their energetic decay products may disrupt\nthe successful calculations of light element\nabundances, which otherwise maintain good agreement with observation.\nThis tension in SUGRA models is known as the {\\it gravitino problem}.\n\nThe second problem is the strong $CP$ problem\\cite{kcreview}. An elegant solution \nto the strong $CP$ problem was proposed by Peccei and Quinn (PQ) many years ago\\cite{pq}.\nThe PQ solution automatically predicts the existence of a new particle (WW)\\cite{ww}: \nthe axion $a$. \nWhile the original PQWW axion was soon ruled out, models of a nearly\n``invisible axion'' were developed in which the PQ symmetry breaking scale\nwas moved up to energies of order $f_a\\sim 10^{9}-10^{12}$ GeV\\cite{ksvz,dfsz}.\nThe axion also turns out to be an excellent candidate \nparticle for CDM in the universe\\cite{absik}.\n\nOf course, it is highly desirable to simultaneously account for\n{\\it both} the strong $CP$ problem and the gauge hierarchy problem.\nIn this case, it is useful to invoke supersymmetric models which\ninclude the PQWW solution to the strong $CP$ problem\\cite{nillesraby}. In a \nSUSY context, the axion field is just one element of an \n{\\it axion supermultiplet}. The axion supermultiplet contains \na complex scalar field, whose real part is the $R$-parity even saxion \nfield $s(x)$, and whose imaginary part is the axion field $a(x)$.\nThe supermultiplet also contains an $R$-parity odd spin-$\\frac{1}{2}$ \nMajorana field, the axino $\\tilde a$\\cite{steffen_rev}.\nThe saxion, while being an $R$-parity even field, nonethless \nreceives a SUSY breaking mass likely of order the weak scale. \nThe axion mass is constrained by\ncosmology and astrophysics to lie in a favored range \n$10^{-2}$ eV$\\stackrel{>}{\\sim} m_a\\stackrel{>}{\\sim} 10^{-5}$ eV. \nThe axino mass is very model dependent\\cite{axmass,rtw,cl,ckkr}, depending\nheavily on the exact form of the superpotential and the mechanism for SUSY breaking. \nIn supergravity models, it may be of\norder the gravitino mass $m_{3\/2}\\sim$ TeV, or as low as $m_{3\/2}^2\/f_a\\sim $keV.\nConditions for realizing these extremes are addressed in \\cite{cl}. \nHere, we will try to avoid explicit model-dependence, and adopt \n$m_{\\tilde a}$ as lying within the general range of keV-GeV, \nas in numerous previous works\\cite{rtw,ckkr,fstw,cmssm,axdm}.\nAn axino in this mass range would likely serve as the lightest\nSUSY particle (LSP), and is also a good candidate particle for\ncold dark matter\\cite{rtw,ckkr}.\n\nIn a previous paper\\cite{axdm}, we investigated supersymmetric models wherein the PQ\nsolution to the strong $CP$ problem is assumed. For definiteness,\nwe restricted the analysis to examining the paradigm minimal\nsupergravity (mSUGRA or CMSSM) model\\cite{msugra}. \nWe were guided in our analysis by considering the possibility of including \na viable mechanism for baryogenesis in the early universe. \nIn order to do so, we needed\nto allow for re-heat temperatures after the inflationary epoch to \nreach values $T_R\\stackrel{>}{\\sim} 10^6$ GeV. We found that in order to sustain\nsuch high re-heat temperatures, as well as generating predominantly \n{\\it cold} dark matter, we were pushed into mSUGRA parameter\nspace regions that are very different from those allowed by\nthe case of thermally produced neutralino dark matter. In addition, we\nfound that very high values of the PQ breaking scale $f_a\/N$ of order\n$10^{11}-10^{12}$ GeV were needed, leading to the mSUGRA model with\n{\\it mainly axion cold dark matter}, but also with a small\nadmixture of thermally produced axinos, and an even smaller\ncomponent of warm axino dark matter arising from neutralino decays.\nThe favored axino mass value is of order 100 keV.\nWe note here recent work on models with dominant axion CDM explore the \npossibility that axions form a cosmic Bose-Einstein condensate, which can \nallow for the solution of several problems associated with large scale\nstructure and the cosmic background radiation\\cite{pierre}.\n\nIn this paper, we will examine the mSUGRA model under the assumption 1. of\nneutralino CDM and 2. that \nmixed axion\/axino DM ($a\\tilde{a}$DM) saturates the WMAP measured \nabundance\\footnote{The possibility of mixed $a\\tilde{a}$CDM was suggested\nin the context of Yukawa-unified SUSY in Ref. \\cite{mix}.}.\nTo compare the two DM scenarios, we will evaluate a measure of\nfine-tuning in the relic abundance\n\\be\n\\Delta_{a_i} \\equiv\\frac{\\partial\\log\\Omega_{DM}h^2}{\\partial\\log a_i}\n\\ee\nwith respect to variations in fundamental parameters $a_i$ of the model.\nSuch a measure of relic abundance fine-tuning was previously calculated in Ref.~\\cite{eo} \nin the context of just neutralino dark matter.\nHere, we will expand upon this and also consider fine-tuning of the relic\ndensity in the case of mixed $a\\tilde{a}$DM.\nOur main conclusion is that the relic abundance of DM is {\\it much less fine-tuned\nin the case of mixed $a\\tilde{a}$CDM, as compared to neutralino CDM}.\nThus, we find that mixed $a\\tilde{a}$CDM is {\\it theoretically preferable to \nneutralino CDM}, at least in the case of the mSUGRA model, and probably also\nin many cases of SUGRA models with non-universal soft SUSY breaking terms. \n\nWe will restrict our work to cases where the lightest neutralino\n$\\tilde\\chi^0_1$ is either the LSP or the next-to-lightest SUSY particle (NLSP) with an\naxino LSP; the case with a stau NLSP and an axino LSP has recently been examined in Ref. \\cite{fstw}.\nRelated previous work on axino DM in mSUGRA can be found in Ref. \\cite{cmssm}.\n\nThe remainder of this paper is organized as follows.\nIn Sec. \\ref{sec:inoDM}, we calculate the neutralino relic abundance fine-tuning \nparameter $\\Delta_{\\tilde\\chi^0_1}$ in the mSUGRA model due to variation in parameters\n$m_0$ and $m_{1\/2}$. \nWe find, in good agreement with Ref. \\cite{eo},\nthat the WMAP allowed regions are all finely-tuned for low values of $\\tan\\beta$. \nFor much higher $\\tan\\beta\\sim 50$, the fine-tuning is much less with respect to $m_0$\nand $m_{1\/2}$, but nevertheless high with respect to $\\tan\\beta$.\nIn Sec. \\ref{sec:inoprob}, we review the gravitino problem, leptogenesis and the \ncosmological production of axion and axino dark matter.\nIn Sec. \\ref{sec:axDM}, we calculate the fine-tuning parameter\n$\\Delta_{a\\tilde a}$ for mixed $a\\tilde{a}$CDM under the assumption of a very light\naxino with $m_{\\tilde a}\\sim 0.1-1$ MeV. The fine-tuning is always quite low,\nfor both cases of mixed axino\/axion CDM and mainly axion CDM. In the case of mainly axino\nCDM, we find the scenario less well-motivated since for high values of \n$T_R\\stackrel{>}{\\sim} 10^6$ GeV, the value of $m_{\\tilde a}\\ll 0.1$ MeV, making the axino \nmainly {\\it warm} DM instead of cold DM.\nIn Sec. \\ref{sec:conclude}, we present a summary and conclusions.\n\n\\section{Fine-tuning in mSUGRA with neutralino cold dark matter}\n\\label{sec:inoDM}\n\n\\subsection{Overview}\n\nWe adopt the mSUGRA model\\cite{msugra} as a template model for examining the\nissue of fine-tuning in cases of neutralino CDM vs. $a\\tilde{a}$CDM.\nThe mSUGRA parameter space is given by\n\\be\nm_0,\\ m_{1\/2},\\ A_0,\\ \\tan\\beta ,\\ sign (\\mu ) ,\n\\ee\nwhere $m_0$ is the unified soft SUSY breaking (SSB) scalar mass \nat the GUT scale, \n$m_{1\/2}$ is the unified gaugino mass at $M_{GUT}$, $A_0$ is\nthe unified trilinear SSB term at $M_{GUT}$ and $\\tan\\beta\\equiv\nv_u\/v_d$ is the ratio of Higgs field vevs at the weak scale.\nThe GUT scale gauge and Yukawa couplings, and the SSB terms are\nevolved using renormalization group equations (RGEs) from\n$M_{GUT}$ to $m_{weak}$, at which point electroweak symmetry is\nbroken radiatively, owing to the large top quark Yukawa coupling. \nAt $m_{weak}$, the various sparticle and Higgs boson mass matrices are \ndiagonalized to find the physical sparticle and Higgs boson masses.\nThe magnitude, but not the sign, of the superpotential $\\mu$ parameter \nis determined by the EWSB minimization conditions.\n\nWe adopt the Isasugra subprogram of Isajet to \ngenerate sparticle mass spectra\\cite{isajet}. \nIsasugra performs an iterative solution of the MSSM two-loop RGEs, and includes\nan RG-improved one-loop effective potential evaluation at an optimized scale,\nwhich accounts for leading two-loop effects\\cite{haber}. Complete\none-loop mass corrections for all sparticles and Higgs boson masses\nare included\\cite{pbmz}. For the neutralino relic density, we use the IsaReD\nsubprogram of Isajet\\cite{isared}.\n\nOur measure of fine-tuning in the neutralino relic density, $\\Delta_{\\tilde\\chi^0_1}$, \nis calculated by constructing a grid of points in $m_0-m_{1\/2}$ space. \nAt each point, the change in $\\Omega_{\\tilde\\chi^0_1}h^2$ corresponding to a change in either \n$m_0$ or $m_{1\/2}$ is calculated for both a positive and negative parameter change, using\n\\begin{equation}\n\\Delta_{a_i}=\\frac{a_i}{\\Omega_{\\tilde{Z}_1}h^2}\\frac{\\partial \\Omega_{\\tilde{Z}_1}h^2}{\\partial a_i}=\n\\frac{a_i}{\\Omega_{\\tilde{Z}_1}h^2}\n\\frac{\\left[\\Omega_{\\tilde{Z}_1}h^2(a_i\\pm\\Delta a_i)-\\Omega_{\\tilde{Z}_1}h^2(a_i)\\right]}{\\Delta a_i}\\;,\n\\end{equation}\nwhere $a_i=m_0$ or $m_{1\/2}$. For each $a_i$, the largest $\\Delta_{a_i}$ is selected from the results for both \nthe positive and negative change. \nTo construct the overall total $\\Delta_{\\tilde\\chi^0_1}$, the individual values are added in quadrature:\n\\begin{equation}\n\\Delta_{\\tilde\\chi^0_1} =\\sqrt{\\Delta_{m_0}^2+\\Delta_{m_{1\/2}}^2}\\;.\n\\end{equation}\nWe may also consider fine-tuning due to variation in $A_0$ and $\\tan\\beta$. \nVariation in $A_0$ yields tiny variations in $\\Omega_{\\tilde\\chi^0_1}h^2$ unless \none moves close to the stop co-annihilation region (see Fig. \\ref{fig:a0} in Sec. \\ref{ssec:a0tanb}). \nVariation in $\\tan\\beta$ gives a slight effect on the relic density unless $\\tan\\beta$ becomes\nvery large. In this case, $m_A$ decreases\\cite{ltanb} to the extent that $m_A\\sim 2m_{\\tilde\\chi^0_1}$, and\nneutralino annihilation rates are greatly increased due to the $A$-resonance.\nThen, variation in $\\tan\\beta$ mainly shifts the {\\it location} of \nthe $A$-resonance in the $m_0\\ vs.\\ m_{1\/2}$ plane. \nMoving on and off the resonance is already accounted for by varying \n$m_0$ and $m_{1\/2}$. \nNevertheless, in Sec. \\ref{ssec:a0tanb} we present results due to including $A_0$ and $\\tan\\beta$ \nin the fine-tuning calculation.\\footnote{\nWe note that Ref. \\cite{eo} consider\nfine-tuning versus variation in $m_b$ and $m_t$. We consider these as fixed\nSM parameters, much as $M_Z$ is fixed.}\n\n\\subsection{Results from variation of $m_0$ and $m_{1\/2}$}\n\\label{ssec:results}\n\nOur first results are shown in Fig. \\ref{fig:ino10}, where we show in frame {\\it a}).\ncontours of $\\Omega_{\\tilde\\chi^0_1}h^2$ in the $m_0\\ vs.\\ m_{1\/2}$ mSUGRA plane for\n$A_0=0$, $\\tan\\beta =10$ and $\\mu >0$. We also take $m_t=172.6$ GeV. The well-known red regions \nare excluded  either due to a stau LSP (left-side) or lack of appropriate EWSB (lower and right side).\nThe gray-shaded region is excluded by LEP2 chargino searches ($m_{\\tilde\\chi_1}>103.5$ GeV), and\nthe green shaded region denotes allowable points with $\\Omega_{\\tilde\\chi^0_1}h^2\\le 0.11$. The region below \nthe orange dashed contour is excluded by LEP2 Higgs searches, which require $m_h>114.4$ GeV; here, \nwe actually require $m_h>111$ GeV to reflect a roughly 3 GeV error on the RGE-improved one-loop\neffective potential calculation of $m_h$. \n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftcomptb10.eps}\n\\caption{In the $m_0\\ vs.\\ m_{1\/2}$ plane of the mSUGRA model for\n$A_0=0$, $\\tan\\beta =10$ and $\\mu >0$, we plot\n{\\it a}). contours of $\\Omega_{\\tilde\\chi^0_1}h^2$ and {\\it b}). regions of \nfine-tuning parameter $\\Delta_{\\tilde\\chi^0_1}$.\n}\\label{fig:ino10}}\n\nThe well-known (green-shaded) hyperbolic branch\/focus point (HB\/FP) region\\cite{hb_fp} stands out on the \nright side, where $\\mu$ becomes small and the $\\tilde\\chi^0_1$ becomes a mixed bino-higgsino state.\nOn the left edge, the very slight stau co-annihilation region\\cite{stau} \nis barely visible.\nWe also plot contours of $\\Omega_{\\tilde\\chi^0_1}h^2$ ranging from 5 to 80. In most of the \nmSUGRA parameter space, the relic abundance is 1-3 orders of magnitude higher than the \nWMAP measured value. The valley in $\\Omega_{\\tilde\\chi^0_1}h^2$ around $m_{1\/2}\\sim 400$ GeV\nis due to the turn-on of the $\\tilde\\chi^0_1\\tilde\\chi^0_1\\rightarrow t\\bar{t}$ annihilation mode.\n\nIn Fig. \\ref{fig:3dn}, we show the neutralino relic density as a 3-d plot in the\n$m_0\\ vs.\\ m_{1\/2}$ plane, to gain extra perspective. The level of fine-tuning corresponds\nto the slope of the surface. We see that in most of parameter space, the slope is relatively small,\n{\\it i.e.} the plateau is nearly flat. However, in this region, the relic density is far too high.\nIn the regions where $\\Omega_{\\tilde\\chi^0_1}h^2\\sim 0.1$, then the slope is extremely steep,\ncorresponding to large fine-tuning: a small variation in fundamental parameters leads to a large\nchange in relic density.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{neutden.eps}\n\\caption{A 3-d plot of neutralino relic density in the $m_0\\ vs.\\ m_{1\/2}$ \nplane of the mSUGRA model for\n$A_0=0$, $\\tan\\beta =10$ and $\\mu >0$.\n}\\label{fig:3dn}}\n\nIn Fig. \\ref{fig:ino10} {\\it b})., we show regions of fine-tuning parameter $\\Delta_{\\tilde\\chi^0_1}$. \nA value of $\\Delta_{\\tilde\\chi^0_1}\\sim 0$ corresponds to no fine-tuning (a flat slope in\n$\\Omega_{\\tilde\\chi^0_1}h^2$ versus variation in all parameters), while higher values of\n$\\Delta_{\\tilde\\chi^0_1}$ give increased fine-tuning in the relic density. We see immediately from the figure\nthat the vast majority of parameter space, where $\\Omega_{\\tilde\\chi^0_1}h^2$ is much too large, \nis also not very fine-tuned. However, the HB\/FP region, where $\\mu\\rightarrow 0$, has a very\nhigh fine-tuning, with $\\Delta_{\\tilde\\chi^0_1}$ ranging from 20-100! There are also regions of\nsubstantial fine-tuning adjacent to the LEP2 chargino mass excluded region,\ndue to rapid changes in $\\Omega_{\\tilde\\chi^0_1}h^2$ as one approaches the $\\tilde\\chi^0_1\\tilde\\chi^0_1\\rightarrow h$ \nannihilation resonance\\cite{hfunnel}, \nand also some fine-tuning at the turn on of $\\tilde\\chi^0_1\\tilde\\chi^0_1\\rightarrow t\\bar{t}$.\nFinally, we see a very narrow region of fine-tuning extending along the stau co-annihilation\nregion.\n\nTo get a better grasp, we plot in Fig. \\ref{fig:slice10} a slice out of\nparameter space at $m_{1\/2}=250$ and 500 GeV, showing in {\\it a}).\n$\\Omega_{\\tilde\\chi^0_1}h^2$ and in {\\it b}). $\\Delta_{\\tilde\\chi^0_1}$ versus $m_0$.\nWe see the slope in $\\Omega_{\\tilde\\chi^0_1}h^2$ is very steep in the HB\/FP region, \nleading to $\\Delta_{\\tilde\\chi^0_1}\\sim 30$ (50) for lower (higher) $m_{1\/2}$ values.\nIn contrast, in the stau co-annihilation region, where $\\Omega_{\\tilde\\chi^0_1}h^2\\sim 0.11$,\nthe value of $\\Delta_{\\tilde\\chi^0_1}\\sim 3$ (12) for low (high) $m_{1\/2}$. The lower\n$m_{1\/2}$ value has only moderate fine-tuning since it is getting close to the \n``bulk'' annihilation region\\cite{bulk}, where $\\tilde\\chi^0_1\\tilde\\chi^0_1$ annihilation is enhanced via light\n$t$-channel slepton exhange diagrams.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftcompslicetb10.eps}\n\\caption{A plot of {\\it a}). $\\Omega_{\\tilde\\chi^0_1} h^2$ and {\\it b}).\n$\\Delta_{\\tilde\\chi^0_1}$ versus $m_0$ for fixed values of\n$m_{1\/2}=250$ GeV (blue) and $m_{1\/2}=500$ GeV (red), in mSUGRA\nwith $A_0=0$, $\\tan\\beta =10$ and $\\mu >0$.\n}\\label{fig:slice10}}\n\nTo gain a better perspective on the stau co-annihilation region, in Fig. \\ref{fig:zoom10}\nwe show a blown-up portrait of the low $m_0$ region of parameter space. The ``turn-around''\nin the green-shaded WMAP allowed region in frame {\\it a}). is due to the impact of the bulk\nannihilation region. Most of this area lies below the $m_h=111$ GeV contour, and thus gives rise \nto Higgs bosons that are too light. In frame {\\it b}). is a blow-up of the fine-tuning parameter\n$\\Delta_{\\tilde\\chi^0_1}$. We see that the major portion of the stau co-annihilation region is\nfine-tuned, with the {\\it possible exception} of the region lying just below the\nLEP2 $m_h$ bound, where mixed bulk\/co-annihilation occurs.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftcompzoomtb10.eps}\n\\caption{A plot of {\\it a}). contours of $\\Omega_{\\tilde\\chi^0_1} h^2$ and {\\it b}).\n$\\Delta_{\\tilde\\chi^0_1}$ in mSUGRA\nwith $A_0=0$, $\\tan\\beta =10$ and $\\mu >0$.\nThis plot zooms in on the stau co-annihilation region.\n}\\label{fig:zoom10}}\n\nIn Fig. \\ref{fig:ino30}, we show contours of relic density and $\\Delta_{\\tilde\\chi^0_1}$\nfor $\\tan\\beta =30$. At higher values of $\\tan\\beta$, the $b$ and $\\tau$ Yukawa couplings\nincrease in magnitude, and enhance neutralino annihilation into $b\\bar{b}$ and $\\tau\\bar{\\tau}$\nfinal states. Overall, we see a similar picture to that shown in Fig. \\ref{fig:ino10}\nin that the HB\/FP region has extreme fine-tuning of the relic density, while the stau\nco-annihilation region is also fine-tuned, but somewhat less so. The bulk annihilation region\nis again excluded by the LEP2 $m_h$ bound.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftcomptb30.eps}\n\\caption{In the $m_0\\ vs.\\ m_{1\/2}$ plane of the mSUGRA model for\n$A_0=0$, $\\tan\\beta =30$ and $\\mu >0$, we plot\n{\\it a}). contours of $\\Omega_{\\tilde\\chi^0_1}h^2$ and {\\it b}). regions of \nfine-tuning parameter $\\Delta_{\\tilde\\chi^0_1}$.\n}\\label{fig:ino30}}\n\nWe plot in Fig. \\ref{fig:ino53} the mSUGRA plane for $\\tan\\beta =53$. In this case, a large\nnew green-shaded region is opening up along the low $m_0$ edge of parameter space. This is due to\nthree effects occuring at large $\\tan\\beta$. 1. The $\\tilde \\tau_1$ mass decreases with $\\tan\\beta$, \nleading to increased annihilation into $\\tau\\bar{\\tau}$ final states; \nthis increases the area of the bulk annihilation region. 2. The tau and $b$ Yukawa couplings \n$f_\\tau$ and $f_b$ increase, thus enhancing annihilation into $\\tau\\bar{\\tau}$ and\n$b\\bar{b}$ final states. \n3. The value of $m_A$ is decreasing while the width $\\Gamma _A$ is increasing \n(due to increasing Yukawa couplings that enter the $A$ decay modes), so that\n$\\tilde\\chi^0_1\\tilde\\chi^0_1\\rightarrow A^{(*)}\\rightarrow b\\bar{b},\\ \\tau\\bar{\\tau}$ increases: {\\it i.e.} we are entering\nthe $A$ resonance annihilation region\\cite{Afunnel}, \nwhich enhances the neutralino annihilation cross section\nin the early universe, thus lowering the relic density. In the case of $\\tan\\beta =53$, we see\nthat the HB\/FP region is still highly fine-tuned. However, broad portions of the low $m_0$ \nmSUGRA parameter space around $m_{1\/2}\\sim 300-600$ have $\\Delta_{\\tilde\\chi^0_1}\\stackrel{<}{\\sim} 3$ due to\nan overlap of bulk annihilation through staus, stau co-annihilation and $A$-resonance annihilation.\nAnother low fine-tuning and relic density consistent region occurs at $m_{1\/2}\\sim 1200$ GeV,\nwhere one sits atop the $A$-resonance.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftcomptb53.eps}\n\\caption{In the $m_0\\ vs.\\ m_{1\/2}$ plane of the mSUGRA model for\n$A_0=0$, $\\tan\\beta =53$ and $\\mu >0$, we plot\n{\\it a}). contours of $\\Omega_{\\tilde\\chi^0_1}h^2$ and {\\it b}). regions of \nfine-tuning parameter $\\Delta_{\\tilde\\chi^0_1}$.\n}\\label{fig:ino53}}\n\nIn Fig. \\ref{fig:ino55}, we show the mSUGRA plane for $\\tan\\beta =55$. Here, the\n$A$-resonance annihilation region is fully displayed, and the $A$ width is even larger.\nWhile much of the HB\/FP region is still very fine-tuned, the regions of\nannihilation though the broad $A$ resonance yield relatively low fine-tuning,\nespecially if one sits right on the resonance, or sits in the\nresonance\/bulk\/co-annihilation overlap region at low\n$m_0$ and low $m_{1\/2}$.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftcomptb55.eps}\n\\caption{In the $m_0\\ vs.\\ m_{1\/2}$ plane of the mSUGRA model for\n$A_0=0$, $\\tan\\beta =55$ and $\\mu >0$, we plot\n{\\it a}). contours of $\\Omega_{\\tilde\\chi^0_1}h^2$ and {\\it b}). regions of \nfine-tuning parameter $\\Delta_{\\tilde\\chi^0_1}$.\n}\\label{fig:ino55}}\n\n\\subsection{Results from variation in $A_0$ and $\\tan\\beta$}\n\\label{ssec:a0tanb}\n\nAs mentioned earlier, including $A_0$ into the measure of fine-tuning typically yields only a small\neffect, unless one is near the top-squark co-annihilation region. This is because variation in $A_0$\nmainly leads to different mixing in the third generation scalar system, and for most of \nmSUGRA parameter space, affects mainly the top squark mass eigenstates. To show this explicitly,\nwe plot in Fig. \\ref{fig:a0}{\\it a}). the value of $\\Omega_{\\tilde\\chi^0_1}h^2$ and in frame {\\it b}).\nthe value of $|\\Delta_{A_0}|$ versus variation in $A_0$ for two cases: \n 1. $m_0=1.5$ TeV and $m_{1\/2}=250$ GeV (blue curves) and 2. $m_0=2$ TeV and \n$m_{1\/2}=750$ GeV (red dashed curve), for $\\tan\\beta =10$ and $\\mu >0$.\nIn these two cases, the slope of $\\Omega_{\\tilde\\chi^0_1}h^2$ is rather mild, leading to a \ncontribution to $|\\Delta_{A_0}|$ of typically 1 or less. The exception comes for the\n$m_{1\/2}=750$ GeV curve around $A_0\\sim -4$ TeV, where indeed the value of $m_{\\tilde t_1}$\nis rapidly becoming lighter, and feeding into the relic density calculation.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{a0slice.eps}\n\\caption{Plot of {\\it a}). $\\Omega_{\\tilde\\chi^0_1}h^2$ and {\\it b}). $|\\Delta_{A_0}|$ \nversus $A_0$ for two slices out of mSUGRA parameter space: \n1. $m_0=1.5$ TeV and $m_{1\/2}=250$ GeV (blue curves) and 2. $m_0=2$ TeV and \n$m_{1\/2}=750$ GeV (red dashed curve), for $\\tan\\beta =10$ and $\\mu >0$.\n}\\label{fig:a0}}\n\nIn Fig. \\ref{fig:tanb}, we show {\\it a}). the relic density and {\\it b}). $|\\Delta_{\\tan\\beta}|$\nversus variation in $\\tan\\beta$ for \n 1. $m_0=1.5$ TeV and $m_{1\/2}=250$ GeV (blue curves) and 2. $m_0=2$ TeV and \n$m_{1\/2}=750$ GeV (red dashed curve), for $A_0=0$ and $\\mu >0$. \nFor most of the $\\tan\\beta$ values, the relic density varies only slowly with $\\tan\\beta$, \nleading to only small contributions to $\\Delta$. When $\\tan\\beta$ becomes of order 50,\nthen $m_A$ is rapidly decreasing, and $\\Gamma_A$ is rapidly increasing, leading to\na high rate of neutralino annihilation through the $A^0$ resonance. In this case, while\nfine-tuning with respect to $m_0$ and $m_{1\/2}$ is low, fine-tuning with respect to $\\tan\\beta$\nis high.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{tbslice.eps}\n\\caption{Plot of {\\it a}). $\\Omega_{\\tilde\\chi^0_1}h^2$ and {\\it b}). $|\\Delta_{\\tan\\beta}|$ \nversus $\\tan\\beta$ for two slices out of mSUGRA parameter space: \n1. $m_0=1.5$ TeV and $m_{1\/2}=250$ GeV (blue curves) and 2. $m_0=2$ TeV and \n$m_{1\/2}=750$ GeV (red dashed curve), for $A_0=0$ and $\\mu >0$.\n}\\label{fig:tanb}}\n\nIn Fig. \\ref{fig:tanbplane}, we show the value of $\\Delta$ including contributions from\nvariation in $m_0$, $m_{1\/2}$ {\\it and} $\\tan\\beta$, for the large values of\n{\\it a}). $\\tan\\beta =53$ and {\\it b}). $\\tan\\beta =55$. \nHere, over essentially all of parameter space, the value of $\\Delta$ has increased to much larger\nvalues than those for $\\Delta_{\\tilde\\chi^0_1}$ as shown in Fig's \\ref{fig:ino53} and \\ref{fig:ino55}. \nThus, inclusion of $\\tan\\beta$ in the \nfine-tuning calculation shows that large values of $\\tan\\beta \\stackrel{>}{\\sim} 50$ result in\nlarge fine-tuning of the relic density.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{dertb5355.eps}\n\\caption{Plot of $\\Delta$ in the $m_0\\ vs.\\ m_{1\/2}$ plane including variation of \n$m_0$, $m_{1\/2}$ and $\\tan\\beta$ for $A_0=0$, $\\mu >0$ and\n{\\it a}). $\\tan\\beta =53$ and {\\it b}). $\\tan\\beta =55$.\n}\\label{fig:tanbplane}}\n\n\n\n\\section{The gravitino problem, leptogenesis, and the re-heat temperature}\n\\label{sec:inoprob}\n\nIn this section, we review the gravitino problem, baryogenesis via\nleptogenesis, and production of mixed axion\/axino dark matter in the \nearly universe. The reader who is familiar with these issues may proceed directly to Sec. \\ref{sec:axDM};\nothers may wish to follow the brief treatment given here and in Ref. \\cite{axdm}.\n\n\\subsection{The gravitino problem}\n\nIn supergravity models, supersymmetry is broken via the superHiggs mechanism.\nThe common scenario is to postulate the existence of a hidden sector which\nis uncoupled to the MSSM sector except via gravity. The superpotential of the \nhidden sector is chosen such that supergravity is broken, which causes \nthe gravitino (which serves as the gauge particle for the superHiggs mechanism)\nto develop a mass $m_{3\/2}\\sim m^2\/M_{Pl}\\sim m_{weak}$. Here, $m$ is a\nhidden sector parameter assumed to be of order $10^{11}$ GeV.\\footnote{\nIn Ref. \\cite{kimnilles}, a link is suggested between \nhidden sector parameters and the PQ breaking scale $f_a$.} \nIn addition to a mass\nfor the gravitino, SSB masses of order $m_{weak}$ are generated for\nall scalar, gaugino, trilinear and bilinear SSB terms.\nHere, we will assume that $m_{3\/2}$ is larger than the lightest\nMSSM mass eigenstate, so that the gravitino essentially decouples from\nall collider phenomenology.\n\nIn all SUGRA scenarios, a potential problem arises for weak-scale\ngravitinos: the gravitino problem. In this case, gravitinos $\\tilde G$ can be\nproduced thermally in the early universe (even though the gravitinos\nare too weakly coupled to be in thermal equilibrium) at a rate which\ndepends on the re-heat temperature $T_R$ of the universe. The produced $\\tilde G$\ncan then decay to various sparticle-particle combinations, with a long \nlifetime of order $1-10^5$ sec (due to the Planck suppressed \ngravitino coupling constant). The late gravitino decays occur\nduring or after BBN, and their energy injection\ninto the cosmic soup threatens to destroy the successful BBN\npredictions of the light element abundances.\nThe precise constraints of BBN on the gravitino mass and $T_R$ are\npresented recently in Ref. \\cite{kohri}. One way to avoid the  \ngravitino problem in the case where $m_{3\/2}\\stackrel{<}{\\sim} 5$ TeV is to maintain \na value of $T_R\\stackrel{<}{\\sim} 10^5$ GeV. Such a low value of $T_R$ rules out many\nattractive baryogenesis mechanisms, and so here instead we assume\nthat $m_{3\/2}\\stackrel{>}{\\sim} 5$ TeV. In this case, the $\\tilde G$ is so heavy that its\nlifetime is of order 1 sec or less, and the $\\tilde G$ decays near the onset of BBN.\nIn this case, values of $T_R$ as large as $10^9$ GeV are allowed.\n\nIn the simplest SUGRA models, one typically finds $m_0= m_{3\/2}$.\nFor more general SUGRA models, the scalar masses are in general\nnon-degenerate and only of order $m_{3\/2}$\\cite{sugmasses}. \nHere for simplicity,\nwe will assume degeneracy of scalar masses, but with $m_0\\ll m_{3\/2}$.\n\n\\subsubsection{Leptogenesis}\n\nOne possible baryogenesis mechanism that requires relatively\nlow $T_R\\sim m_{weak}$ is electroweak baryogenesis. However,\ncalculations of successful electroweak baryogenesis within \nthe MSSM context seem to require sparticle mass spectra\nwith $m_h\\stackrel{<}{\\sim} 120$ GeV, and $m_{\\tilde t_1}\\stackrel{<}{\\sim} 125$ GeV\\cite{cnqw}. \nThe latter requirement\nis difficult (though not impossible) to achieve in the MSSM, and\nis also partially excluded by collider searches for light\ntop squarks\\cite{tev+stop}. \nWe will not consider this possibility further.\n\nAn alternative attractive mechanism-- especially in light of recent\nevidence for neutrino mass-- is thermal leptogenesis\\cite{leptog}. \nIn this scenario,\nheavy right-handed neutrino states $N_i$ ($i=1-3$) \ndecay asymmetrically to leptons\nversus anti-leptons in the early universe. \nThe lepton-antilepton asymmetry is converted to a baryon-antibaryon \nasymmetry via sphaleron effects. The measured baryon abundance can be achieved\nprovided the re-heat temperature $T_R$ exceeds $\\sim 10^9$ GeV\\cite{buchm}. \nThe high $T_R$ value needed here apparently puts this mechanism into \nconflict with the gravitino problem in SUGRA theories.\n\nA related leptogenesis mechanism called non-thermal leptogenesis\ninvokes an alternative to thermal production of heavy neutrinos \nin the early universe. \nIn non-thermal leptogenesis\\cite{NTlepto}, it is possible to have lower\nre-heat temperatures, since the $N_i$ may be generated via inflaton decay.\nThe Boltzmann equations for the $B-L$ asymmetry have been solved numerically \nin Ref. \\cite{imy}.\nThe $B-L$ asymmetry is then converted to a baryon asymmetry via sphaleron \neffects as usual.\nThe baryon-to-entropy ratio is calculated in \\cite{imy}, where it is found\n\\be\n\\frac{n_B}{s}\\simeq 8.2\\times 10^{-11}\\times \\left(\\frac{T_R}{10^6\\ {\\rm GeV}}\\right)\n\\left(\\frac{2M_{N_1}}{m_\\phi}\\right) \\left(\\frac{m_{\\nu_3}}{0.05\\ {\\rm eV}}\\right) \\delta_{eff} ,\n\\ee\nwhere $m_\\phi$ is the inflaton mass and $\\delta_{eff}$ is an effective $CP$ violating phase\nwhich may be of order 1.\nComparing calculation with data (the measured value of $n_B\/s\\simeq 0.9\\times 10^{-10}$), \na lower bound $T_R\\stackrel{>}{\\sim} 10^6$ GeV may be \ninferred for viable non-thermal leptogenesis via inflaton decay.\n\nA fourth mechanism for baryogenesis is Affleck-Dine\\cite{ad} \nleptogenesis\\cite{my}. In this approach, a flat direction \n$\\phi_i =(2H\\ell_i)^{1\/2}$ is identified in the scalar potential,\nwhich may have a large field value in the early universe. When the \nexpansion rate becomes comparable to the SSB terms, the field oscillates,\nand since the field carries lepton number, coherent oscillations about\nthe potential minimum will develop a lepton number asymmetry. The lepton\nnumber asymmetry is then converted to a baryon number asymmetry by sphalerons\nas usual. Detailed calculations\\cite{my} find that the\nbaryon-to-entropy ratio is given by\n\\be\n\\frac{n_B}{s}\\simeq\\frac{1}{23}\\frac{|\\langle H\\rangle |^2  T_R}\n{m_\\nu M_{Pl}^2}\n\\ee\nwhere $\\langle H\\rangle$ is the Higgs field vev, $m_\\nu$ is the mass of the\nlightest neutrino and $M_{Pl}$ is the Planck scale. To obtain the observed \nvalue of $n_B\/s$, values of $T_R\\sim 10^6-10^8$ are allowed for\n$m_{\\nu}\\sim 10^{-9}-10^{-7}$ eV.\n\nThus, to maintain accord with either non-thermal or Affleck-Dine\nleptogenesis, along with constraints from the gravitino problem, we will\naim for $a\\tilde{a}$DM scenarios with $T_R\\sim  10^6-10^8$ GeV.\n\n\\subsection{Mixed axion\/axino dark matter}\n\n\\subsubsection{Relic axions}\n\nAxions can be produced via various mechanisms\nin the early universe. Since their lifetime \n(they decay via $a\\rightarrow\\gamma\\gamma$) turns out to\nbe longer than the age of the universe, \nthey can be a good candidate for dark matter.\nAs we will be concerned here with re-heat temperatures \n$T_R\\stackrel{<}{\\sim} 10^9\\ {\\rm GeV }<f_a\/N$ \n(to avoid overproducing gravitinos in the early universe), \nthe axion production mechanism relevant\nfor us here is just one: \nproduction via vacuum mis-alignment\\cite{absik}. In this mechanism, the axion field\n$a(x)$ can have any value $\\sim f_a$ at temperatures $T\\gg \\Lambda_{QCD}$. As the temperature\nof the universe drops,\nthe potential turns on, and the axion field oscillates and settles to its minimum \nat $-\\bar{\\theta} f_a\/N$ (where $\\bar{\\theta}=\\theta +arg(det\\ m_q)$,\n$\\theta$ is the fundamental strong $CP$ violating Lagrangian parameter and $m_q$ is\nthe quark mass matrix).\nThe difference in axion field before and after potential turn-on corresponds to\nthe vacuum mis-alignment: it produces an axion number density\n\\be\nn_a(t)\\sim {\\frac{1}{2}}m_a(t)\\langle a^2(t)\\rangle ,\n\\ee\nwhere $t$ is  the time near the QCD phase transition.\nRelating the number density to the entropy density allows one to determine the\naxion relic density today\\cite{absik}:\n\\be\n\\Omega_a h^2\\simeq {\\frac{1}{4}}\\left(\\frac{6\\times 10^{-6}\\ {\\rm eV}}{m_a}\\right)^{7\/6}\\theta_i^2\n\\simeq\\frac{1}{4}\\left(\\frac{f_a\/N}{10^{12}\\ \\mathrm{GeV}}\\right)^{7\/6}\\theta_i^2 ,\n\\label{eq:axrd}\n\\ee\nwhere $\\theta_i$ is the initial vacuum mis-alignment angle, with $-\\pi\\stackrel{<}{\\sim} \\theta_i\\stackrel{<}{\\sim}\\pi$.               \nAn error estimate of the axion relic density from vacuum mis-alignment is\nplus-or-minus a factor of three.\nAxions produced via vacuum mis-alignment would constititute {\\it cold} dark matter.\nHowever, in the event that $\\langle a^2(t)\\rangle$ is\ninadvertently small, then much lower values of axion relic density could be allowed.\nAdditional entropy production at $t>t_{QCD}$ can also lower the axion relic abundance.\nTaking the value of Eq.~(\\ref{eq:axrd}) literally, along with $\\theta_i\\simeq 1$,\nand comparing to the WMAP5 measured abundance of CDM in the universe,\none gets an upper bound $f_a\/N\\stackrel{<}{\\sim} 5\\times 10^{11}$ GeV, or a lower bound\n$m_a\\stackrel{>}{\\sim} 10^{-5}$ eV. If we take the axion\nrelic density a factor of three lower, then the bounds change\nto $f_a\/N \\stackrel{<}{\\sim} 1.2\\times 10^{12}$ GeV, and $m_a\\stackrel{>}{\\sim} 4\\times 10^{-6}$ eV.\n\n\\subsubsection{Axinos from neutralino decay}\n\nIf the $\\tilde a$ is the lightest SUSY particle, then the $\\tilde\\chi^0_1$ will no longer\nbe stable, and can decay via $\\tilde\\chi^0_1\\rightarrow \\tilde a\\gamma$.\nThe relic abundance of axinos from neutralino decay\n(non-thermal production, or $NTP$) is given simply by\n\\be\n\\Omega_{\\tilde a}^{NTP}h^2 =\\frac{m_{\\tilde a}}{m_{\\tilde\\chi^0_1}}\\Omega_{\\tilde\\chi^0_1}h^2 ,\n\\label{eq:Oh2_NTP}\n\\ee\nsince in this case the axinos inherit the thermally produced\nneutralino number density.\nThe neutralino-to-axino decay offers a mechanism to shed\nlarge factors of relic density. For a case where $m_{\\tilde\\chi^0_1}\\sim 100$\nGeV and $\\Omega_{\\tilde\\chi^0_1}h^2\\sim 10$ (as can occur in the mSUGRA model\nat large $m_0$ values)\nan axino mass of less than 1 GeV reduces the DM abundance to below\nWMAP-measured levels.\n\nThe lifetime for these decays has been calculated,\nand it is typically in the range of $\\tau (\\tilde\\chi^0_1\\rightarrow \\tilde a\\gamma )\\sim 0.01-1$ sec\\cite{ckkr}.\nThe photon energy injection from $\\tilde\\chi^0_1\\rightarrow\\tilde a\\gamma$ decay\ninto the cosmic soup occurs typically before\nBBN, thus avoiding the constraints that plague the case of a gravitino LSP\\cite{kohri}.\nThe axino DM arising from neutralino decay is generally\nconsidered warm or even hot dark matter for cases with\n$m_{\\tilde a}\\stackrel{<}{\\sim} 1-10$ GeV\\cite{jlm}.\nThus, in the mSUGRA scenario considered here, where $m_{\\tilde a}\\stackrel{<}{\\sim} 1-10$ GeV, we usually get {\\it warm} axino DM from neutralino decay.\n\n\\subsubsection{Thermal production of axinos}\n\nEven though axinos may not be in thermal equilibrium in the early universe, \nthey can still be produced thermally via scattering and decay processes in the cosmic soup.\nThe axino thermally produced (TP) relic abundance has been\ncalculated in Ref. \\cite{ckkr,steffen}, and is given in Ref. \\cite{steffen} using\nhard thermal loop resummation as\n\\be\n\\Omega_{\\tilde a}^{TP}h^2\\simeq 5.5 g_s^6\\ln\\left(\\frac{1.211}{g_s}\\right)\n\\left(\\frac{10^{11}\\ {\\rm GeV}}{f_a\/N}\\right)^2\n\\left(\\frac{m_{\\tilde a}}{0.1\\ {\\rm GeV}}\\right)\n\\left(\\frac{T_R}{10^4\\ {\\rm GeV}}\\right)\n\\label{eq:Oh2_TP}\n\\ee\nwhere $g_s$ is the strong coupling evaluated at $Q=T_R$ and $N$ is the\nmodel dependent color anomaly of the PQ symmetry, of order 1.\nFor reference, we take $g_s(T_R=10^6\\ {\\rm GeV})=0.932$ \n(as given by Isajet 2-loop $g_s$ evolution in mSUGRA), with $g_s$ at \nother values of $T_R$ given by the 1-loop MSSM running value.\nThe thermally produced axinos qualify as {\\it cold} dark matter as long as\n$m_{\\tilde a}\\stackrel{>}{\\sim} 0.1$ MeV\\cite{ckkr,steffen}.\n\n\\section{Fine-tuning in mSUGRA with mixed axion\/axino CDM}\n\\label{sec:axDM}\n\nIn this section, we calculate the fine-tuning parameter for the\ndark matter relic density in models with mixed $a\\tilde{a}$DM: $\\Delta_{a\\tilde a}$.\nContributions to $\\Delta_{a\\tilde a}$ are calculated from both the axion relic density and the \nthermally produced axino relic density. \nWe do not include the non-thermally produced axino relic density as it makes a tiny\ncontribution to the total for the values of $m_{\\tilde a}\\sim 1$ MeV considered here\n(see Figs. 2 and 3 of Ref. \\cite{axdm}). We take the total relic density to be\n\\begin{align}\n\\Omega_{a\\tilde{a}}h^2&=\\Omega_ah^2+\\Omega^{TP}_{\\tilde{a}}h^2\\\\\n\\label{eq:fulldel}&=\\frac{1}{4}\\left(\\frac{f_a\/N}{10^{12}\\ \\mathrm{GeV}}\\right)^{7\/6}\\theta_i^2\n+5.5g^6_s\\ln{\\left(\\frac{1.211}{g_s}\\right)}\\left(\\frac{10^{11}\\ \\mathrm{GeV}}{f_a\/N}\\right)^2\\left(\\frac{m_{\\tilde{a}}}{0.1\\ \\mathrm{GeV}}\\right)\\left(\\frac{T_R}{10^4\\ \\mathrm{GeV}}\\right)\n\\end{align}\nand calculate the total $\\Delta_{a\\tilde a}$ exactly by differentiating \\eqref{eq:fulldel} with respect to \n$f_a\/N$, $T_R$ and $m_{\\tilde{a}}$.\\footnote{Here, one objection may be that the value of $T_R$\ndoes not appear as an explicit Lagrangian parameter. However, in the standard inflationary cosmology,\nthe reheat temperature is related to the inflaton decay width via\n$T_R\\simeq(3\/\\pi^3)^{1\/4}g_*^{-1\/4}(M_{Pl}\\Gamma_\\phi )^{1\/2}$\\cite{kolbturner}, \nwhere $\\Gamma_\\phi$ depends on the inflaton mass and couplings to matter. \nIn this case, a detailed model including the inflaton field $\\phi$ \nwould provide $T_R$ in terms of inflaton Lagrangian parameters. We do not wish to bring such model-dependence\ninto our calculations, so instead just adopt the value of $T_R$ as a fundamental parameter. Also, \nthe value of $m_{\\tilde a}$ will appear as a Lagrangian parameter in the weak scale effective Lagrangian, after the\neffects of SUSY breaking and PQ breaking are taken into account.\n} \nWe find:\n\\bea\n\\Delta_{T_R}&=\\frac{T_R}{\\Omega_{a\\tilde{a}}h^2}\\frac{\\partial \\Omega_{a\\tilde{a}}h^2}{\\partial T_R}=\\frac{T_R}{\\Omega_{a\\tilde{a}}h^2}5.5g^6_s\\ln{\\left(\\frac{1.211}{g_s}\\right)}\\left(\\frac{10^{11}\\ \\mathrm{GeV}}{f_a\/N}\\right)^2\\left(\\frac{m_{\\tilde{a}}}{0.1\\ \\mathrm{GeV}}\\right)\\left(\\frac{1}{10^4\\ \\mathrm{GeV}}\\right)\\\\\n\\Delta_{m_{\\tilde{a}}}&=\\frac{m_{\\tilde{a}}}{\\Omega_{a\\tilde{a}}h^2}\\frac{\\partial \\Omega_{a\\tilde{a}}h^2}{\\partial m_{\\tilde{a}}}=\\frac{m_{\\tilde{a}}}{\\Omega_{a\\tilde{a}}h^2}5.5g^6_s\\ln{\\left(\\frac{1.211}{g_s}\\right)}\\left(\\frac{10^{11}\\ \\mathrm{GeV}}{f_a\/N}\\right)^2\\left(\\frac{1}{0.1\\ \\mathrm{GeV}}\\right)\\left(\\frac{T_R}{10^4\\ \\mathrm{GeV}}\\right)\n\\eea\nand\n\\begin{align}\n\\Delta_{f_a\/N}&=\\frac{f_a\/N}{\\Omega_{a\\tilde{a}}h^2}\\frac{\\partial \\Omega_{a\\tilde{a}}h^2}{\\partial f_a\/N}\\\\\\begin{split}\n&=\\frac{f_a\/N}{\\Omega_{a\\tilde{a}}h^2}\\left[\\frac{7}{24}\\left(\\frac{1}{10^{12}\\ \\mathrm{GeV}}\\right)^{7\/6}\\left(f_a\/N\\right)^{1\/6}\\theta_i^2\\right.\\\\\n&\\qquad\\qquad-\\left. 11g^6_s\\ln{\\left(\\frac{1.211}{g_s}\\right)}\\left(10^{11}\\ \\mathrm{GeV}\\right)^2\\left(\\frac{1}{f_a\/N}\\right)^3\\left(\\frac{m_{\\tilde{a}}}{0.1\\ \\mathrm{GeV}}\\right)\\left(\\frac{T_R}{10^4\\ \\mathrm{GeV}}\\right)\\right] ,\n\\end{split}\\end{align}\nand\n\\be\n\\Delta_{\\theta_i}=\\frac{\\theta_i}{\\Omega_{a\\tilde{a}}h^2}\\frac{\\partial \\Omega_{a\\tilde{a}}h^2}{\\partial \\theta_i}=2\\frac{\\Omega_ah^2}{\\Omega_{a\\tilde{a}}h^2}.\n\\ee\nThe total fine tuning parameter is then given by\n\\begin{equation}\n\\Delta_{a\\tilde a} =\\sqrt{\\Delta_{T_R}^2+\\Delta_{m_{\\tilde{a}}}^2+\\Delta_{f_a\/N}^2+\\Delta_{\\theta_i}^2} .\n\\end{equation}\n\nWe plot our first results in the $f_a\/N\\ vs.\\ T_R$ plane, keeping $m_{\\tilde a}$ fixed at 1 MeV:\nsee Fig. \\ref{fig:max-3}.\nIn frame {\\it a})., we show contours of $\\Omega_{a\\tilde a}h^2$. The green region gives\n$\\Omega_{a\\tilde a}<0.11$, and so is consistent with WMAP. In frame {\\it b})., we show regions\nof fine-tuning $\\Delta_{a\\tilde a}$. The scale is shown on the right edge of the plot. Note in this case\nthe entire plane has $\\Delta_{a\\tilde a}<2.5$, so there is very little fine-tuning across the entire\nplane of parameter space. The left region, color-coded dark blue, is the region of \ndominantly thermally produced {\\it axino CDM}, whilst the right-most region, color-coded lighter blue,\nis dominantly {\\it axion CDM}. In this region, the fine-tuning parameter $\\Delta_{a\\tilde a}\\simeq 2.3$.\nThe intermediate region, shaded by yellow and purple bands, is the region of mixed $a\\tilde{a}$CDM: \nthe purple band has very low fine-tuning, with $\\Delta_{a\\tilde a}<1.4$. The contour\nwhere mixed $a\\tilde{a}$CDM saturate the WMAP measured value is shown by the green dashed line.\nThe region where the highest values of $T_R$ are found coincide with the region of lowest fine-tuning, \nwith a nearly equal mix of axion and axino CDM.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftma10m3.eps}\n\\caption{Plot of {\\it a}). contours of axion\/axino relic density\n$\\Omega_{a\\tilde a}h^2$ in the $f_a\/N\\ vs.\\ T_R$ plane\nand {\\it b}). regions of fine-tuning $\\Delta_{a\\tilde a}$ for\nfixed axino mass $m_{\\tilde a}=1$ MeV.\nThe green region in {\\it a}). has $\\Omega_{a\\tilde a}h^2\\le 0.11$. The green dashed line\nin frame {\\it b}). is where $\\Omega_{a\\tilde a}h^2= 0.11$.\n}\\label{fig:max-3}}\n\nTo gain additional perspective, in Fig. \\ref{fig:3da} we show the mixed axion\/axino relic density \nas a 3-d plot in the $f_a\/N\\ vs.\\ T_R$ plane for $m_{\\tilde a}=1$ MeV. The level of fine-tuning, corresponding\nto the slope of the surface, is rather low throughout, since there are no regions with a steep slope.\nThe fine-tuning is minimal along the trough running through the right-center of the plot. \n\\FIGURE[t]{\n\\includegraphics[width=14cm]{axioden.eps}\n\\caption{A 3-d plot of axion\/axino relic density in the $f_a\/N\\ vs.\\ T_R$ \nplane of the $a\\tilde a$ augmented mSUGRA model.\n}\\label{fig:3da}}\n\nTo better understand the situation with mixed $a\\tilde{a}$CDM, we show in Fig. \\ref{fig:slice_max-3}\na slice of our Fig. \\ref{fig:max-3} with constant $T_R=10^5$ GeV.\nIn frame {\\it a})., we see that the value of $\\Omega_{a\\tilde a}h^2$ initially drops as\n$f_a\/N$ increases. This is in the region of dominant axino CDM, and increasing $f_a\/N$\ndecreases the axino coupling strength, and hence suppresses its thermal production in the early universe.\nAs $f_a\/N$ increases further, the relic abundance of axions steadily increases, until\naround $f_a\/N\\sim 2\\times 10^{11}$ GeV there is an upswing in the relic abundance. This is the \nstable fine-tuning region since small fluctuations of parameters about this point do not substantially alter the\naxino\/axino relic density. The fine-tuning parameter $\\Delta_{a\\tilde a}$ is shown in frame {\\it b})..\nHere, we see that the fine-tuning is slightly high in the region of mainly axino CDM, with low\n$f_a\/N$, but reaches a minimum at the point of equal admixture. The value of $\\Delta_{a\\tilde a}$ doesn't extend\nall the way to zero, in spite of the zero slope shown, because $\\Delta_{a\\tilde a}$ still varies with \n$m_{\\tilde a}$, $T_R$ and $\\theta_i$. The fine-tuning parameter increases to the analytic value of $2.3$ as\n$f_a\/N$ increases further, into the region of mainly axion CDM.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{slicefatma10m3tr105.eps}\n\\caption{Plot of {\\it a}). axion\/axino relic density\n$\\Omega_{a\\tilde a}h^2$ \nand {\\it b}). fine-tuning parameter $\\Delta_{a\\tilde a}$ \nversus $f_a\/N$ for fixed axino mass $m_{\\tilde a}=1$ MeV and fixed re-heat temperature\n$T_R=10^5$ GeV.\n}\\label{fig:slice_max-3}}\n\nA similar plot to Fig. \\ref{fig:max-3} is shown in Fig. \\ref{fig:max-4}, but in this case taking\n$m_{\\tilde a}=0.1$ MeV. Note that this value yields the approximate dividing line given in Refs. \\cite{ckkr,steffen}\nbelow which the thermally produced axinos would be mainly warm DM instead of cold DM. In any case, in frame {\\it a}).,\nwe see that the WMAP allowed region has expanded, and now values of $T_R$ as high as $5\\times10^{6}$ GeV are allowed, \nmaking the scenario consistent with at least non-thermal leptogenesis. The region of maximal $T_R$ also\ncoincides with the region of least fine-tuning, with a roughly equal admixture of axion and thermally produced axino DM.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{ftma10m4.eps}\n\\caption{Plot of {\\it a}). contours of axion\/axino relic density\n$\\Omega_{a\\tilde a}h^2$ in the $f_a\/N\\ vs.\\ T_R$ plane\nand {\\it b}). regions of fine-tuning $\\Delta_{a\\tilde a}$ for\nfixed axino mass $m_{\\tilde a}=0.1$ MeV.\nThe green region in {\\it a}). has $\\Omega_{a\\tilde a}h^2\\le 0.11$. The green dashed line\nin frame {\\it b}). is where $\\Omega_{a\\tilde a}h^2= 0.11$.\n}\\label{fig:max-4}}\n\nIn Fig. \\ref{fig:ata_fafix1} we show the contours of relic density $\\Omega_{a\\tilde a}h^2$ in the \n$m_{\\tilde a}\\ vs.\\ T_R$ plane for fixed value of $f_a\/N=4.88\\times 10^{11}$ GeV. The large value of\n$f_a\/N$ yields a scenario with mainly axion CDM when the WMAP measured abundance is saturated.\nThe green shaded region in frame {\\it a}). is WMAP-allowed. The red dashed line shows the approximate\ndividing line between warm and cold thermally produced axinos. In this case, the demarcation line is\nlargely irrelevant, since if $\\Omega_{a\\tilde a}h^2\\simeq 0.11$, almost all the DM is composed of cold axions,\nand a tiny admixture of warm axinos would be allowed. \nIn frame {\\it b})., we show the regions of fine-tuning $\\Delta_{a\\tilde a}$. Since the WMAP-allowed region\ncoincides with mainly axion CDM, the fine-tuning along the green dashed line is always low:\n$\\Delta_{a\\tilde a}\\sim 2.3$.\nNote that in the scenario with mainly axion CDM, the value of $T_R$ can easily reach to well over \n$10^7$ GeV, allowing for non-thermal or Affleck-Dine leptogenesis.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{matfa4p88.eps}\n\\caption{Plot of {\\it a}). contours of axion\/axino relic density\n$\\Omega_{a\\tilde a}h^2$ in the $m_{\\tilde a}\\ vs.\\ T_R$ plane\nand {\\it b}). regions of fine-tuning $\\Delta_{a\\tilde a}$ for\nfixed $f_a\/N=4.88\\times 10^{11}$ GeV (which gives mainly axion CDM along the line\nof $\\Omega_{a\\tilde a}=0.11$).\nThe green region in {\\it a}). has $\\Omega_{a\\tilde a}h^2\\le 0.11$. The green dashed line\nin frame {\\it b}). is where $\\Omega_{a\\tilde a}h^2= 0.11$.\nThe region to the left of red-dashed line gives thermally produced {\\it warm} \naxino dark matter.\n}\\label{fig:ata_fafix1}}\n\nIn Fig. \\ref{fig:ata_fafix2} we show the $m_{\\tilde a}\\ vs.\\ T_R$ plane for $f_a\/N=3\\times 10^{11}$ GeV, \nwhich gives roughly an equal admixture of axion and thermally produced axino DM. In this case, \nthe value of $T_R$ reaches beyond $10^8$ GeV, although for very low values of $m_{\\tilde a}$ where\nit is expected that the axino will be warm DM. For this scenario, it is unclear how much mixture of\nwarm and cold dark matter is cosmologically allowed. To answer the question, the velocity profile of the\nwarm axinos would have to be fed into $n-body$ simulations of large scale structure formation, \nto see how well such a mixed warm\/cold DM scenario fits the data. At present, we are unaware of such studies.\nIn frame {\\it b})., we see that the line of WMAP-saturated abundance lies nearly on top of the region of lowest\nfine-tuning, with $\\Delta_{a\\tilde a}<1.4$. \n\\FIGURE[t]{\n\\includegraphics[width=14cm]{matfa3.eps}\n\\caption{Plot of {\\it a}). contours of axion\/axino relic density\n$\\Omega_{a\\tilde a}h^2$ in the $m_{\\tilde a}\\ vs.\\ T_R$ plane\nand {\\it b}). regions of fine-tuning $\\Delta_{a\\tilde a}$ for\nfixed $f_a\/N=3\\times 10^{11}$ GeV (which gives a 50-50 mix of axion\/axino DM along the line\nof $\\Omega_{a\\tilde a}=0.11$).\nThe green region in {\\it a}). has $\\Omega_{a\\tilde a}h^2\\le 0.11$. The green dashed line\nin frame {\\it b}). is where $\\Omega_{a\\tilde a}h^2= 0.11$.\nThe region to the left of red-dashed line gives thermally produced {\\it warm} \naxino dark matter.\n}\\label{fig:ata_fafix2}}\n\nIn Fig. \\ref{fig:ata_fafix3}, we show again the $m_{\\tilde a}\\ vs.\\ T_R$ plane, but this time for\n$f_a\/N=1\\times 10^{11}$ GeV, which gives mainly {\\it thermally produced axino} DM. In this case\nthe region to the left of the red-dashed line should likely be disallowed, since the dominant\nform of DM will be warm, rather than cold. The region to the right of the $m_{\\tilde a}=0.1$ MeV line,\nin the WMAP-allowed region, only allows for $T_R$ to reach a max of $10^6$ GeV. Furthermore,\nfrom frame {\\it b})., we see that the fine-tuning parameter in this case for the\nWMAP-saturated region along the green dashed line is somewhat higher, reaching $\\Delta_{a\\tilde a}\\sim 2$.\n\\FIGURE[t]{\n\\includegraphics[width=14cm]{matfa1.eps}\n\\caption{Plot of {\\it a}). contours of axion\/axino relic density\n$\\Omega_{a\\tilde a}h^2$ in the $m_{\\tilde a}\\ vs.\\ T_R$ plane\nand {\\it b}). regions of fine-tuning $\\Delta_{a\\tilde a}$ for\nfixed $f_a\/N=1\\times 10^{11}$ GeV (which gives mainly axino DM along the line\nof $\\Omega_{a\\tilde a}=0.11$).\nThe green region in {\\it a}). has $\\Omega_{a\\tilde a}h^2\\le 0.11$. The green dashed line\nin frame {\\it b}). is where $\\Omega_{a\\tilde a}h^2= 0.11$.\nThe region to the left of red-dashed line gives thermally produced {\\it warm} \naxino dark matter.\n}\\label{fig:ata_fafix3}}\n\n\n\n\\section{Summary and conclusions}\n\\label{sec:conclude}\n\nIn this paper, we have examined the fine-tuning associated \nwith the relic density of dark matter in the minimal\nsupergravity model. We have calculated a measure of fine-tuning assuming two\nscenarios for SUSY dark matter: 1. neutralino dark matter with\nfine-tuning parameter $\\Delta_{\\tilde\\chi^0_1}$, and 2. mixed axion\/axino dark matter\nwith fine-tuning parameter $\\Delta_{a\\tilde a}$. \n\nIn the case of neutralino dark matter, we find that the WMAP-allowed regions of\nmSUGRA such as the stau co-annihilation region, the HB\/FP region and the \nlight Higgs $h$-resonance annihilation region, are all rather highly fine-tuned, \nespecially the HB\/FP region, where $\\Delta_{\\tilde\\chi^0_1}$ ranges from 20-100. \nOnly mild fine-tuning is found in the low $m_0$, low $m_{1\/2}$ region \nwhere stau co-annihilation and bulk annihilation through $t$-channel slepton\nexchange overlap. If one moves to large $\\tan\\beta\\sim 50$, then larger\nregions of parameter space which are consistent with WMAP occur. These large\n$\\tan\\beta$ regions have modest fine-tuning versus $m_0$ and $m_{1\/2}$, but very \nlarge fine-tuning versus $\\tan\\beta$.\n\nIf instead we assume that dark matter is composed of an axion\/axino\nadmixture, rather than neutralinos, then we find that the relic density \nfine-tuning parameter is generically much lower: $\\Delta_{a\\tilde a}\\sim 1.3-2.5$\nthroughout parameter space. \nHere, we have assumed the existence of a light axino with mass $m_{\\tilde a}\\sim$ keV-MeV.\nSuch a light axino opens up all of mSUGRA parameter space to being WMAP allowed, \nsince now the neutralino decays via $\\tilde\\chi^0_1\\rightarrow \\tilde a\\gamma$. If the DM is dominated by\nthermally produced axinos, then the re-heat temperature $T_R$ is generally lower than\n$10^6$ GeV unless the axinos are actually warm dark matter ($m_{\\tilde a}\\stackrel{<}{\\sim} 100$ keV), \nso this scenario seems rather unlikely. However, if the PQ breaking scale $f_a\/N$ is large, then\nthe DM can be either a nearly equal axion\/axino admixture, in which case fine-tuning is lowest\n($\\Delta_{a\\tilde a}\\sim 1.3$), or a dominantly axion mixture (in which case\n$\\Delta_{a\\tilde a}\\sim 2.3$). Either scenario easily admits $T_R>10^6$ GeV, which can allow for\nnon-thermal leptogenesis to occur.\n\nThe consequences of the mixed $a\\tilde{a}$CDM scenario for future dark matter searches is as follows.\nFor collider searches, we expect much the same collider signatures as in the mSUGRA model with\nneutralino dark matter, since we assume the $\\tilde\\chi^0_1$ is the NLSP, and decays far outside the collider \ndetectors. \nHowever, {\\it all} of mSUGRA parameter space is now WMAP-allowed, instead of just the special\nco-annihilation, HB\/FP region and resonance annihilation regions. As shown in\nRef. \\cite{axdm}, the regions of WMAP-allowed neutralino CDM yield the lowest\nvalues of $T_R$, and so the stau co-annihilation, HB\/FP region and $h$ resonance\nannihilation regions are most dis-favored for the case of mixed $a\\tilde a$CDM.\n\nAs far as WIMP searches go, in the mixed \n$a\\tilde{a}$CDM scenario, we expect no positive signals if $m_{\\tilde\\chi^0_1}>m_{\\tilde a}$. If \n$m_{\\tilde a}>m_{\\tilde\\chi^0_1}$, then the $\\tilde\\chi^0_1$ would still be stable (assuming $R$-parity conservation) and WIMP\ndirect and indirect detection signals are still possible\\cite{njp}. In the case of large axion relic abundance,\nwhich appears to us to be the favored scenario, then a positive signal at relic axion search experiments\nsuch as ADMX might be expected\\cite{admx}, although solar axion searches are less likely to achieve positive results,\nsince large values of $f_a\/N$ are favored, leading to small axion\/axino couplings.\n\nOur analysis has been based on the admittedly subjective basis of fine-tuning of the\nrelic density of dark matter relative to model input parameters. We note here that\nthe mSUGRA model already needs substantial fine-tuning in the electroweak sector in order to\naccomodate the relatively light $Z$ boson mass in the face of limits on the soft \nSUSY breaking parameters\\cite{ewft} (the little hierarchy problem). Our philosophy here is \nthat less fine-tuning is better, and high fine-tuning in one sector is better than \nhigh fine-tuning in two sectors, {\\it e.g.} electroweak and dark matter sectors.\n\nWhile our analysis has been restricted to the mSUGRA SUSY model, one might ask \nhow general our conclusions might be.\nIn SUSY models based on gravity mediation, with a neutralino LSP, the\nDM relic density is {\\it generically too high} unless some special mechanism is acting to\nenhance the neutralino annihilation cross section in the early universe.\\footnote{\nDiscussion on numerous different SUGRA models with non-universality has been explored in \nRef. \\cite{wtn}.}\nFor instance, in SUSY models with non-universality, instead of stau or stop co-annihilation, \none might have sbottom or sneutrino co-annihilation, or\nbino-wino co-annihilation: in any case, the mass gap between co-annihilating \nparticles must be fine-tuned to obtain agreement with the measured dark matter abundance. \nIn non-universal models with a {\\it well-tempered neutralino}\\cite{wtn}, where\nthe neutralino bino-higgsino or bino-wino composition is adjusted to fit the \nmeasured relic density, \nother parameters (Higgs soft masses, gaugino masses) must be fine-tuned to get just the\nright ``tempering'', as occurs in the mSUGRA HB\/FP region. \nIn other models, Higgs soft mass terms can be adjusted to allow $2m_{\\tilde\\chi^0_1}$ to sit atop the \n$A$ resonance; but again, in this case, parameters must be\nfine-tuned (unless $\\tan\\beta$ is large, which also occurs in mSUGRA). The case where the SUSY \nneutralino abundance is not fine-tuned has long been noted: it is where squarks and sleptons\nare so light that $t$-channel annihilation channels are large. \nHowever, LEP2 search limits now essentially exclude all these regions.\nThus, although we restrict our analysis here to the mSUGRA model, we feel this model provides a sort of\nmicrocosm for general SUSY models, in that it illustrates many of the \nfeatures common to all SUSY models.\n\nOur main conclusion is this. In the world HEP community, a tremendous effort is underway to explore\nfor WIMP cold dark matter, based partly on the view that SUSY models naturally \ngive rise to the ``WIMP-miracle'', and an excellent WIMP candidate for CDM. \nWe have shown here that at least for the\nparadigm SUSY model-- mSUGRA-- usually a large overabundance of neutralino CDM is produced, unless one\nlies along a region of very high fine-tuning, where a slight change in model parameters leads to a large\nchange in relic density: this equates to a high degree of relic density fine-tuning. Alternatively, if one\nassumes the PQWW solution to the strong CP problem within SUSY models, and a very light axino with\n$m_{\\tilde a}$ of the order of MeV, then along with an elegant solution to the strong CP problem, one obtains a \nmixed axion\/axino relic density with much less fine-tuning. Given our results, we would advocate that a \nmuch increased share of HEP resources be given to relic axion searches, where the global search effort \nhas been much more limited.\n\n\n\\acknowledgments\n\nWe thank H. Summy for discussions.\nThis research was supported in part by the U.S. Department of Energy.\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nThe light field imaging and 3D display markets are growing rapidly. In recent years, we have seen autostereoscopic displays emerge as a potential alternative to stereoscopic 3D displays as it supports stereopsis and motion parallax from different viewing directions~\\cite{surman2014towards, balogh2007holovizio,pseudo3d, glassesfree3d}. Parallax-based and lenticular-based 3D display technology has still not reached up to the standards to simultaneously provide direction-dependent outputs without losing out on the resolution in reconstructing dense light fields. Emerging multi-layered light field displays provide continuous motion parallax, greater depth-of-field, and wider field-of-view, which are critical for reproducing realistic 3D vision. Tensor or multi-layered displays can accurately reproduce multi-view images or light fields simultaneously with high resolution using just a few light attenuating layers \\cite{li2020light, wetzstein2012tensor,sharma2016novel}.\n\nThe typical structure of a multi-layered display is shown in Fig.~\\ref{fig:backlight layers}. It consists of light-attenuating pixelized layers (\\textit{e.g.}, LCD panels) stacked in front of a backlight. On each layer, the transmittance of pixels can be controlled independently by carrying out multiplicative operations. Fig.~\\ref{fig:multiplicative layers config} illustrates the light rays which pass through different combinations of pixels in stacked layers depending on the viewing directions. Efficient representation and coding of light rays in multi-layered 3D displays are essential for adaptation on different auto-stereoscopic platforms. \n\n\\begin{figure}\n\\centering \n      \\includegraphics[width=0.34\\textwidth]{.\/Figures\/backlight.PNG}\n      \\caption{\\footnotesize Structure of layered light field display}\n      \\label{fig:backlight layers}\n\\end{figure}\n\n\nExisting light field coding approaches are based on raw lenslet image~\\cite{RwlensletRef1_li2014efficient,RwlensletRef2_perra2016high,RwlensletRef3_li2016compression,RwlensletRef4_monteiro2017light,RwlensletRef5_liu2019content}, geometry information~\\cite{RwDispRef2_jiang2017light,RwDispRef1_zhao2017light,RwEpiRef2_ahmad2020shearlet,RwEpiRef3_chen2020light}, scene content information \\cite{RwCbRef1_hu2020adaptive}, disparity information~\\cite{RwDispRef1_zhao2017light,RwDispRef2_jiang2017light,RwDispRef3_dib2020local}, epipolar plane image-based and multiplane image-based \\cite{RwEpiRef1_vagharshakyan2017light,RwEpiRef2_ahmad2020shearlet,RwEpiRef3_chen2020light}, view prediction based learning schemes \\cite{RwVsRef3_huang2019light,RwVsRef4_heriard2019light, RwDeepRef6_schiopu2019deep,RwDeepRef8_liu2021view} or methods considering light field data as a pseudo video sequence \\cite{RwPsvRef1_liu2016pseudo,RwPsvRef3_ahmad2017interpreting,RwPsvRef4_ahmad2019computationally,RwPsvRef5_gu2019high}. These compression approaches are not specifically designed for tensor or multi-layered displays.\n\n\n\nOur previous work on light field coding exploits the spatial and temporal correlations among light field multiplicative layers~\\cite{ravishankar2021flexible}. We handled the inherent redundancies in light fields by approximating multiplicative layers in the image-based spatial domain. In this work, we propose a novel hierarchical coding scheme for light field compression based on a hybrid multiplicative layers \\cite{maruyama2020comparison} and Fourier disparity layers representation \\cite{le2019fourier}. The current approach efficiently deals with elimination of spatial, temporal and non-linear redundancies present in the light field view subsets while working in an integrated spatial and Fourier domain representation. This is a generalized coding scheme applicable to a variety of autostereoscopic displays, in particular, useful for tensor or multi-layer light field displays~\\cite{surman2014towards, balogh2007holovizio,pseudo3d, glassesfree3d}. It offers much more bitrate savings and adaptability to different coding approaches and also achieves the goal of covering a range of multiple bitrates in a single unified system. Apart from supporting multi-view and layered 3D displays, our model can complement other light field coding schemes based on learning networks to support various bitrates as well~\\cite{RwDeepRef3_bakir2018light, RwDeepRef4_zhao2018light,RwDeepRef5_wang2019region,RwDeepRef6_schiopu2019deep, RwDeepRef7_jia2018light,RwDeepRef8_liu2021view}.\n\nIn the proposed coding scheme, the input light field is divided into view subsets based on pre-defined Circular and Hierarchical prediction orders (Fig.~\\ref{fig:view subset orders}). Three optimized multiplicative layers for each view subset using convolutional neural networks (CNN) are constructed. The key idea in our proposed coding scheme is to reduce the dimensionality of stacked multiplicative layers using the randomized Block-Krylov singular value decomposition (BK-SVD) \\cite{musco2015randomized}. Factorization derived from BK-SVD effectively exploits the high spatial correlation between the multiplicative layers and approximates the light field subsets for varying low ranks. Encoding of these low-rank approximated subset multiplicative layers using HEVC codec \\cite{sullivan2012overview} is performed to eliminate intra-view and inter-view redundancies further. Thus, our scheme approximates multiplicative layers of target light field view subsets for multiple ranks and quantization parameters (QPs) in the first stage. The view subsets are then reconstructed from the decoded layers. \n\n\n\n\\begin{figure}[t!]\n    \\centering \n      \\includegraphics[width=0.38\\textwidth]{.\/Figures\/mult_layers.PNG}\n      \\caption{\\footnotesize Configuration of multiplicative layers}\n      \\label{fig:multiplicative layers config}\n\\end{figure}\n\nIn the second stage, the processing of the entire approximated light field is done in the Fourier domain, following a hierarchical coding procedure. A Fourier Disparity Layer (FDL) calibration is performed to estimate disparity values and angular coordinates of each light field view \\cite{dib2019light}. These essential parameters provide additional information for the FDL construction and view prediction. They are then transmitted to the decoder as metadata. Next, we split the approximated light field into subsets as identified by four scanning or prediction orders. The first set of views are encoded and utilized for constructing the FDL representation. This FDL representation synthesizes the succeeding view subsets. The remaining correlations between the prediction residue of synthesized views and the approximated subset are further eliminated by encoding the residual signal using HEVC. The set of views obtained from decoding the residual are employed to refine the FDL representation and predict the next subset of views with improved accuracy. This hierarchical procedure continues iterating until all light field views are coded. The critical advantages of the proposed hybrid layered representation and coding scheme are:\n\n\\begin{itemize} \n\\item{Our scheme efficiently exploits spatial, temporal, and non-linear redundancies between adjacent sub-aperture images in the light field structure within a single integrated framework. The scheme leverages the benefits of hybrid multiplicative layers and Fourier disparity layers representation in our hierarchical coding and prediction model for gaining superior compression efficiency without compromising reconstruction quality. In the first stage, we process light field in the spatial domain and remove the intra-view and inter-view redundancies among subset multiplicative layers. In the second stage, while working in the Fourier domain, our scheme ensures to eliminate the non-linear redundancies among adjacent views in both horizontal and vertical directions that exhibit high similarities. Experiments with various real light fields following different scanning orders demonstrate superior compression performance of our proposed model.\n}\n\\item{The scheme is versatile to realize a range of multiple bitrates within a single integrated system trained using few convolutional neural networks. This characteristic of the proposed model complements existing light field coding systems or methods which usually support only specific bitrates during compression using multiple networks. Our coding model is adaptable to support various computational, multi-view auto-stereoscopic platforms, table-top, head-mounted displays, or mobile platforms by optimizing the bandwidth for a given target bitrate.\n}\n\\end{itemize}\n\nA shorter version of this work has been accepted for publication at IEEE SMC 2021~\\cite{DBLP:journals\/corr\/abs-2104-09378}. The current journal version is an extension that elaborates on two more light field scanning patterns for view subsets. It includes new detailed results and extensive analysis of the performance of the proposed light field coding model. The rest of this article is organized into four major sections. Section~\\ref{rw} describes various existing light field compression approaches and their shortcomings. The proposed layered representation and coding scheme for multi-view displays is discussed in Section~\\ref{pm} in detail. We have elaborated our experiments specifying the implementation, results, and analysis in Section~\\ref{ra}. Lastly, the conclusion with comprehensive findings of our proposed scheme and implications of future work are presented in Section~\\ref{con}.\n\n\n\n\\begin{figure*}\n\\centering \n \\begin{subfigure}{0.4\\linewidth}\n     \\includegraphics[width=\\linewidth]{.\/Figures\/order_c2.png}\n     \\caption{\\footnotesize Circular-2}\n \\end{subfigure}\n \\begin{subfigure}{0.4\\linewidth} \n     \\includegraphics[width=\\linewidth]{.\/Figures\/order_c4.png}\n     \\caption{\\footnotesize Circular-4}\n \\end{subfigure} \n \\begin{subfigure}{0.4\\linewidth}\n     \\includegraphics[width=\\linewidth]{.\/Figures\/order_h2.png}\n     \\caption{\\footnotesize Hierarchical-2}\n \\end{subfigure}\n \\begin{subfigure}{0.4\\linewidth} \n     \\includegraphics[width=\\linewidth]{.\/Figures\/order_h4.png}\n     \\caption{\\footnotesize Hierarchical-4}\n \\end{subfigure} \n \\caption{\\footnotesize The light field view prediction orders $C_2$, $C_4$, $H_2$ \\& $H_4$. Views in blue, green, yellow \\& orange form the first, second, third and fourth subset respectively.}\n \\label{fig:view subset orders}\n\\end{figure*}\n\n\\section{Related Works}\\label{rw}\n\nThe light field image or plenoptic image contains information about the intensity and direction of light rays in space \\cite{levoy1996light,gortler1996lumigraph}. It involves a large volume of data. Thus, the storage and transmission requirements for light fields are tremendous. Efficient compression techniques leveraging the data redundancy in both the spatial and angular domains of light fields are essential. Compression or coding techniques for light fields are often broadly classified into two categories; lenslet-based or approaches based on perspective sub-aperture images. \n\nThe direct compression using raw light field lenslet image captured by plenoptic camera exploits the spatial redundancy between the microimages~\\cite{RwlensletRef1_li2014efficient,RwlensletRef2_perra2016high,RwlensletRef3_li2016compression,RwlensletRef4_monteiro2017light,RwlensletRef5_liu2019content}. This approach is usually based on the existing image\/video coding standards, such as JPEG \\cite{jpeg_pennebaker1992jpeg} or high efficiency video coding (HEVC) \\cite{hevc_sullivan2012overview}. The unique light field structural characteristics make it harder to predict the lenslet image regions with complex textures accurately. Moreover, there are numerous microimages in the raw light field image which need careful handling. The microimages are of low resolution and require customized reshaping before being fed into an HEVC encoder. Ideally, the lenslet-based compression solutions need to transmit camera parameters for further processing, and thus increase the coding burden on the compressed data stream. These drawbacks encouraged researchers to extract the light field sub-aperture images (SAIs) from raw plenoptic images and explore compression possibilities on these views. \n\n\nThere are various coding methods that directly take light field SAIs as the input. They are broadly categorized as content-based compression \\cite{RwCbRef1_hu2020adaptive}, disparity-based \\cite{RwDispRef1_zhao2017light,RwDispRef2_jiang2017light,RwDispRef3_dib2020local}, epipolar plane image-based and multiplane image-based \\cite{RwEpiRef1_vagharshakyan2017light,RwEpiRef2_ahmad2020shearlet,RwEpiRef3_chen2020light}, pseudo sequence based \\cite{RwPsvRef1_liu2016pseudo,RwPsvRef2_li2017pseudo,RwPsvRef3_ahmad2017interpreting,RwPsvRef4_ahmad2019computationally,RwPsvRef5_gu2019high}, view synthesis based \\cite{RwVsRef1_senoh2018efficient,RwVsRef2_huang2018view,RwVsRef3_huang2019light,RwVsRef4_heriard2019light}, and learning-based compression methods \\cite{RwDeepRef3_bakir2018light, RwDeepRef4_zhao2018light,RwDeepRef5_wang2019region,RwDeepRef6_schiopu2019deep, RwDeepRef7_jia2018light,RwDeepRef8_liu2021view}.\n\nDisparity-based compression methods approximate particular views by the weighted sum of other views \\cite{RwDispRef1_zhao2017light} or apply the homography-based low-rank approximation method called HLRA \\cite{RwDispRef2_jiang2017light}. This approach depends on how much the disparities across views vary and may not optimally reduce the low-rank approximation error for light fields with large baselines. Dib et al.~\\cite{RwDispRef3_dib2020local} proposed a novel parametric disparity estimation method to support the low-rank approximation using super rays. They efficiently exploit redundancy across the different views compared to the homography-based alignment. A shearlet transform-based method presented in \\cite{RwEpiRef2_ahmad2020shearlet} categorizes the SAIs into key and decimated views. The scheme performs well under low bitrates. The decimated views are predicted from the compressed key views and a residual bitstream. Chen et al. \\cite{RwEpiRef3_chen2020light} used multiplane representation and strongly reduced the calculation burden on the decoder.\n\n\n\\begin{figure*}\n    \\centering \n    \\begin{subfigure}{\\linewidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/workflow_enc.jpg}\n      \\caption{\\footnotesize}\n      \\label{fig:workflow_encoding}\n    \\end{subfigure}\n     \\begin{subfigure}{0.6\\linewidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/workflow_dec.PNG}\n      \\caption{\\footnotesize}\n      \\label{fig:workflow_decoding}\n    \\end{subfigure}\n\\caption{\\footnotesize Three components of proposed light field coding scheme. (a) Overview of the encoding scheme (b) Overview of the decoding scheme. }\n\\label{fig:main workflow}\n\\end{figure*}\n\n\n\nThe views are reordered as pseudo sequences in predictive compression methods. Liu et al. \\cite{RwPsvRef1_liu2016pseudo} in their symmetric 2D hierarchical order, compresses the central view first as the I-frame (intra frame) followed by the remaining views as P-frames. Their prediction structure ensures inter-view prediction from adjacent views.  Another proposal by Li et al. \\cite{RwPsvRef2_li2017pseudo} involves dividing all the light field views into four quadrants and adopting a hierarchical coding structure within each quadrant. Ahmad et al. \\cite{RwPsvRef3_ahmad2017interpreting} utilize the multi-view extension of HEVC (MV-HEVC) to compress the light field in the form of a multi-view sequence. They interpret each row of the sub-aperture views as a single view of a multi-view video sequence and propose a two-dimensional prediction and rate allocation scheme. The extension of their work \\cite{RwPsvRef4_ahmad2019computationally} uses hierarchical levels, where views belonging to higher levels are assigned with better quality. The higher-level views are used as references to predict the lower-level views. Synthesized virtual reference frames are generated from Adaptive Separable Convolution Network (ASCN) in another SAIs based technique~\\cite{RwPsvRef5_gu2019high}. Such frames are considered as extra reference candidates in a hierarchical coding structure for MV-HEVC to further exploit intrinsic similarities in light field images.\n\nA block-basis estimation of views from translated reference views is proposed in \\cite{RwVsRef4_heriard2019light}. The residuals of estimated views are also transmitted to the decoder along with the rest of the view estimation parameters. Other view synthesis compression schemes estimate image depth maps from a subset of reference light field views. The Multi-view Video plus Depth (MVD) structure is adopted for depth image-based rendering to synthesize the intermediate SAIs \\cite{RwVsRef2_huang2018view}. A pair of steps to generate noise-refined depth maps for selected perspective views is elaborated in \\cite{RwVsRef3_huang2019light}.\n\nBakir et al.~\\cite{RwDeepRef3_bakir2018light} presented a deep learning-based scheme on the decoder side to improve the reconstruction quality of sub-aperture images. Similarly, Zhao et al. \\cite{RwDeepRef4_zhao2018light} encoded only sparsely sampled SAIs, while the remaining SAIs are synthesized using a CNN from the decoded sampled SAIs as priors. However, these methods require large-scale and diverse training samples, and high quality of reconstructed views is only obtained if at least half the SAIs are taken as reference. Schiopu et al. \\cite{RwDeepRef6_schiopu2019deep} proposed a novel network that synthesizes the entire light field image as an array of synthesized macro pixels in one step. Wang et al.~\\cite{RwDeepRef5_wang2019region} identify a region of interest (ROI), a complex non-ROI, and a smooth non-ROI to compress light field videos framewise. A generative adversarial network (GAN) is proposed in \\cite{RwDeepRef7_jia2018light} for unsampled SAIs generation. Liu et al. \\cite{RwDeepRef8_liu2021view} also adopt a GAN framework to boost the light field compression. They use an image group-based sampling method to reduce more SAI redundancy and maintain the reconstructed SAI quality. A perceptual quality-based loss function is also proposed by considering the PSNR of synthetic SAIs and adversarial loss. The above-mentioned coding techniques are not explicitly designed for the representation used in multi-layer-based light field displays. They also usually train a system (or network) to support only specific bitrates during the compression. \n\n\n\n\\section{Proposed Methodology}\\label{pm}\n\nThe complete workflow of our proposed representation and coding scheme with three main components is illustrated in Fig.~\\ref{fig:main workflow}. In COMPONENT I, input light field images are divided into view subsets depending on the specific prediction orders. To efficiently exploit the intrinsic redundancies in light field data, the proposed scheme constructs multiplicative layers from each view subset by employing a CNN. BLOCK I represents a CNN that converts the light field views of the input subsets into three multiplicative layers \\cite{maruyama2020comparison}. In BLOCK II, we removed the intrinsic redundancy present in subset views by exploiting the hidden low-rank structure of multiplicative layers on a Krylov subspace \\cite{musco2015randomized}. The low-rank approximation of each subset multiplicative layers is performed using Block-Krylov singular value decomposition (BK-SVD) by choosing various ranks~\\cite{musco2015randomized}. The intra-frame and inter-frame redundancies are further eliminated by encoding the approximated layers with HEVC~\\cite{sullivan2012overview}. We reconstruct the approximated views of each subset from their respective decoded multiplicative layers in BLOCK III. At the end of COMPONENT I of our scheme, we obtain approximated light field data at various ranks and quantization parameters.\n\nIn the next second phase, approximated subsets are used to construct Fourier disparity layer (FDL) representation of light fields~\\cite{le2019fourier}. The processing of approximated light field in COMPONENT II of our scheme is shown in Fig.~\\ref{fig:workflow_encoding}. There exist non-linear correlations between neighboring sub-aperture views in both horizontal and vertical directions in the light field structure. We particularly target these redundancies between adjacent light field views by processing in the Fourier domain as specified by different scanning or predication orders. The light field is then iteratively reconstructed by the FDL representation in a hierarchical fashion. We find the angular coordinates and disparity values of each view of the low-rank approximated light field (at different ranks and QPs) through FDL calibration and directly transmit them to the decoder (COMPONENT III) as metadata~\\cite{dib2019light}. The approximated light field is divided into subsets specified by four different prediction orders. The initial subset is used in construction of the FDL representation, which is further employed to synthesize the subsequent subsets of views. The correlations in prediction residue are removed, and a more accurate FDL representation is constructed from previously encoded subsets. Thus, we iteratively refine the FDL representation in COMPONENT II until all the approximated light field views are encoded. The decoding scheme is depicted in COMPONENT III (Fig.~\\ref{fig:workflow_decoding}). Here, angular coordinates and disparity values of the low-rank approximated light field, along with the encoded bitstreams of approximated subsets are utilized for the final light field reconstruction.  \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.17]{.\/Figures\/Parameters.jpg}\n    \\caption{\\footnotesize The light ray is parameterized by point of intersection with the $(u,v)$ plane and the $(s,t)$ plane located at a depth $x$.}\n    \\label{fig:lf planes}\n\\end{figure}\n\n\n\\subsection{Approximation of light field at different ranks}\n\nThe input light field is divided into view subsets in COMPONENT I, depending on the specified scanning orders, and a low-rank approximated representation of the subset multiplicative layers is obtained in the Krylov subspace. BLOCK I to III in Fig.~\\ref{fig:workflow_encoding} illustrate the three sub-blocks involved in this step. The details of each step are described in the following sub-sections.\n\n\n\n\\subsubsection{View Subsets of Light Field}\\label{sec:view_subsets}\n\nThe proposed scheme divides the light field into different view subsets based on predefined scanning orders. We adopt a hierarchical pattern configuration and a circular view prediction order described in the work of Dib et al.~\\cite{dib2019light}. The four chosen patterns are Circular-2 ($C_2$), Circular-4 ($C_4$), Hierarchical-2 ($H_2$) and Hierarchical-4 ($H_4$). For a 9$\\times$9 light field, the $C_2$ and $H_2$ patterns contain two view subsets and $C_4$ and $H_4$ patterns have four subsets. The exact coding orders of each subset of these four chosen scanning orders are shown in Fig.~\\ref{fig:view subset orders}. In all subsets of these patterns, the light field views form a circle that spiral out from the center. Generally, the corner views of light fields are of lower quality, and thus we choose to form a circle rather than a square while scanning the views. Our proposed workflow begins with partitioning of the input light field into subsets based on $C_2$, $C_4$, $H_2$ or $H_4$ patterns (Fig.~\\ref{fig:main workflow}).\n\n\n\n\\begin{figure*}\n    \\centering \n    \\begin{subfigure}{0.29\\textwidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/dragon_layer00.png}\n      \\caption{\\footnotesize Layer -1}\n    \\end{subfigure}\n     \\begin{subfigure}{0.29\\textwidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/dragon_layer01.png}\n      \\caption{\\footnotesize Layer 0}\n    \\end{subfigure}\n     \\begin{subfigure}{0.29\\textwidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/dragon_layer02.png}\n      \\caption{\\footnotesize Layer +1}\n    \\end{subfigure}\n\\caption{\\footnotesize The three optimal multiplicative layers produced by the CNN for  the \\textit{Dragon and Bunnies} light field~\\cite{mitdragon}. }\n\\label{fig:dragon_layers}\n\\end{figure*}\n\n\n\n\\subsubsection{Light Field Views to Stacked Multiplicative Layers} \n\nThe light field  $L(s,t,u,v)$ characterizes radiance along light rays as a 4D function~\\cite{levoy1996light,gortler1996lumigraph}. In $L(s,t,u,v)$, the parameters $(s,t)$ specify the spatial coordinates  and $(u,v)$ denote the angular coordinates that represent intersection of the rays with two parallel planes (Fig.~\\ref{fig:lf planes}). A light field can be interpreted as a set of directional views and can be produced using stacked layers that carry out multiplicative light ray operations~\\cite{maruyama2020comparison}. Such multiplicative layers could be realized with few light-attenuating panels stacked in equally spaced intervals in front of a backlight (Fig.~\\ref{fig:backlight layers}).\n\nThe normalized intensity of a light ray emitted from light field display can be described as \n\\begin{equation}\n\\centering\n    L_{m}(s,t,u,v)= \\prod_{x \\in X}T_{x}(u+xs,v+xt)\n    \\label{eq:mult layers}\n\\end{equation}\nwhere, $T_{x}(u,v)$ is the transmittance of the layer present at disparity $x$ and $X$ denotes the set of disparities among directional views. We determined the multiplicative layers for each subset of different view scanning order. The experiments are performed by considering the light field display composed of three layers located at $X = \\{ -1, 0, 1\\}$.\n\nThe three multiplicative layers obtained from each subset are optimized using a data-driven CNN based approach. The network architecture is depicted in Fig.~\\ref{fig:workflow_encoding}. The objective is to optimize the layer patterns. Mathematically, it is expressed as \n\\begin{equation}\n    \\underset{T_{x}|x \\in X}{\\mathrm{argmin}} \\sum_{s,t,u,v}\\left \\| L(s,t,u,v)-L_{m}(s,t,u,v) \\right \\|^{2}\n    \\label{eqn:opteqn}\n\\end{equation}\nThe optimized multiplicative layers obtained for $5 \\times 5$ \\textit{Dragon and Bunnies} light field is shown in Fig.~\\ref{fig:dragon_layers}.\n\n\n\n\\subsubsection{Low-Rank Representation of Subset Multiplicative Layers on Krylov Subspace} \\label{sec:bksvd}\n\nThe randomized Block-Krylov SVD method introduced by Cameron and Christopher can optimally achieve the low-rank approximation of a matrix within $(1 + \\epsilon)$ of optimal for spectral norm error~\\cite{musco2015randomized}. The algorithm quickly converges in $\\tilde{O}(\\frac{1}{\\sqrt{\\epsilon}})$ iterations for any matrix. In our proposed scheme, the optimized multiplicative layers are compactly represented on a Krylov subspace in order to remove the intrinsic redundancy.\n\nWe denote each multiplicative layer pattern obtained from the CNN as $T_{x} \\in \\mathbb{R}^{m \\times n \\times 3}$, where $ x \\in \\{ -1, 0, 1\\}$. The red, green, and blue colour channels of the layer $x$ are denoted as $T^{r}_{x}$, $T^{g}_{x}$, and $T^{b}_{x}$ respectively. We construct three matrices $A^{ch} \\in \\mathbb{R}^{3m \\times n}$, $ch \\in \\{ r, g, b \\}$ as \n\\begin{equation}\n\\centering\nA^{ch}= \\left( \\begin{array}{cc}\n    \\: \\:T^{ch}_{-1} \\\\ \n    T^{ch}_{0} \\\\ \n    T^{ch}_{1} \\end{array} \\right)\n\\end{equation}\nThe BK-SVD low-rank approximation in a Krylov subspace for each $A^{ch}$ can remove the intrinsic redundancies in multiplicative layers of the subsets. For simplicity, we will denote $A^{ch}$ as $A$ henceforth. To approximate $A$, the objective is to  achieve a subspace that closely captures the variance of $A$'s top singular vectors and avoid the gap dependence in singular values. \nThe spectral norm low-rank approximation error of $A$ is \n\\begin{equation}\n    \\left \\| A - D D^{T} A  \\right \\|_{2} \\leq \\left ( 1 + \\epsilon  \\right ) \\left \\|A - A_{k}  \\right \\|_{2}\n    \\label{eq:spectral norm error}\n\\end{equation}\nOnly top $k$ singular vectors of $A$ are used in its rank-$k$ approximation $A_{k}$. If $D$ is a rank-$k$ matrix with orthonormal columns, the spectral norm guarantee ensures that $D D^{T} A$ recovers $A$ up to a threshold $\\epsilon$.\n\nBlock Krylov iteration is performed working with the Krylov subspace\n\\begin{equation}\nK =[\\Pi \\hspace{8pt} A\\Pi  \\hspace{8pt} A^{2} \\Pi \\hspace{8pt} A^{3}\\Pi \\cdot \\cdot \\cdot A^{q}\\Pi ]\n\\label{eq: Kspace} \n\\end{equation}\n\nAs analyzed in~\\cite{musco2015randomized}, we choose to work on low degree polynomials that allow  faster computation in fewer powers of $A$, and thus enabling convergence of the BK-SVD algorithm in fewer iterations. From subspace $K$, we construct $p_{q}(A)\\Pi$ for any polynomial $p_{q}(\\cdot)$ of degree $q$, where $\\Pi \\sim N (0,1)^{d \\times k}$. The approximation of $A$ done by projecting it onto the span of $p_{q}(A)\\Pi$ atleast matches the best $k$ rank approximation of $A$ lying in the span of Krylov space $K$~\\cite{musco2015randomized}. Further, we orthonormalize columns of $K$ to obtain $Q \\in \\mathbb{R}^{c \\times qk}$ using QR decomposition method~\\cite{gu1996efficient}. After computing  SVD of matrix $S = Q^{T} B B^{T} Q$, we found matrix $\\bar{U}_{k}$ containing the top $k$ singular vectors of $S$. Thus, the rank-$k$ approximation of $A$ is matrix $D$, which is computed as \n\\begin{equation}\n    D = Q \\bar{U}_{k}\n\\end{equation}\n\nConsequently, the rank-$k$ block Krylov approximation of matrices $A^{r}$ , $A^{g}$ and $A^{b}$ are ${D}^{r}$, ${D}^{g}$, and ${D}^{b}$ respectively. Matrix $D^{ch} \\in \\mathbb{R}^{y \\times z}$ for every colour channel $ch$.\nTo obtain the approximated layers $\\bar{T}_{x}$, we extract colour channels from the approximated ${D}^{ch}$ matrices by sectioning out the rows uniformly as \n\\begin{flalign*}\n    & \\bar{T}^{ch}_{-1} = {D}^{ch}[1 : y \\: \\:,\\: \\: 1 : z \\: \\:] \\\\\n    & \\bar{T}^{ch}_{0} = {D}^{ch}[y+1 :2y \\:,\\: \\: 1 : z \\: \\:]   \\\\\n    & \\bar{T}^{ch}_{1} = {D}^{ch}[2y+1 : 3y \\:,\\: \\: 1 : z \\: \\:] \n\\end{flalign*}\nThe red, green, and blue colour channels are combined to form each approximated layer $\\bar{T}_{-1}$, $\\bar{T}_{0}$, and $\\bar{T}_{1}$. Thus, factorization derived from BK-SVD exploits the spatial correlation in multiplicative layers of the subset views for varying low ranks. The three block Krylov approximated layers for each subset are subsequently encoded into a bitstream using HEVC for various QPs to further eliminate intra and inter layer redundancies in the low-rank approximated representation. The low-rank representation and coding of stacked multiplicative layers on Krylov subspace is depicted in BLOCK II of Fig.~\\ref{fig:workflow_encoding}.\n\n\n\\begin{figure*}\n    \\centering \n    \\begin{subfigure}{0.3\\textwidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/o_b.png}\n      \\caption{\\footnotesize Bikes}\n    \\end{subfigure}\n     \\begin{subfigure}{0.3\\textwidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/o_s.png}\n      \\caption{\\footnotesize Stone pillars outside}\n    \\end{subfigure}\n     \\begin{subfigure}{0.3\\textwidth}\n      \\includegraphics[width=\\linewidth]{.\/Figures\/o_f.png}\n      \\caption{\\footnotesize Fountain \\& Vincent 2}\n    \\end{subfigure}\n\\caption{\\footnotesize Central views of the three datasets. }\n\\label{fig:orglfs}\n\\end{figure*}\n\n\n\\subsubsection{Decoding and Reconstruction of the Light Field Subsets}\n\nThe decoding procedure of compressed layers and the reconstruction of light field subsets is shown in BLOCK III of Fig.~\\ref{fig:workflow_encoding}. We decoded three multiplicative layers from the bitstream as $\\grave{T}_{x}$, $x \\in \\{ -1, 0, 1 \\}$. We consider the integers $s^{*}, t^{*}$, which vary depending on the number of views in each subset. The view $I_{(s^{*},t^{*})}$ is obtained by translating the decoded layers to $\\widehat{\\textit{T}}_{x}$. This step is followed by an element-wise product of colour channels of the translated layers. For a particular view $(s^{*},t^{*})$, the translation of every $x^{th}$ layer $\\grave{T}_{x}$, to $\\widehat{T}_{x}$ is carried out as  \n\\begin{equation}\n     \\widehat{T}_{x(s^{*},t^{*})}(u,v) = \\grave{T}_{x}(u+xs^{*}, v+xt^{*})\n\\end{equation}\nThus, the three translated layers for every viewpoint $(s^{*},t^{*})$ is computed as  \n\\begin{flalign*}\n    & \\widehat{\\textit{T}}_{-1(s^{*},t^{*})}(u,v) = \\grave{T}_{-1}(u-s^{*}, v-t^{*}) \\\\\n    & \\widehat{\\textit{T}}_{0 (s^{*},t^{*})}(u,v)\\:\\:\\: = \\grave{T}_{0}\\:(u, v)  \\\\\n    & \\widehat{\\textit{T}}_{1 (s^{*},t^{*})}(u,v)\\:\\:\\: = \\grave{T}_{1}(u+s^{*}, v+t^{*})\n\\end{flalign*}\nAn element-wise product of each colour channel $ch \\in \\{r, g, b \\}$ of the translated layers give the corresponding colour channel of the subset view.\n\\begin{equation}\nI^{ch}_{(s^{*},t^{*})}= \\widehat{\\textit{M}}^{ch}_{-1(s^{*},t^{*})} \\odot  \\widehat{\\textit{M}}^{ch}_{0(s^{*},t^{*})} \\odot \\widehat{\\textit{M}}^{ch}_{1(s^{*},t^{*})}\n\\end{equation}\nThe combined red, green and blue colour channels output the reconstructed light field subset at the viewpoint $(s^{*},t^{*})$ as $I_{(s^{*},t^{*})}$. At last, we merged the view subsets according to $C_2$, $C_4$, $H_2$ or $H_4$ patterns.\n\nThus, the spatial correlation in multiplicative layers of the subset views is exploited for different low ranks in COMPONENT I of the proposed scheme. Intra and inter-layer redundancies in the low-rank approximated representation are also well removed. We further compressed the light field by eliminating intrinsic similarities caused by non-linear correlations among neighboring views in horizontal and vertical directions as specified by $C_2$, $C_4$, $H_2$ and $H_4$ patterns using the Fourier Disparity Layers (FDL) representation. The following section describes the processing of approximated light fields in the Fourier domain.\n\n\n\n\\subsection{Fourier Disparity Layers Representation \\& Light Field Processing}\n\nThe Fourier Disparity Layers representation \\cite{le2019fourier} samples the input BK-SVD approximated light field in the disparity dimension by decomposing it as a discrete sum of layers. The layers are constructed from the approximated light field sub-aperture views through a regularized least square regression performed independently at each spatial frequency in the Fourier domain. The FDL representation has been shown to be effective for numerous light field processing applications \\cite{le2019fourier,dib2019light, le2020hierarchical, le2020high}. We have summarised the use of FDL and encoding of low-rank approximated light field as COMPONENT II in Fig.~\\ref{fig:workflow_encoding}. The corresponding decoding and reconstruction of the light field subsets are illustrated as COMPONENT III in Fig.~\\ref{fig:workflow_decoding}. The Fourier Disparity Layer calibration, subset view synthesis, and prediction are described in the following subsections.\n\n\n\n\\subsubsection{FDL Calibration}\n\nWithout loss of generality, we have considered one spatial coordinate $s$ and one angular coordinate $u$ of 4D light field to present the notations in a simple manner. The approximated light field view $L_{u_{o}}$ at angular position $u_{o}$ can be defined as $L_{u_{o}}(s)=L(s,u_{o})$. We can obtain Fourier coefficients of the $j^{th}$ input light field view using $n$ disparity values $\\{d_{k}\\}_{k \\in  \\llbracket 1,n  \\rrbracket }$~\\cite{dib2019light}. The Fourier transform of $L_{u_{o}}$ at spatial frequency $f_{s}$ is\n\\begin{equation}\\label{eq:ft}\n    \\hat{L}_{u_{o}}(f_{s})=\\sum_{k} e^{+2i\\pi u_{o}d_{k}f_{s}}\\hat{L}^{k}(f_{s})\n\\end{equation}\nHere, the Fourier coefficients of the disparity layers for a particular frequency $f_{s}$ is defined by \n\\begin{equation}\n    \\hat{L}^{k}(f_{s})= \\int_{\\Omega_{k}}e^{-2i\\pi s f_{s}}L(s,0)dx\n\\end{equation}\nEach of such Fourier coefficients can be understood as the Fourier transform of the central light field view (as $u_{o}=0$ by just considering the spatial region $\\Omega_{k}$ of disparity $d_{k}$.\n\nThe angular coordinates $u_j$ of the input views and the disparity values of the layers $d_{k}$ are estimated by minimizing over all frequency components $f^{q}_{s}$, $q \\in \\llbracket 1,Q\\rrbracket$, where $Q$ is the number of pixels in each input image~\\cite{le2019fourier}. By computing Fourier transforms of all $m$ approximated light field views as $\\hat{L}_{u_{j}} (j \\in \\llbracket 1,m\\rrbracket)$, the FDL representation is learnt by solving a linear regression problem for each frequency $f_{s}$.\n\nThe optimization problem is formulated as $\\textbf{Ax} = \\textbf{b}$ with Tikhonov regularization, where $\\textbf{A} \\in \\mathbb{R}^{m \\times n}$, $\\textbf{x} \\in \\mathbb{R}^{n \\times 1}$ and $\\textbf{b} \\in \\mathbb{R}^{m \\times 1}$. Elements of matrix \\textbf{A} are $\\textbf{A}_{jk} = e^{+2i\\pi u_{j}d_{k}f_{s}}$ and \\textbf{x} contains the Fourier coefficients of disparity layers $\\textbf{x}_{k} = \\hat{L}^{k}(f_{s})$. The vector $\\textbf{b}$ contains the Fourier coefficients of $j^{th}$ input view, $\\textbf{b}_{j} = \\hat{L}_{u_{j}}(f_{s})$. The solution of the optimization problem results in the disparity values $d_k$ and view positions $u_j$ that are passed as metadata information to the decoder in COMPONENT III.\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s1_f00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s2_f00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s1_f00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s2_f00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s3_f00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s4_f00.png}\n\\end{subfigure}\n\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s1_f01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s2_f01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s1_f01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s2_f01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s3_f01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s4_f01.png}\n\\end{subfigure}\n\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s1_f02.png}\n  \\caption{\\footnotesize $C_{2}-S1$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s2_f02.png}\n  \\caption{\\footnotesize $C_{2}-S2$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s1_f02.png}\n  \\caption{\\footnotesize $C_{4}-S1$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s2_f02.png}\n  \\caption{\\footnotesize $C_{4}-S2$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s3_f02.png}\n  \\caption{\\footnotesize $C_{4}-S3$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s3_f02.png}\n  \\caption{\\footnotesize $C_{4}-S4$}\n\\end{subfigure}\n\n\\caption{\\footnotesize The multiplicative layers of each view subset (subset 1 to 4 denoted as S1 to S4) of the Circular-2 ($C_2$) and Circular-4 ($C_4)$ scanning patterns. The first, second and third rows illustrate multiplicative layers -1. 0 and 1 respectively. }\n\\label{fig:fountlayers}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s1_b00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s2_b00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s1_b00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s2_b00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s3_b00.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s4_b00.png}\n\\end{subfigure}\n\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s1_b01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s2_b01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s1_b01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s2_b01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s3_b01.png}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s4_b01.png}\n\\end{subfigure}\n\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s1_b02.png}\n  \\caption{\\footnotesize $H_{2}-S1$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s2_b02.png}\n  \\caption{\\footnotesize $H_{2}-S2$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s1_b02.png}\n  \\caption{\\footnotesize $H_{4}-S1$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s2_b02.png}\n  \\caption{\\footnotesize $H_{4}-S2$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s3_b02.png}\n  \\caption{\\footnotesize $H_{4}-S3$}\n\\end{subfigure}\n\\begin{subfigure}{0.13\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s3_b02.png}\n  \\caption{\\footnotesize $H_{4}-S4$}\n\\end{subfigure}\n\n\\caption{\\footnotesize The multiplicative layers of each view subset (subset 1 to 4 denoted as S1 to S4) of the Hierarchical-2 ($H_2$) and Hierarchical-4 ($H_4)$ scanning patterns. The first, second and third rows illustrate multiplicative layers -1. 0 and 1 respectively. }\n\\label{fig:bikeslayers}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_bikes.jpg}\n\\end{subfigure} \n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_stone.jpg}\n\\end{subfigure}\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_fount.jpg}\n\\end{subfigure} \n\n\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_bikes.jpg}\n\\end{subfigure} \n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_stone.jpg}\n\\end{subfigure}\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_fount.jpg}\n\\end{subfigure} \n\n\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_bikes.jpg}\n\\end{subfigure} \n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_stone.jpg}\n\\end{subfigure}\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_fount.jpg}\n\\end{subfigure} \n\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_bikes.jpg}\n    \\caption{\\footnotesize Bikes}\n\\end{subfigure} \n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_stone.jpg}\n  \\caption{\\footnotesize Stone pillars Outside}\n\\end{subfigure}\n\\begin{subfigure}{0.32\\linewidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_fount.jpg}\n  \\caption{\\footnotesize  Fountain \\& Vincent 2}\n\\end{subfigure} \n\n\\caption{\\footnotesize Bitrate vs YUV-PSNR curves for the proposed compression scheme and Dib et al. for the three datasets in the $C_2$, $C_4$, $H_2$ and $H_4$ patterns. \\textit{(Kindly expand for better clarity)}}\n\\label{fig:graphs}\n\\end{figure*}\n\n\n\n\\subsubsection{FDL View Synthesis and Prediction}\n\nIn section~\\ref{sec:view_subsets}, two basic scanning orders, circular and hierarchical, are discussed for the synthesis and coding of light field views. Each of these chosen orders has two or four view subsets. This results in four patterns $C_2$, $C_4$, $H_2$ and $H_4$. In all four cases of the view prediction orders, the images are arranged in a spiral order starting from the center of the light field for each subset. The initial view subset of every pattern is always the first subset as specified by the scanning order. This subset is directly encoded first in COMPONENT II. For example, in Fig.~\\ref{fig:view subset orders}, the blue coloured subset in $C_2$ is the first subset.\n\nThe angular coordinates $u_j$ and disparity values $d_k$ are determined by the Fourier Disparity Layer calibration~\\cite{le2019fourier}. These are required in the further FDL construction and view predictions. This additional information is transmitted to the decoder in COMPONENT III as metadata~\\cite{dib2019light}. The initial view subset is used in the basic construction of FDL representation. This aids in the synthesis of succeeding view subsets. The residual signal is also encoded to account for the remaining correlations in the prediction residue of synthesized views. The FDL representation is then refined before prediction and encoding of the next subset of views. Thus, the FDL representation is iteratively fine-tuned at every stage, after encoding every view subset, until all the approximated input light field views are encoded.\n\n\n\n\\section{Results and Analysis}\\label{ra}\nThe performance of the proposed compression scheme is evaluated on real light fields captured by plenoptic cameras. The experiments are performed with  \\textit{Bikes}, \\textit{Fountain \\& Vincent 2}, and \\textit{Stone pillars outside} light field datasets from the EPFL Lightfield JPEG Pleno database~\\cite{rerabek2016new}. The central views of the chosen light field images are shown in Fig.~\\ref{fig:orglfs}. The raw plenoptic images are extracted into 15$\\times$15 sub-aperture views using the Matlab Light field toolbox~\\cite{dansereau2013decoding}.\n\nThe patterns $C_2$, $C_4$, $H_2$ and $H_4$ are constructed from inner 9$\\times$9 light field views for our experiments. Subsets 1 and 2 of $C_2$ contain 24 and 57 light field views respectively. The first and second subsets $H_2$ contain 25 and 56 views respectively. In $C_4$, subsets 1, 2, 3 and 4 have 4, 16, 12 and 49 views respectively. Lastly, subsets 1, 2, 3 and 4 of $H_4$ contain 4, 5, 16 and 56 light field views respectively. The exact scanning orders of the patterns and their subsets are specified in Fig.~\\ref{fig:view subset orders}.\n\n\n\\begin{table}\n\\centering\n\\caption{\\footnotesize Total number of bytes for each subset of Circular-2 pattern using our proposed coding scheme and Dib et al. \\cite{dib2019light}.}\n\\label{table_bytesC2}\n\\resizebox{6cm}{!}{\n\\makebox[\\linewidth]{\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{} & \\multicolumn{2}{|c|}{Bikes} & \\multicolumn{2}{|c|}{Stone pillars outside}  & \\multicolumn{2}{|c|}{Fountain \\& Vincent 2} \\\\\n\\hline\nQP & Scheme & Subset 1 & Subset 2 & Subset 1 & Subset 2 & Subset 1 & Subset 2\\\\\n\\hline\n\\multirow{8}{*}{2} & Dib et al.\t&\t6373276\t&\t9269248\t&\t6321523\t&\t7928095\t&\t6912472\t&\t8363327\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t1634865\t&\t285016\t&\t1358448\t&\t243122\t&\t1827059\t&\t335430\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t1877550\t&\t349216\t&\t1512085\t&\t283331\t&\t2115774\t&\t380847\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t2177689\t&\t417606\t&\t1847285\t&\t351680\t&\t2519102\t&\t446949\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t2438540\t&\t486569\t&\t2142502\t&\t418877\t&\t2806767\t&\t494263\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t2674089\t&\t532163\t&\t2350849\t&\t487484\t&\t3008060\t&\t545947\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t2749844\t&\t561758\t&\t2401859\t&\t503039\t&\t3063768\t&\t535674\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t2805401\t&\t581882\t&\t2448047\t&\t527789\t&\t3113991\t&\t535619\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{6} &\tDib et al.\t&\t4350810\t&\t5663692\t&\t4354754\t&\t4732850\t&\t4869817\t&\t4963030\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t602334\t&\t124074\t&\t493765\t&\t104272\t&\t728339\t&\t149454\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t741293\t&\t165605\t&\t571942\t&\t130267\t&\t893240\t&\t178163\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t916846\t&\t211107\t&\t761895\t&\t173837\t&\t1138839\t&\t220365\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t1086119\t&\t247005\t&\t925727\t&\t218830\t&\t1349095\t&\t252514\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t1252891\t&\t282873\t&\t1051703\t&\t256343\t&\t1499318\t&\t268165\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t1311143\t&\t290011\t&\t1084796\t&\t268153\t&\t1547468\t&\t279455\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t1354933\t&\t310371\t&\t1118995\t&\t281491\t&\t1583718\t&\t284981\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{10} &\tDib et al.\t&\t4350810\t&\t5663692\t&\t4354754\t&\t4732850\t&\t4869817\t&\t4963030\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t602334\t&\t124074\t&\t493765\t&\t104272\t&\t728339\t&\t149454\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t741293\t&\t165605\t&\t571942\t&\t130267\t&\t893240\t&\t178163\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t916846\t&\t211107\t&\t761895\t&\t173837\t&\t1138839\t&\t220365\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t1086119\t&\t247005\t&\t925727\t&\t218830\t&\t1349095\t&\t252514\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t1252891\t&\t282873\t&\t1051703\t&\t256343\t&\t1499318\t&\t268165\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t1311143\t&\t290011\t&\t1084796\t&\t268153\t&\t1547468\t&\t279455\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t1354933\t&\t310371\t&\t1118995\t&\t281491\t&\t1583718\t&\t284981\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{14} &\tDib et al.\t&\t1635624\t&\t1555027\t&\t1697516\t&\t1227782\t&\t1879832\t&\t1231959\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t85418\t&\t47842\t&\t60119\t&\t36583\t&\t124338\t&\t51127\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t129326\t&\t64433\t&\t81540\t&\t45979\t&\t177036\t&\t66168\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t197901\t&\t80959\t&\t136933\t&\t64691\t&\t279887\t&\t79125\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t271565\t&\t98486\t&\t200957\t&\t79562\t&\t376807\t&\t94429\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t347606\t&\t112747\t&\t255491\t&\t93448\t&\t449040\t&\t103222\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t372263\t&\t119795\t&\t267999\t&\t97639\t&\t466826\t&\t103998\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t389902\t&\t127835\t&\t280325\t&\t102454\t&\t483184\t&\t107398\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{20} &\tDib et al.\t&\t1635624\t&\t1555027\t&\t695581\t&\t292728\t&\t713369\t&\t335902\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t85418\t&\t47842\t&\t25871\t&\t19602\t&\t54329\t&\t25826\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t129326\t&\t64433\t&\t34316\t&\t23444\t&\t76408\t&\t33557\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t197901\t&\t80959\t&\t55115\t&\t31584\t&\t116924\t&\t40566\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t271565\t&\t98486\t&\t77665\t&\t38192\t&\t160911\t&\t47636\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t347606\t&\t112747\t&\t99503\t&\t44221\t&\t187776\t&\t51037\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t372263\t&\t119795\t&\t104642\t&\t46549\t&\t195728\t&\t52069\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t389902\t&\t127835\t&\t109416\t&\t48116\t&\t203384\t&\t51973\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{26} &\tDib et al.\t&\t191049\t&\t125748\t&\t180485\t&\t36164\t&\t213553\t&\t66399\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t18109\t&\t15295\t&\t12194\t&\t10550\t&\t26763\t&\t14884\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t26471\t&\t20342\t&\t16372\t&\t13010\t&\t35126\t&\t18824\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t36035\t&\t24566\t&\t22220\t&\t15525\t&\t46554\t&\t19651\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t46328\t&\t28090\t&\t28818\t&\t17922\t&\t60959\t&\t23291\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t55291\t&\t29746\t&\t35061\t&\t20507\t&\t70030\t&\t24332\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t58525\t&\t30969\t&\t37056\t&\t20533\t&\t73670\t&\t24606\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t61024\t&\t33335\t&\t39291\t&\t21200\t&\t75358\t&\t25193\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{38} &\tDib et al.\t&\t23065\t&\t7880\t&\t12613\t&\t1880\t&\t22582\t&\t2155\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t2737\t&\t3283\t&\t2289\t&\t3043\t&\t5098\t&\t3887\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t4208\t&\t4244\t&\t2719\t&\t3314\t&\t6606\t&\t4240\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t5752\t&\t5095\t&\t3197\t&\t3572\t&\t7206\t&\t4708\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t6864\t&\t5757\t&\t3888\t&\t3914\t&\t8261\t&\t4785\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t7852\t&\t6246\t&\t4442\t&\t3969\t&\t9137\t&\t5104\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t7946\t&\t6244\t&\t4437\t&\t4024\t&\t9294\t&\t5126\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t8559\t&\t6592\t&\t4553\t&\t4038\t&\t9630\t&\t5197\t\\\\\t\\cline{1-8}\n\\hline\n\\end{tabular} }\n}\n\\end{table}\n\n\n\n\\begin{table}\n\\centering\n\\caption{\\footnotesize Total number of bytes for each subset of Hierarchical-2 pattern using our proposed coding scheme and Dib et al. \\cite{dib2019light}. }\n\\label{table_bytesH2}\n\\resizebox{6cm}{!}{\n\\makebox[\\linewidth]{\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{} & \\multicolumn{2}{|c|}{Bikes} & \\multicolumn{2}{|c|}{Stone pillars outside}  & \\multicolumn{2}{|c|}{Fountain \\& Vincent 2} \\\\\n\\hline\nQP & Scheme & Subset 1 & Subset 2 & Subset 1 & Subset 2 & Subset 1 & Subset 2\\\\\n\\hline\n\\multirow{8}{*}{2} &\tDib et al.\t&\t8149470\t&\t9362963\t&\t8187441\t&\t7746889\t&\t8467600\t&\t8311549\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t1752364\t&\t327571\t&\t1544489\t&\t277441\t&\t2000768\t&\t399280\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t1923491\t&\t425176\t&\t1668430\t&\t336414\t&\t2197190\t&\t475908\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t2132305\t&\t534173\t&\t1904622\t&\t452160\t&\t2413070\t&\t576114\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t2307124\t&\t626191\t&\t2137972\t&\t550384\t&\t2589097\t&\t663419\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t2466765\t&\t728164\t&\t2348147\t&\t638116\t&\t2701529\t&\t709257\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t2531862\t&\t732311\t&\t2415703\t&\t670368\t&\t2728626\t&\t732779\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t2572006\t&\t759606\t&\t2470431\t&\t700779\t&\t2741756\t&\t740021\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{6} &\tDib et al.\t&\t5968201\t&\t5584732\t&\t5943824\t&\t4566311\t&\t6317422\t&\t4734277\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t696307\t&\t141510\t&\t600337\t&\t110556\t&\t870560\t&\t169079\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t796855\t&\t191515\t&\t665513\t&\t143058\t&\t991937\t&\t208585\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t928516\t&\t259814\t&\t809014\t&\t209917\t&\t1119092\t&\t266975\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t1044367\t&\t309978\t&\t953162\t&\t267569\t&\t1237456\t&\t314759\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t1151050\t&\t364678\t&\t1079187\t&\t314645\t&\t1320242\t&\t341091\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t1195073\t&\t376134\t&\t1122665\t&\t333719\t&\t1345621\t&\t355264\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t1224508\t&\t388297\t&\t1153867\t&\t350318\t&\t1356529\t&\t359455\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{10} &\tDib et al.\t&\t3906437\t&\t2991907\t&\t3972408\t&\t2394424\t&\t4256777\t&\t2365804\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t178206\t&\t74529\t&\t139283\t&\t54783\t&\t278393\t&\t83747\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t234038\t&\t103767\t&\t176282\t&\t74433\t&\t350782\t&\t106751\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t313521\t&\t143968\t&\t251942\t&\t112280\t&\t442029\t&\t138069\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t389826\t&\t175682\t&\t336035\t&\t146248\t&\t523367\t&\t166752\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t465603\t&\t213126\t&\t423251\t&\t172780\t&\t582988\t&\t184091\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t493156\t&\t213184\t&\t449828\t&\t182623\t&\t600907\t&\t189758\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t515966\t&\t221252\t&\t476680\t&\t192152\t&\t606755\t&\t192677\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{14} &\tDib et al.\t&\t2265550\t&\t1479998\t&\t2432694\t&\t1068158\t&\t2505792\t&\t1058776\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t80756\t&\t43910\t&\t56613\t&\t32851\t&\t125237\t&\t48892\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t108524\t&\t61888\t&\t74445\t&\t41825\t&\t162045\t&\t61496\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t150025\t&\t85871\t&\t111220\t&\t63932\t&\t212590\t&\t78435\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t191702\t&\t105961\t&\t151212\t&\t82963\t&\t264760\t&\t93719\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t228876\t&\t121783\t&\t194620\t&\t96725\t&\t297942\t&\t104203\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t243903\t&\t126699\t&\t209660\t&\t101918\t&\t309631\t&\t106855\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t254844\t&\t129017\t&\t222970\t&\t106454\t&\t314274\t&\t109546\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{20} &\tDib et al.\t&\t842135\t&\t459805\t&\t946011\t&\t221956\t&\t926805\t&\t286174\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t31602\t&\t20923\t&\t21663\t&\t15803\t&\t50142\t&\t23547\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t43511\t&\t29373\t&\t28169\t&\t19788\t&\t63745\t&\t29074\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t60423\t&\t40866\t&\t40732\t&\t28521\t&\t83392\t&\t35879\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t76647\t&\t49200\t&\t53703\t&\t35594\t&\t103900\t&\t42129\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t89653\t&\t56477\t&\t66649\t&\t41210\t&\t118608\t&\t45865\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t94969\t&\t59779\t&\t71693\t&\t42630\t&\t121445\t&\t47138\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t98186\t&\t61299\t&\t75923\t&\t44694\t&\t124309\t&\t47441\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{26} &\tDib et al.\t&\t242763\t&\t111266\t&\t213782\t&\t28940\t&\t272168\t&\t50062\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t15705\t&\t11349\t&\t9646\t&\t7203\t&\t20645\t&\t11951\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t20006\t&\t14619\t&\t12592\t&\t9004\t&\t27722\t&\t14167\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t26014\t&\t19172\t&\t15947\t&\t12784\t&\t32178\t&\t15899\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t30615\t&\t22252\t&\t19394\t&\t14768\t&\t39019\t&\t17819\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t34692\t&\t25694\t&\t23190\t&\t16965\t&\t42684\t&\t19317\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t36820\t&\t26082\t&\t25005\t&\t17802\t&\t44419\t&\t19885\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t37464\t&\t26717\t&\t26359\t&\t18185\t&\t45207\t&\t19866\t\\\\\t\\cline{1-8}\n\\hline\n\\multirow{8}{*}{38} &\tDib et al.\t&\t26295\t&\t6202\t&\t13132\t&\t2001\t&\t22971\t&\t1975\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 4)\t&\t2485\t&\t2434\t&\t2065\t&\t2188\t&\t4019\t&\t3065\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 8)\t&\t3475\t&\t3205\t&\t2465\t&\t2484\t&\t4976\t&\t3523\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 16)\t&\t4143\t&\t3742\t&\t2736\t&\t2707\t&\t5456\t&\t3699\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 28)\t&\t4872\t&\t4364\t&\t3002\t&\t2868\t&\t6114\t&\t4137\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 44)\t&\t5280\t&\t4572\t&\t3121\t&\t2984\t&\t6327\t&\t4141\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 52)\t&\t5604\t&\t4810\t&\t3174\t&\t2980\t&\t6384\t&\t4192\t\\\\\t\\cline{2-8}\n&\tProposed (Rank 60)\t&\t5704\t&\t4866\t&\t3205\t&\t3033\t&\t6467\t&\t4291\t\\\\\t\\cline{1-8}\n\\hline\n\\end{tabular} }\n}\n\\end{table}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{\\footnotesize Total number of bytes for each subset of Circular-4 pattern using our proposed coding scheme and Dib et al. \\cite{dib2019light}. }\n\\label{table_bytesC4}\n\\resizebox{6cm}{!}{\n\\makebox[\\linewidth]{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{} & \\multicolumn{4}{|c|}{Bikes} & \\multicolumn{4}{|c|}{Stone pillars outside}  & \\multicolumn{4}{|c|}{Fountain \\& Vincent 2} \\\\\n\\hline\nQP & Scheme & Subset 1 & Subset 2 & Subset 3 & Subset 4 & Subset 1 & Subset 2 & Subset 3 & Subset 4 & Subset 1 & Subset 2 & Subset 3 & Subset 4\\\\\n\\hline\n\\multirow{8}{*}{2} &\tDib E et al.\t&\t1416866\t&\t3109571\t&\t5037933\t&\t12979009\t&\t1435611\t&\t2966575\t&\t4753968\t&\t12051303\t&\t1441765\t&\t2912377\t&\t4845501\t&\t12249516\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t388099\t&\t258152\t&\t440152\t&\t768518\t&\t323881\t&\t173763\t&\t298852\t&\t569007\t&\t448166\t&\t341166\t&\t624766\t&\t1047411\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t455053\t&\t324627\t&\t558934\t&\t968901\t&\t370490\t&\t225967\t&\t392536\t&\t740275\t&\t521873\t&\t472205\t&\t817319\t&\t1322771\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t539142\t&\t479586\t&\t839084\t&\t1367088\t&\t472612\t&\t361397\t&\t632487\t&\t1084026\t&\t630974\t&\t695930\t&\t1211480\t&\t1864056\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t620618\t&\t615680\t&\t1076100\t&\t1771137\t&\t570270\t&\t515012\t&\t906551\t&\t1475466\t&\t711522\t&\t887817\t&\t1536556\t&\t2377277\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t701312\t&\t733863\t&\t1277639\t&\t2111200\t&\t651481\t&\t705884\t&\t1222296\t&\t1979756\t&\t773943\t&\t1039250\t&\t1785773\t&\t2679246\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t732590\t&\t796888\t&\t1366241\t&\t2220068\t&\t675245\t&\t749842\t&\t1245301\t&\t1983765\t&\t792259\t&\t1177083\t&\t1968347\t&\t2854259\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t755916\t&\t864119\t&\t1454924\t&\t2314135\t&\t696777\t&\t789366\t&\t1336327\t&\t2114408\t&\t806779\t&\t1256248\t&\t2078412\t&\t2981898\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{6} &\tDib E et al.\t&\t1086423\t&\t2038371\t&\t3237999\t&\t8276077\t&\t1106257\t&\t1486520\t&\t2490436\t&\t6394743\t&\t1117362\t&\t2219082\t&\t3574509\t&\t8740608\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t214122\t&\t110706\t&\t188813\t&\t331799\t&\t175603\t&\t67554\t&\t116720\t&\t226090\t&\t256811\t&\t156184\t&\t289118\t&\t471852\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t251850\t&\t151013\t&\t267156\t&\t453856\t&\t199275\t&\t93701\t&\t165247\t&\t308654\t&\t303608\t&\t234658\t&\t409134\t&\t641950\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t309322\t&\t245463\t&\t439774\t&\t698158\t&\t269341\t&\t170374\t&\t302438\t&\t513257\t&\t385171\t&\t394167\t&\t678570\t&\t1001574\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t370101\t&\t337259\t&\t595617\t&\t933555\t&\t341876\t&\t266334\t&\t474772\t&\t752003\t&\t454738\t&\t520966\t&\t898310\t&\t1310081\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t434134\t&\t425824\t&\t737800\t&\t1151208\t&\t404438\t&\t384524\t&\t655959\t&\t1001388\t&\t507159\t&\t631820\t&\t1073918\t&\t1528251\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t459408\t&\t462759\t&\t794905\t&\t1220896\t&\t425346\t&\t432563\t&\t711515\t&\t1073548\t&\t524083\t&\t727147\t&\t1192872\t&\t1641494\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t481777\t&\t511679\t&\t855678\t&\t1289391\t&\t440675\t&\t457735\t&\t766217\t&\t1149042\t&\t535587\t&\t785724\t&\t1271113\t&\t1730093\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{10} &\tDib E et al.\t&\t752723\t&\t1222920\t&\t1913852\t&\t4856266\t&\t777216\t&\t801876\t&\t1330787\t&\t3442540\t&\t782679\t&\t1345211\t&\t2155105\t&\t5205636\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t70222\t&\t49881\t&\t89829\t&\t166182\t&\t52813\t&\t29982\t&\t50755\t&\t103849\t&\t114218\t&\t79685\t&\t149995\t&\t246078\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t103595\t&\t83335\t&\t144000\t&\t247091\t&\t73658\t&\t44585\t&\t77578\t&\t148404\t&\t152447\t&\t128770\t&\t220654\t&\t346457\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t150921\t&\t135894\t&\t243209\t&\t385532\t&\t123883\t&\t87714\t&\t154861\t&\t265680\t&\t216590\t&\t228011\t&\t389982\t&\t561801\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t200848\t&\t192706\t&\t340303\t&\t523869\t&\t178811\t&\t140370\t&\t254368\t&\t402085\t&\t271876\t&\t309157\t&\t522290\t&\t743004\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t252927\t&\t249671\t&\t430367\t&\t652158\t&\t225984\t&\t223766\t&\t375786\t&\t553262\t&\t313974\t&\t379414\t&\t635701\t&\t880929\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t272641\t&\t273179\t&\t463991\t&\t690749\t&\t241537\t&\t245429\t&\t399996\t&\t582554\t&\t327708\t&\t442106\t&\t715530\t&\t955029\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t289867\t&\t299732\t&\t497046\t&\t726508\t&\t253575\t&\t272199\t&\t437409\t&\t627469\t&\t336198\t&\t477148\t&\t759510\t&\t1002492\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{14}  &\tDib E et al.\t&\t480892\t&\t679589\t&\t1050351\t&\t2626996\t&\t521633\t&\t383952\t&\t638700\t&\t1648073\t&\t500828\t&\t742109\t&\t1187800\t&\t2801625\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t40679\t&\t29652\t&\t51278\t&\t98485\t&\t29001\t&\t15388\t&\t25291\t&\t56627\t&\t65006\t&\t43589\t&\t81743\t&\t136374\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t61694\t&\t44851\t&\t77249\t&\t140007\t&\t40184\t&\t23177\t&\t39845\t&\t80289\t&\t88848\t&\t66867\t&\t116614\t&\t189391\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t91688\t&\t75524\t&\t135330\t&\t219789\t&\t69269\t&\t46427\t&\t79815\t&\t142707\t&\t131125\t&\t127103\t&\t212521\t&\t310005\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t124841\t&\t108047\t&\t190712\t&\t296426\t&\t102889\t&\t75180\t&\t133133\t&\t214414\t&\t171095\t&\t175954\t&\t294598\t&\t419739\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t158055\t&\t140028\t&\t240595\t&\t365750\t&\t133740\t&\t116563\t&\t195948\t&\t288671\t&\t198596\t&\t215928\t&\t358409\t&\t496160\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t172129\t&\t153785\t&\t259051\t&\t385398\t&\t143679\t&\t127010\t&\t209194\t&\t303887\t&\t207304\t&\t252547\t&\t401949\t&\t532371\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t183237\t&\t167289\t&\t274776\t&\t401261\t&\t152195\t&\t136239\t&\t222340\t&\t319578\t&\t213372\t&\t275681\t&\t430690\t&\t564375\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{20} &\tDib E et al.\t&\t213090\t&\t254203\t&\t381661\t&\t936738\t&\t237383\t&\t78987\t&\t150403\t&\t386715\t&\t220461\t&\t274582\t&\t434129\t&\t997828\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t20074\t&\t11978\t&\t20998\t&\t45282\t&\t13422\t&\t5936\t&\t9649\t&\t25904\t&\t30830\t&\t17895\t&\t33796\t&\t60029\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t29699\t&\t18356\t&\t30789\t&\t60782\t&\t18083\t&\t9076\t&\t14808\t&\t34361\t&\t41525\t&\t28044\t&\t48058\t&\t81843\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t43104\t&\t29966\t&\t52919\t&\t92344\t&\t29269\t&\t17566\t&\t29273\t&\t57232\t&\t60581\t&\t47269\t&\t78364\t&\t121263\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t58382\t&\t42027\t&\t73297\t&\t119860\t&\t42490\t&\t27054\t&\t49257\t&\t83411\t&\t79556\t&\t66517\t&\t110634\t&\t163393\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t73197\t&\t53595\t&\t90516\t&\t142227\t&\t56027\t&\t39443\t&\t64624\t&\t100884\t&\t93770\t&\t81035\t&\t133102\t&\t188734\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t78841\t&\t58178\t&\t96110\t&\t147314\t&\t60266\t&\t44362\t&\t71646\t&\t108650\t&\t98119\t&\t93685\t&\t147415\t&\t200453\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t83557\t&\t62421\t&\t100162\t&\t151331\t&\t63644\t&\t46378\t&\t73782\t&\t110035\t&\t100506\t&\t98744\t&\t154522\t&\t208763\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{26} &\tDib E et al.\t&\t79458\t&\t76052\t&\t111845\t&\t267957\t&\t74881\t&\t8010\t&\t18414\t&\t52494\t&\t81501\t&\t76202\t&\t124228\t&\t276944\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t9863\t&\t5121\t&\t8570\t&\t23184\t&\t6554\t&\t2967\t&\t4983\t&\t13967\t&\t15128\t&\t7839\t&\t15117\t&\t28897\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t14088\t&\t8012\t&\t13369\t&\t33342\t&\t8505\t&\t4174\t&\t6644\t&\t17217\t&\t18704\t&\t11299\t&\t19552\t&\t35754\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t19741\t&\t11666\t&\t19789\t&\t38861\t&\t12568\t&\t7266\t&\t12305\t&\t24932\t&\t25352\t&\t15455\t&\t26860\t&\t44990\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t25182\t&\t15000\t&\t25496\t&\t46274\t&\t16190\t&\t9206\t&\t15165\t&\t29982\t&\t32582\t&\t21640\t&\t35738\t&\t57038\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t30210\t&\t17987\t&\t29681\t&\t50997\t&\t20472\t&\t10873\t&\t19081\t&\t34787\t&\t37997\t&\t30815\t&\t46664\t&\t68833\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t32119\t&\t19219\t&\t31116\t&\t52433\t&\t22260\t&\t11543\t&\t20113\t&\t36221\t&\t39422\t&\t27733\t&\t44203\t&\t65498\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t33288\t&\t19430\t&\t31422\t&\t52858\t&\t23179\t&\t11864\t&\t20961\t&\t37382\t&\t40662\t&\t28354\t&\t44886\t&\t66797\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{38} &\tDib E et al.\t&\t13365\t&\t5614\t&\t8039\t&\t19199\t&\t9983\t&\t737\t&\t1149\t&\t2903\t&\t13141\t&\t3418\t&\t6028\t&\t13529\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t2303\t&\t1128\t&\t1700\t&\t5087\t&\t1466\t&\t787\t&\t1257\t&\t3905\t&\t2733\t&\t1701\t&\t2922\t&\t6111\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t3256\t&\t1399\t&\t2215\t&\t6353\t&\t1831\t&\t796\t&\t1249\t&\t3973\t&\t3418\t&\t1875\t&\t3098\t&\t6911\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t4087\t&\t1792\t&\t2941\t&\t6919\t&\t2306\t&\t919\t&\t1475\t&\t4642\t&\t3888\t&\t2140\t&\t3599\t&\t7383\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t5124\t&\t2025\t&\t3500\t&\t8088\t&\t2666\t&\t1017\t&\t1765\t&\t5011\t&\t4503\t&\t2291\t&\t4025\t&\t8008\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t5542\t&\t2349\t&\t3864\t&\t8427\t&\t2949\t&\t1157\t&\t1992\t&\t5248\t&\t5240\t&\t2339\t&\t3980\t&\t7964\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t5754\t&\t2458\t&\t3969\t&\t8508\t&\t3069\t&\t1177\t&\t2105\t&\t5587\t&\t5296\t&\t2432\t&\t4122\t&\t8436\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t5863\t&\t2536\t&\t4048\t&\t8389\t&\t3101\t&\t1149\t&\t1991\t&\t5345\t&\t5558\t&\t2435\t&\t4312\t&\t8577\t\\\\\t\\cline{1-14}\n\\hline\n\\end{tabular} }\n}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{\\footnotesize Total number of bytes for each subset of Hierarchical-4 pattern using our proposed coding scheme and Dib et al. \\cite{dib2019light}. }\n\\label{table_bytesH4}\n\\resizebox{6cm}{!}{\n\\makebox[\\linewidth]{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{} & \\multicolumn{4}{|c|}{Bikes} & \\multicolumn{4}{|c|}{Stone pillars outside}  & \\multicolumn{4}{|c|}{Fountain \\& Vincent 2} \\\\\n\\hline\nQP & Scheme & Subset 1 & Subset 2 & Subset 3 & Subset 4 & Subset 1 & Subset 2 & Subset 3 & Subset 4 & Subset 1 & Subset 2 & Subset 3 & Subset 4\\\\\n\\hline\n\\multirow{8}{*}{2} &\tDib E et al.\t&\t1440724\t&\t1452789\t&\t5165455\t&\t14131746\t&\t1441216\t&\t1362428\t&\t4475726\t&\t12790584\t&\t1491971\t&\t1436217\t&\t4898061\t&\t13465870\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t369210\t&\t134008\t&\t323631\t&\t711432\t&\t319254\t&\t115673\t&\t265421\t&\t578809\t&\t427823\t&\t168575\t&\t540513\t&\t976300\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t449987\t&\t189225\t&\t491920\t&\t944449\t&\t365506\t&\t138944\t&\t354594\t&\t733985\t&\t497979\t&\t236676\t&\t758310\t&\t1293087\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t527356\t&\t284122\t&\t740612\t&\t1320068\t&\t461363\t&\t199211\t&\t550438\t&\t1042584\t&\t587856\t&\t308720\t&\t1036070\t&\t1653996\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t594826\t&\t330343\t&\t938636\t&\t1687787\t&\t542214\t&\t264184\t&\t790867\t&\t1374410\t&\t669761\t&\t373958\t&\t1307700\t&\t2043902\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t654269\t&\t396269\t&\t1161377\t&\t1988746\t&\t607986\t&\t334920\t&\t1018052\t&\t1718058\t&\t730933\t&\t513132\t&\t1660083\t&\t2511724\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t675815\t&\t414816\t&\t1235023\t&\t2107314\t&\t632929\t&\t353228\t&\t1118538\t&\t1877849\t&\t757197\t&\t551166\t&\t1824076\t&\t2674922\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t696291\t&\t488035\t&\t1460949\t&\t2399511\t&\t650310\t&\t381130\t&\t1185575\t&\t1980309\t&\t776640\t&\t561102\t&\t1783246\t&\t2694231\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{6} &\tDib E et al.\t&\t1111819\t&\t1038264\t&\t3514681\t&\t9168841\t&\t1108750\t&\t960856\t&\t2990048\t&\t8187927\t&\t1165584\t&\t1034953\t&\t3260693\t&\t8419196\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t204317\t&\t62436\t&\t141660\t&\t311661\t&\t170346\t&\t51465\t&\t108216\t&\t241181\t&\t237931\t&\t96343\t&\t276107\t&\t473329\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t245169\t&\t101036\t&\t243053\t&\t467936\t&\t197249\t&\t71588\t&\t166252\t&\t344009\t&\t283215\t&\t137939\t&\t410716\t&\t663983\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t294934\t&\t153306\t&\t400324\t&\t701366\t&\t255205\t&\t110146\t&\t291160\t&\t544525\t&\t348866\t&\t195417\t&\t615257\t&\t934284\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t344636\t&\t195036\t&\t539529\t&\t919037\t&\t314895\t&\t162077\t&\t451320\t&\t755049\t&\t410858\t&\t245260\t&\t809494\t&\t1190873\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t395944\t&\t243143\t&\t691427\t&\t1120180\t&\t365865\t&\t208893\t&\t597212\t&\t959611\t&\t463453\t&\t324537\t&\t1030603\t&\t1465281\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t415335\t&\t260202\t&\t745264\t&\t1191375\t&\t388987\t&\t224727\t&\t666304\t&\t1052010\t&\t488418\t&\t354981\t&\t1146980\t&\t1579657\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t431383\t&\t300783\t&\t885964\t&\t1357140\t&\t399568\t&\t236506\t&\t701288\t&\t1100371\t&\t504481\t&\t383573\t&\t1159853\t&\t1615488\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{10} &\tDib E et al.\t&\t775655\t&\t666938\t&\t2142004\t&\t5440908\t&\t775397\t&\t625993\t&\t1827285\t&\t4832009\t&\t827917\t&\t670702\t&\t1897787\t&\t4726968\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t62635\t&\t37584\t&\t75744\t&\t172831\t&\t50046\t&\t25982\t&\t50848\t&\t126123\t&\t100594\t&\t57683\t&\t146200\t&\t258515\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t95314\t&\t60742\t&\t135199\t&\t271180\t&\t69850\t&\t35405\t&\t79321\t&\t176440\t&\t135538\t&\t85910\t&\t228956\t&\t369946\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t137298\t&\t96352\t&\t230464\t&\t411368\t&\t113623\t&\t64943\t&\t154415\t&\t293640\t&\t185272\t&\t126344\t&\t364217\t&\t548547\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t179852\t&\t124355\t&\t318267\t&\t535924\t&\t160773\t&\t100419\t&\t257622\t&\t431745\t&\t235083\t&\t160710\t&\t494057\t&\t711798\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t222286\t&\t157319\t&\t416397\t&\t663668\t&\t199480\t&\t129589\t&\t348407\t&\t554301\t&\t277377\t&\t207598\t&\t626034\t&\t870952\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t237984\t&\t167861\t&\t448807\t&\t703844\t&\t213343\t&\t143137\t&\t394288\t&\t609943\t&\t293737\t&\t229906\t&\t711584\t&\t949928\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t252828\t&\t190614\t&\t532675\t&\t790999\t&\t224197\t&\t145857\t&\t405564\t&\t625081\t&\t305891\t&\t247286\t&\t714102\t&\t964923\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{14}  &\tDib E et al.\t&\t500049\t&\t388083\t&\t1191431\t&\t2963845\t&\t513183\t&\t378425\t&\t1007183\t&\t2594115\t&\t541926\t&\t378184\t&\t988038\t&\t2383130\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t37151\t&\t21951\t&\t41475\t&\t104839\t&\t28674\t&\t16217\t&\t30015\t&\t74285\t&\t56057\t&\t35279\t&\t82651\t&\t149587\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t57510\t&\t37209\t&\t75972\t&\t161312\t&\t39181\t&\t21571\t&\t43882\t&\t103726\t&\t78222\t&\t52274\t&\t125153\t&\t213294\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t84762\t&\t59234\t&\t128341\t&\t243684\t&\t63968\t&\t37448\t&\t83762\t&\t167620\t&\t110915\t&\t78720\t&\t204995\t&\t319141\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t112659\t&\t77934\t&\t182347\t&\t317020\t&\t91763\t&\t55481\t&\t130123\t&\t240029\t&\t144823\t&\t100782\t&\t282460\t&\t414813\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t139924\t&\t97313\t&\t239016\t&\t389102\t&\t116872\t&\t76489\t&\t191692\t&\t313499\t&\t171103\t&\t121884\t&\t353503\t&\t497967\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t150018\t&\t104436\t&\t256212\t&\t409471\t&\t126814\t&\t82765\t&\t212476\t&\t335104\t&\t181934\t&\t144534\t&\t412929\t&\t555576\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t159098\t&\t115299\t&\t305078\t&\t457263\t&\t132712\t&\t88355\t&\t222964\t&\t347488\t&\t190145\t&\t145263\t&\t397671\t&\t546693\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{20} &\tDib E et al.\t&\t220871\t&\t155877\t&\t442338\t&\t1069809\t&\t221668\t&\t144280\t&\t323891\t&\t808099\t&\t240428\t&\t132454\t&\t317605\t&\t742520\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t18693\t&\t9386\t&\t16817\t&\t49155\t&\t14681\t&\t6261\t&\t12388\t&\t38873\t&\t27684\t&\t17171\t&\t34590\t&\t69842\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t28822\t&\t17725\t&\t32732\t&\t79384\t&\t18981\t&\t10397\t&\t19570\t&\t51678\t&\t37767\t&\t23972\t&\t50490\t&\t96406\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t42123\t&\t27257\t&\t52383\t&\t112254\t&\t29375\t&\t17001\t&\t34307\t&\t76671\t&\t53174\t&\t35808\t&\t79916\t&\t136134\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t54836\t&\t35292\t&\t71902\t&\t136347\t&\t40267\t&\t21487\t&\t45165\t&\t93643\t&\t69187\t&\t45105\t&\t111945\t&\t176694\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t66687\t&\t42676\t&\t92133\t&\t161840\t&\t51066\t&\t29424\t&\t66927\t&\t119603\t&\t80368\t&\t54863\t&\t136769\t&\t200474\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t71351\t&\t45196\t&\t97830\t&\t166948\t&\t55199\t&\t30240\t&\t71069\t&\t121892\t&\t83943\t&\t59049\t&\t146132\t&\t205276\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t74991\t&\t47735\t&\t114919\t&\t183393\t&\t58046\t&\t33606\t&\t77786\t&\t128307\t&\t86632\t&\t61772\t&\t151950\t&\t212506\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{26} &\tDib E et al.\t&\t80816\t&\t51560\t&\t127832\t&\t306862\t&\t70635\t&\t35903\t&\t66479\t&\t163981\t&\t88935\t&\t34759\t&\t71299\t&\t167645\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t9615\t&\t3901\t&\t7759\t&\t27704\t&\t7357\t&\t2136\t&\t4879\t&\t18330\t&\t13811\t&\t7502\t&\t13052\t&\t31445\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t14339\t&\t7033\t&\t13249\t&\t40246\t&\t9352\t&\t3875\t&\t7448\t&\t22526\t&\t17877\t&\t9379\t&\t19699\t&\t43266\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t19865\t&\t10767\t&\t19976\t&\t53272\t&\t13320\t&\t5723\t&\t10509\t&\t30017\t&\t23372\t&\t12724\t&\t25181\t&\t52259\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t25090\t&\t13451\t&\t25385\t&\t58593\t&\t16450\t&\t6534\t&\t14970\t&\t35654\t&\t29631\t&\t15747\t&\t34124\t&\t63501\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t29298\t&\t15253\t&\t30770\t&\t63818\t&\t19658\t&\t8361\t&\t16303\t&\t40251\t&\t33967\t&\t17384\t&\t37712\t&\t67354\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t30893\t&\t15025\t&\t29329\t&\t62347\t&\t20976\t&\t9012\t&\t19664\t&\t41450\t&\t35155\t&\t18862\t&\t42001\t&\t67561\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t31911\t&\t15795\t&\t32651\t&\t66267\t&\t22010\t&\t9312\t&\t20796\t&\t42488\t&\t35670\t&\t19892\t&\t43887\t&\t69423\t\\\\\t\\cline{1-14}\n\\hline\n\\multirow{8}{*}{38} &\tDib E et al.\t&\t12808\t&\t5091\t&\t10029\t&\t22576\t&\t9452\t&\t2567\t&\t3519\t&\t6282\t&\t13878\t&\t1650\t&\t2798\t&\t5890\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 4)\t&\t2070\t&\t697\t&\t1396\t&\t5099\t&\t1470\t&\t511\t&\t1120\t&\t4115\t&\t3161\t&\t1300\t&\t2589\t&\t7469\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 8)\t&\t3312\t&\t1179\t&\t2248\t&\t7874\t&\t1817\t&\t604\t&\t1350\t&\t5095\t&\t3839\t&\t1278\t&\t2615\t&\t7893\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 16)\t&\t4414\t&\t1452\t&\t2788\t&\t9105\t&\t2418\t&\t768\t&\t1562\t&\t5528\t&\t4534\t&\t1754\t&\t3159\t&\t8763\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 28)\t&\t5149\t&\t1711\t&\t3140\t&\t9691\t&\t2826\t&\t777\t&\t1460\t&\t6012\t&\t5114\t&\t1616\t&\t3265\t&\t9168\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 44)\t&\t5836\t&\t1842\t&\t3717\t&\t11701\t&\t3085\t&\t1173\t&\t2064\t&\t6325\t&\t5833\t&\t2084\t&\t4167\t&\t10721\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 52)\t&\t6056\t&\t2071\t&\t3957\t&\t11951\t&\t3333\t&\t1056\t&\t1998\t&\t6683\t&\t5905\t&\t1891\t&\t4093\t&\t10013\t\\\\\t\\cline{2-14}\n&\tProposed (Rank 60)\t&\t6124\t&\t1844\t&\t3440\t&\t10214\t&\t3445\t&\t1113\t&\t2127\t&\t6741\t&\t5953\t&\t1987\t&\t3931\t&\t10061\t\\\\\t\\cline{1-14}\n\\hline\n\\end{tabular} }\n}\n\\end{table*}\n\n\n\\subsection{Experimental Settings and Implementation Details}\\label{subsec41}\n\nThe proposed scheme is implemented on a single high-end HP OMEN X 15-DG0018TX system with 9th Gen i7-9750H, 16 GB RAM, RTX 2080 8 GB Graphics, and Windows 10 operating system. The multiplicative layers for each view subset of the four scanning orders are optimized using their corresponding convolutional neural network (CNN). The CNN contains twenty 2D convolutional layers stacked in a sequence. They have constant spatial size of tensors throughout and only the number of channels is varied. The first layer in each CNN corresponds to the tensor $\\mathbf{L}$, which contains all the pixels of subset $L(u,v,s,t)$ being handled. $\\mathbf{L}$ has number of channels equal to the number of viewpoints in respective subset and tensor $\\mathbf{T}$ (that contained all the pixels of $T_{x}(u,v)$) has 3 channels corresponding to the 3 light field subset multiplicative layers. Intermediate convolutional layers comprise of 64 channels each. In Fig.~\\ref{fig:workflow_encoding}, 'ch' refers to the channels in the convolutional layers. We train each CNN model for 25 epochs at a learning rate of 0.001 and a batch size of 15. The entire networks are implemented using the Python-based framework, Chainer (version 7.7.0).\n\n\nThe resultant output multiplicative layers for each subset of  $C_2$, $C_4$, $H_2$ and $H_4$ patterns are obtained from the trained CNN models. Fig.~\\ref{fig:fountlayers} illustrates the three multiplicative layers produced for each view subset of Circular-2 and Circular-4 patterns in the \\textit{Fountain \\& Vincent 2} data. The three multiplicative layers of each subset of Hierarchical-2 and Hierarchical-4 are shown in Fig.~\\ref{fig:bikeslayers} for the \\textit{Bikes} data.  We rearranged the colour channels of these multiplicative layers as described in section~\\ref{sec:bksvd} and then applied BK-SVD for ranks 4, 8, 16, 28, 44, 52 and 60. The approximated matrices are then arranged back into layers and compressed using HEVC (BLOCK II of COMPONENT I). We use quantization parameters 2, 6, 10, 14, 20, 26, and 38 to test both high and low bitrate cases of HEVC. The decoding and reconstruction of BK-SVD approximated subsets for all four patterns are then performed.\n\n\nLow-rank approximated subsets are then utilized to form the FDL representation of light fields. The number of layers in the FDL method are fixed to $n = 30$. Views in approximated Subset 1 construct the initial FDL representation. The subsequent view subsets are predicted from this FDL representation and the residues iteratively refine the FDL representation. We used HEVC  to perform the encoding in COMPONENT II, choosing quantization parameters 2, 6, 10, 14, 20, 26 and 38. The final reconstructed light field central views at the end of COMPONENT III for the \\textit{Stone pillars outside} data is illustrated in Fig.~\\ref{fig:stone_recon} for the BK-SVD ranks 28, 44 and 60.\n\n\n\n\\begin{table*}\n\\centering\n\\caption{\\footnotesize Bjontegaard percentage rate savings for the proposed compression scheme with respect to Dib et al. \\cite{dib2019light}. Negative values represent gains.}\n\\label{table_bdpsnr}\n\\resizebox{6cm}{!}{\n\\makebox[\\linewidth]{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n &  & \\multicolumn{2}{|c|}{Circular-2} & \\multicolumn{2}{|c|}{Hierarchical-2} & \\multicolumn{4}{|c|}{Circular-4} & \\multicolumn{4}{|c|}{Hierarchical-4} \\\\\n\\hline\nScene & Rank & Subset 1 & Subset 2 & Subset 1 & Subset 2 & Subset 1 & Subset 2 & Subset 3 & Subset 4 & Subset 1 & Subset 2 & Subset 3 & Subset 4\\\\\n\\hline\n\\multirow{7}{*}{Bikes} &\t4\t&\t-97.56541214\t&\t-99.29882593\t&\t-98.3907291\t&\t-99.54258115 &\t-96.20857709\t&\t-99.34681784\t&\t-99.32842039\t&\t-99.34121406\t&\t-96.7351185\t&\t-99.00868427\t&\t-99.47500508\t&\t-99.34379191\t\\\\\t\\cline{2-14}\n&\t8\t&\t-95.76892379\t&\t-99.03123267\t&\t-97.66246664\t&\t-99.29956835 &\t-93.72289918\t&\t-98.65123258\t&\t-98.60472387\t&\t-98.82635815\t&\t-94.28513258\t&\t-97.74418081\t&\t-98.647811\t&\t-98.71977919\t\\\\\t\\cline{2-14}\n&\t16\t&\t-93.17474979\t&\t-98.45339444\t&\t-96.63095962\t&\t-98.95497164 &\t-89.76525824\t&\t-97.16858635\t&\t-96.92654865\t&\t-98.03975782\t&\t-90.71898512\t&\t-95.38280704\t&\t-97.04688734\t&\t-97.88951207\t\\\\\t\\cline{2-14}\n&\t28\t&\t-90.3477664\t&\t-98.11361299\t&\t-95.58497187\t&\t-98.67266268 &\t-84.94581006\t&\t-95.34857594\t&\t-94.83394299\t&\t-97.03298825\t&\t-86.86256935\t&\t-92.89572507\t&\t-95.15379369\t&\t-97.0052932\t\\\\\t\\cline{2-14}\n&\t44\t&\t-87.34955344\t&\t-97.75859383\t&\t-94.60404741\t&\t-98.07437089 &\t-79.85797265\t&\t-92.77852898\t&\t-92.20674636\t&\t-95.85782231\t&\t-82.89202874\t&\t-90.05487069\t&\t-92.67313576\t&\t-96.06675045\t\\\\\t\\cline{2-14}\n&\t52\t&\t-86.18089698\t&\t-97.59830221\t&\t-94.16664723\t&\t-97.93405439 &\t-77.60680642\t&\t-91.69706112\t&\t-91.16454361\t&\t-95.4561184\t&\t-81.31504874\t&\t-89.00949178\t&\t-91.89758467\t&\t-95.79899808\t\\\\\t\\cline{2-14}\n&\t60\t&\t-85.37903466\t&\t-97.4382535\t&\t-93.89176465\t&\t-97.88383496 &\t-75.79473857\t&\t-90.6698963\t&\t-90.38054322\t&\t-95.1466604\t&\t-80.07701534\t&\t-87.21412008\t&\t-90.0680445\t&\t-95.07426992\t\\\\\t\\cline{1-14}\n\\hline\n\n\\multirow{7}{*}{Stone pillars outside } &\t4\t&\t-98.28738998\t&\t-99.5671618\t&\t-98.54643236\t&\t-99.69494577 &\t-96.70390985\t&\t-99.63514874\t&\t-99.63502461\t&\t-99.59656973\t&\t-96.57281486\t&\t-99.34377591\t&\t-99.50760509\t&\t-99.37042389\t\\\\\t\\cline{2-14}\n&\t8\t&\t-97.48146218\t&\t-99.45696678\t&\t-97.88804729\t&\t-99.65675017 &\t-96.83115691\t&\t-99.35667758\t&\t-99.32191491\t&\t-99.35202452\t&\t-96.76795633\t&\t-99.10596584\t&\t-98.99132784\t&\t-99.12125107\t\\\\\t\\cline{2-14}\n&\t16\t&\t-95.3611157\t&\t-98.79333481\t&\t-96.93436888\t&\t-99.10493755 &\t-93.99292616\t&\t-98.16597465\t&\t-98.2962672\t&\t-98.53366947\t&\t-94.36508582\t&\t-98.03801948\t&\t-98.47550354\t&\t-98.38338342\t\\\\\t\\cline{2-14}\n&\t28\t&\t-93.25557142\t&\t-98.75966092\t&\t-95.85758052\t&\t-98.75257079 &\t-90.4631812\t&\t-96.53677355\t&\t-96.49717131\t&\t-97.50938121\t&\t-91.45018488\t&\t-97.04731002\t&\t-97.62269234\t&\t-98.45732096\t\\\\\t\\cline{2-14}\n&\t44\t&\t-91.33970709\t&\t-98.53538189\t&\t-94.86352824\t&\t-98.5749095\t&\t-86.68637432\t&\t-93.88697432\t&\t-94.29482711\t&\t-97.03694131\t&\t-88.48088951\t&\t-95.14677527\t&\t-95.98370386\t&\t-97.95248612 \\\\\t\\cline{2-14}\n&\t52\t&\t-91.07697181\t&\t-98.45815322\t&\t-94.57758068\t&\t-98.46705075 &\t-85.22044971\t&\t-93.13489515\t&\t-93.67714426\t&\t-96.75798999\t&\t-87.20785431\t&\t-94.70593446\t&\t-95.65455146\t&\t-97.72227925\t\\\\\t\\cline{2-14}\n&\t60\t&\t-90.76161372\t&\t-98.35636369\t&\t-94.30087553\t&\t-98.40819422 &\t-84.15256024\t&\t-93.01769387\t&\t-93.33095286\t&\t-96.66548892\t&\t-86.25854702\t&\t-94.27613508\t&\t-95.06018928\t&\t-97.5857605\t\\\\\t\\cline{1-14}\n\\hline\n\n\\multirow{7}{*}{Fountain \\& Vincent 2} &\t4\t&\t-96.27656828\t&\t-99.26603902\t&\t-96.9530291\t&\t-99.57099646 &\t-93.24324684\t&\t-98.55291445\t&\t-98.36932301\t&\t-98.98906935\t&\t-94.92713556\t&\t-97.50272257\t&\t-98.13359141\t&\t-98.93827929\t\\\\\t\\cline{2-14}\n&\t8\t&\t-94.48002919\t&\t-98.96179277\t&\t-96.34841383\t&\t-99.08720888 &\t-90.2813932\t&\t-97.37199047\t&\t-97.20967603\t&\t-98.62957034\t&\t-92.52170343\t&\t-95.77161689\t&\t-96.4265349\t&\t-98.44379923\t\\\\\t\\cline{2-14}\n&\t16\t&\t-91.14745649\t&\t-99.02010069\t&\t-95.17447177\t&\t-99.23270338 &\t-84.41060243\t&\t-94.13934209\t&\t-93.86215798\t&\t-97.62140921\t&\t-88.53676951\t&\t-92.8003062\t&\t-94.58597524\t&\t-97.81833936\t \\\\\t\\cline{2-14}\n&\t28\t&\t-87.71507969\t&\t-98.92810522\t&\t-93.90335982\t&\t-98.97503776 &\t-78.21793701\t&\t-92.83126167\t&\t-91.91796171\t&\t-96.58176205\t&\t-83.95868385\t&\t-91.25156494\t&\t-92.28424221\t&\t-96.94553427\t\\\\\t\\cline{2-14}\n&\t44\t&\t-85.20018338\t&\t-98.74257926\t&\t-92.71888896\t&\t-98.94441882 &\t-73.07216196\t&\t-91.43574166\t&\t-90.58850443\t&\t-95.97145607\t&\t-80.11693215\t&\t-86.79417239\t&\t-88.96051247\t&\t-95.75589133\t\\\\\t\\cline{2-14}\n&\t52\t&\t-84.44629524\t&\t-98.77812886\t&\t-92.50450918\t&\t-98.83289017 &\t-71.5870472\t&\t-89.37972401\t&\t-88.61773338\t&\t-95.48946614\t&\t-78.80228046\t&\t-85.31225663\t&\t-87.39959326\t&\t-95.3181428\\\\\t\\cline{2-14}\n&\t60\t&\t-83.73495541\t&\t-98.74446465\t&\t-92.34241435\t&\t-98.90331735 &\t-70.27253249\t&\t-88.33184967\t&\t-87.93389356\t&\t-95.17732486\t&\t-77.72435271\t&\t-85.15958083\t&\t-87.64733413\t&\t-95.43459988\t\\\\\t\\cline{1-14}\n\n\\hline\n\n\\end{tabular} }\n}\n\\end{table*}\n\n\\subsection{Results and Comparative Analysis}\n\nWe compare the proposed scheme with Dib et al.~\\cite{dib2019light} light field coding algorithm. The proposed coding scheme outperforms by large margins. The total number of bytes taken by our approach is comparatively far lesser than the work by Dib et al.~\\cite{dib2019light} for all ranks and QPs. The corresponding results are depicted in Table~\\ref{table_bytesC2} for $C_2$, Table~\\ref{table_bytesH2} for $H_2$, Table~\\ref{table_bytesC4} for $C_4$ and Table~\\ref{table_bytesH4} for $H_4$ patterns. The bitrate vs YUV-PSNR graphs of three datasets in $C_2$, $C_4$, $H_2$ and $H_4$ configurations are illustrated in Fig.~\\ref{fig:graphs}. For all four scanning patterns, the proposed scheme has significantly better rate-distortion results considering both subset-wise light field and entire light field. \n\nFurther, we analyze  bitrate reduction (BD-rate) of the proposed scheme with respect to Dib et al.~\\cite{dib2019light} using Bjontgaard metric~\\cite{bjontegaard2001calculation}. The average percent difference in rate change is estimated over a range of QPs for seven chosen ranks. A comparison of the percentage of bitrate savings of our proposed coding scheme with respect to the anchor method for three chosen light field datasets is shown in Table~\\ref{table_bdpsnr}. For $C_2$ pattern, the proposed scheme achieves $94.53\\%$, $96.39\\%$, and $93.96\\%$ bitrate reduction compared to Dib et al.~\\cite{dib2019light} for light fields \\textit{Bikes}, \\textit{Stone pillars outside}, and \\textit{Fountain \\& Vincent 2} respectively. For pattern $H_2$, the proposed scheme achieves $97.23\\%$, $97.54\\%$, and $96.67\\%$ bitrate reduction compared to Dib et al.~\\cite{dib2019light} for the light fields \\textit{Bikes}, \\textit{Stone pillars outside}, and \\textit{Fountain \\& Vincent 2} respectively. Lastly, the $C_4$ and $H_4$ patterns have $93.09\\%$, $95.29\\%$, $90.71\\%$, and $93.18\\%$, $96.02\\%$, $91.25\\%$ bitrate reduction respectively in the proposed scheme over Dib et al.~\\cite{dib2019light} for \\textit{Bikes}, \\textit{Stone pillars outside}, and \\textit{Fountain \\& Vincent 2}.\n\n\n\n\\begin{figure*}\n\\centering\n\n\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s_r28_q2.png}\n\\end{subfigure} \n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s_r28_q2.png}\n\\end{subfigure}\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c2_s_r28_q2.png}\n\\end{subfigure} \n\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s_r28_q2.png}\n\\end{subfigure} \n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s_r28_q2.png}\n\\end{subfigure}\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/c4_s_r28_q2.png}\n\\end{subfigure} \n\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s_r28_q2.png}\n\\end{subfigure} \n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s_r28_q2.png}\n\\end{subfigure}\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h2_s_r28_q2.png}\n\\end{subfigure} \n\n\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s_r28_q2.png}\n    \\caption{\\footnotesize Rank 28}\n\\end{subfigure} \n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s_r28_q2.png}\n  \\caption{\\footnotesize Rank 44}\n\\end{subfigure}\n\\begin{subfigure}{0.2\\textwidth}\n  \\includegraphics[width=\\linewidth]{.\/Figures\/h4_s_r28_q2.png}\n  \\caption{\\footnotesize Rank 60}\n\\end{subfigure} \n\\caption{\\footnotesize The central views of the reconstructed \\textit{Stone pillars outside} data for ranks 28, 44 and  60, encoded with a quantization of 2. The The first, second, third and fourth rows show results of Circular-2, Circular-4, Hierarchical-2 and Hierarchical-4 respectively.}\n\\label{fig:stone_recon}\n\\end{figure*}\n\n\\section{Conclusion}\\label{con}\nWe have proposed a novel hierarchical hybrid coding scheme for light fields based on transmittance patterns of low-rank multiplicative layers and Fourier Disparity Layers. Typical pseudo sequence based light field compression schemes \\cite{RwPsvRef1_liu2016pseudo,RwPsvRef3_ahmad2017interpreting,RwPsvRef4_ahmad2019computationally,RwPsvRef5_gu2019high} do not efficiently consider the similarities between horizontal and vertical views of a light field. Our proposed scheme not only exploits the spatial correlation in multiplicative layers of the subset views for varying low ranks, but also removes the temporal intra and inter-layer redundancies in the low-rank approximated representation of view subsets. The approximated light field is further compressed by eliminating intrinsic similarities among neighboring views of Circular-2, Circular-4, Hierarchical-2 and Hierarchical-4 patterns using Fourier Disparity Layers representation. This integrated compression achieves excellent bitrate reductions without compromising the quality of the reconstructed light field.\n\nOur scheme offers flexibility to cover a range of multiple bitrates using just a few trained CNNs to obtain a layered representation of the light field subsets. This critical feature sets our proposed scheme apart from other existing light field coding methods, which usually train a system (or network) to support only specific bitrates during the compression. Besides, existing coding approaches are not explicitly designed to target layered displays and are usually only classified to work for lenslet-based formats or sub-aperture images based pseudo-sequence light field representation. The proposed flexible coding scheme can support not just multi-layered light field displays, but is also adaptable to table-top~\\cite{maruyama2018implementation} or other variety of autostereoscopic displays.\n\n\nIn our future work, we plan to extend the proposed idea to light field displays with more than three light attenuating layers. Proof-of-concept experiments with our scheme also pave the way to form a more profound understanding in the rank-analysis of a light-field using other mathematically valid tensor-based models~\\cite{wetzstein2012tensor,kobayashi2017focal} and coded mask cameras~\\cite{maruyama2019coded}. We also aim to verify our scheme with physical light field display hardware. Further, we wish to adapt the proposed scheme with display device availability and optimize the bandwidth for a target bitrate. This would enable deploying the concepts of layered displays on different display platforms that deliver 3D contents with limited hardware resources and thus best meet the viewers' preferences for depth impression and visual comfort.\n\n\\section*{Author contributions}\n\\textbf{Joshitha Ravishankar:} Methodology, Software, Validation, Formal analysis, Investigation, Data Curation, Writing- Original Draft, Visualization. \\textbf{Mansi Sharma:} Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Writing- Original Draft, Supervision, Project administration.\n\n\n\\section*{Acknowledgement}\nThe scientific efforts leading to the results reported in this paper have been carried out under the supervision of Dr. Mansi Sharma, INSPIRE Hosted Faculty, IIT Madras. This work has been supported, in part, by the Department of Science and Technology, Government of India project \\textit{``Tools and Processes for Multi-view 3D Display Technologies''}, DST\/INSPIRE\/04\/2017\/001853.\n\nWe would like to thank Nikitha Varma Sunchu and P Sai Shankar Pavan Srinivas for their help in running the codes for experiments. \n\n\n\\section*{Declaration of interests}\nThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOur main interest is  the following  system of semilinear stochastic PDE with value in  $\\mathbb{R}^k$,\n\\small\n\\begin{equation}\n\\begin{split}\n\\label{SPDE1}   du_t (x) + [\\mathcal {L}u_t (x) +f_t(x,u_t (x),\\nabla u_t\\sigma (x))] dt +  h_t(x,u_t(x),\\nabla u_t\\sigma(x))\\cdot d\\overleftarrow{W}_t  = 0, \\,\n\\end{split}\n\\end{equation}\nover the time interval $[0,T]$. The final condition is given by $u_T = \\Phi$, $f,$\n$h$ are  non-linear random functions  and  $\\mathcal L$ is the second order differential operator  which is defined\n component by component by\n \\begin{equation}\\label{operator conv}\n \\begin{array}{lll}\n {\\mathcal L}\\varphi(x)&=&\\displaystyle\\sum_{i=1}^{d}b^i(x)\\frac{\\partial}{\\partial\n x_i}\\varphi(x)+\\frac{1}{2}\\sum_{i,j=1}^{d}a^{ij}(x)\\frac{\\partial^2}{\\partial\n x_i\\partial x_j}\\varphi(x).\n\\end{array}\n\\end{equation}\nThe integral term with $d\\overleftarrow{W}_t$\nrefers to the backward stochastic integral with respect to a $d$-dimensional Brownian motion on\n$\\big(\\Omega, \\mathcal{F},\\mathbb{P}, (W_t)_{t\\geq 0} \\big)$. We use the backward notation  because in the proof we will employ the backwards doubly stochastic framework introduced by Pardoux and Peng \\cite{pp1994}.\\\\\n Such SPDEs appear in various applications like pathwise stochastic control problems, the Zakai equations in filtering and stochastic control with partial observations.  It is well known now that  BSDEs give a probabilistic interpretation for the solution of a class of semi-linear PDEs. \nBy introducing in  standard BSDEs a second nonlinear term driven by an external noise, we obtain Backward Doubly SDEs (BDSDEs in short) \\cite{pp1994} (see also \\cite{BM01}, \\cite{MS02}), which can be seen as Feynman-Kac representation of SPDEs and provide a powerful tool for probabilistic  numerical  schemes \\cite{matouetal13}  for such SPDEs. Several generalizations to investigate more general nonlinear SPDEs have been developed following different approaches of the notion of weak solutions, namely,  Sobolev's solutions  \\cite{DS04, GR00, Krylov99, SSV03, Walsh},  and  stochastic viscosity  solutions \\cite{lion:soug:98, lion:soug:00, lion:soug:01, buck:ma:10a, buck:ma:10b}. \\\\\n\n Given  a  convex domain $ D$ in $ \\mathbb{R}^k$, our paper is concerned with the study of  weak solutions  to the reflection problem for  multidimensional SPDEs  \\eqref{SPDE1}   in $D$ by introducing the associated BDSDE. \\\\ \nInspired by the variational  formulation  of the obstacle problem for SPDEs and  Menaldi's work  \\cite{M83}  on  reflected diffusion, we consider the  solution of the  refection problem for the SPDEs \\eqref{SPDE1}  as a  pair $ (u, \\nu)$, where $ \\nu$ is a random regular measure and $ u \\in \\mathbf{L}^2 \\big(\\Omega \\times [0,T]; H^1 (\\mathbb R^d)\\big)$ satisfies the following relations :\n\\small\n\\begin{equation}\n\\begin{split}\n\\label{OSPDE1}\n& (i) \\; \\;    u_t(x) \\in \\bar D , \\quad d\\mathbb{P}\\otimes dt\\otimes dx - \\mbox{a.e.},  \\\\\n& (ii)\\;\\; du_t (x) + \\big[\n \\;  \\mathcal{L} u_t (x)  + f_t(x,u_t (x),\\nabla u_t\\sigma(x)) \\, \\big]\\,  dt +  h_t(x,u_t(x),\\nabla u_t\\sigma(x))\\cdot d\\overleftarrow{W}_t  = - \\nu (dt,dx), \\quad a.s.,  \\\\\n& (iii)\\; \\; \\nu(u\\notin \\partial  D)=0 , a.s.,\\\\\n & (iv)  \\; \\;  u_T = \\Phi, \\quad dx-\\mbox{a.e.}.\n\\end{split}\n\\end{equation}\n$\\nu$  is a random measure which  acts only when  the process $u$ reaches the boundary of  the domain $ D$.\n The rigorous  sense of the relation $(iii)$ will be based on the  probabilistic representation of the measure $\\nu$  in term of the bounded variation processes $K$, a component of the associated  solution of the reflected  BDSDE in the domain $D$. This  problem is  well known as a Skorohod  problem for SPDEs.\\\\  \nIn the case of diffusion processes  in a domain,  the reflection problem has been investigated by severals authors (see  \\cite{S61}, \\cite{W71},  \\cite{ElkChM80},   \\cite{LS84}). In the case of a convex domain this reflection problem was treated by Tanaka \\cite{T79} and Menaldi \\cite{M83} by using the variational inequality and the convexity properties of the domain.\\\\\nIn the one dimensional case, the reflection problem  for nonlinear PDEs (or SPDEs) has been studied by using different approaches.  \n\n The work of El Karoui et al \\cite{Elk2} treats the obstacle problem for viscosity solution of  deterministic semilinear PDEs  within the framework of backward stochastic differential equations (BSDEs in short). \nThis increasing process determines in fact the measure from the relation $(ii)$.  \nBally et al \\cite{BCEF} (see also Matoussi and Xu \\cite{MX08})  point out that the continuity of this process allows the classical notion of strong variational solution to be extended (see Theorem 2.2 of \\cite{BensoussanLions78} p.238) and express the solution to the obstacle as a pair $(u, \\nu)$ where $ \\nu$ is supported by the set  $\\{u=g\\}$. \\\\\nMatoussi and Stoica \\cite{MS10} have proved an existence and uniqueness result for the obstacle problem of backward quasilinear stochastic PDE on the whole space $\\mathbb{R}^d $ and driven by a finite dimensional Brownian motion. The method is based on the probabilistic interpretation of the solution by us- ing the backward doubly stochastic differential equation. They have also proved that the solution $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\\nu$ is a random regular measure satisfying the minimal Skohorod condition. In particular, they gave for the regular measure $\\nu$ a probabilistic interpretation in terms of the continuous increasing process $K$ where $(Y,Z,K)$ is the solution of a reflected generalized BDSDE.\n\nOn the other hand, M.Pierre \\cite{Pierre, Pierre80}  has studied parabolic PDEs with obstacles using the parabolic potential as a tool. He proved that the solution uniquely exists and is quasi-continuous with respect to the analytical capacity.  Moreover he gave a representation of the reflected measure $\\nu$ in terms of the associated regular potential and the approach used  is based  on analytical quasi-sure analysis.  More recently,  Denis, Matoussi and  Zhang \\cite{DMZ12} have extended this approach \nfor the obstacle problem of quasilinear SPDEs when the obstacle is regular in some sense and controlled by the solution of a SPDE.\\\\\n\nNualart and Pardoux \\cite{Nualart} have studied the obstacle problem for a nonlinear heat equation on the spatial interval $[0,1]$ with Dirichlet boundary conditions, driven by an additive space-time white noise. They proved the existence and uniqueness of the solution and their method relied heavily on the results for a deterministic variational inequality. Donati- Martin and Pardoux \\cite{Donati-Pardoux} generalized the model of Nualart and Pardoux. The nonlinearity appears both in the drift and in the diffusion coefficients. They proved the existence of the solution by penalization method but they did not obtain the uniqueness result. And then in 2009, Xu and Zhang solved the problem of the uniqueness, see \\cite{XZ09}.  We note also that Zhang established in \\cite{Z11} the existence and uniqueness of solutions of system \\eqref{OSPDE1} in the forward case when $x$ belongs to $[0,1]$. He approximated the system of SPDEs by a penalized system and used a number of a priori estimates to prove the convergence of the solutions.\nHowever, in all their models,  they do not consider the case where the coefficients depend on $\\nabla u$.\\\\\n\nOur contributions in this paper are as following: first of all,  reflected BDSDEs in the convex domain $D$ are introduced and results of existence and uniqueness of  such RBDSDEs are established. Next, the existence and uniqueness results of the solution $(u, \\nu)$ of the reflection  problem for \\eqref{SPDE1} are given in Theorem \\ref{existence:RSPDE}. Indeed, a probabilistic method based on reflected BDSDEs and stochastic flow technics   are investigated in our context (see e.g. \\cite{BM01}, \\cite{MS02}, \\cite{K94a, K94b}  for these flow technics). \nThe key element in \\cite{BM01} is to use  \nthe inversion of stochastic flow which transforms the variational\nformulation of the SPDEs to the associated BDSDEs. Thus it plays the same role as It\\^o's formula in the case of the classical solution of SPDEs.  \\\\\nWe also mention the works \\cite{CEK11}, \\cite{HZ10} and  \\cite{HT10} where they have studied  a  Reflected BSDEs  with oblique reflection in multi-dimensional case and their relations   to switching  problems.\\\\\n  \n  This paper is organized as following: in Section 2, first the basic assumptions and the definitions of the solutions\nfor  Reflected BDSDE in a convex domain are presented.  Then,  existence and  uniqueness of  solution of RBDSDE (Theorem \\ref{existence:RBDSDE}) is given under only convexity assumption for the domain without any regularity on the boundary. This result  is  proved  by using  penalization approximation. Thanks to the convexity properties we prove several technical lemmas, in particular the  fundamental Lemma \\ref{fundamental:lemma}. In Section 3, we study  semilinear SPDE's in a convex domain.  We first  provide useful results on stochastic flow associated with the forward SDEs, then in this\n setting  as in Bally and Matoussi \\cite{BM01}, an equivalence norm result associated to the diffusion process  is given. The main result of this section  Theorem \\ref{existence:RSPDE} is the existence and uniqueness results\n   of the solution of reflected SPDEs in a convex domain. The proof of this result is  based on the probabilistic interpretation via the Reflected Forward-BDSDEs.  The uniqueness is a consequence of the variational formulation of the SPDEs  written with random test functions and the uniqueness of the solution of the Reflected  FBSDE.\n    The existence of the solution is established by an approximation penalization  procedure, a priori estimates \n    and the equivalence norm results.  In the Appendix, technical lemmas for the existence of the solution of the Reflected BDSDEs and SPDEs in a convex domain are given.\n\n\n\n\\section{Backward Doubly Stochastic Differential Equations in a domain}\n\\subsection{Hypotheses and preliminaries}\n\\label{hypotheses}\nThe euclidean norm of a vector $x\\in\\mathbb{R}^k$ will be denoted by $|x|$, and for a $k\\times k$ matrix $A$, we define $\\|A\\|=\\sqrt{Tr AA^*}$. In what folllows let us fix a positive number $T>0$.\\\\\nLet $(\\Omega, {\\cal F},\\mathbb{P})$ be a probability product space, and let $\\{W_s, 0\\leq s\\leq T\\}$ and $\\{B_s, 0\\leq s\\leq T\\}$ be two mutually independent standard Brownian motion processes, with values respectively in $\\mathbb{R}^d$ and in $\\mathbb{R}^l$.\nFor each $t\\in[0,T]$, we define\n$${\\cal F}_t:={\\cal F}_t^B\\vee{\\cal F}_{t,T}^W\\vee {\\cal N}$$\nwhere ${\\cal F}_t^B=\\sigma\\{B_r, 0\\leq r\\leq t\\}$, ${\\cal F}_{t,T}^W=\\sigma\\{W_r-W_t, t\\leq r\\leq T\\}$ and ${\\cal N}$ the class of $\\mathbb{P}$ null sets of ${\\cal F}$.\nNote that the collection $\\{{\\cal F}_t, t\\in[0,T]\\}$ is neither increasing nor decreasing, and it does not constitute a filtration.\n\\subsubsection{Convexity results}\n\\label{convexity:subsubsection}\nBesides, we need to  recall properties related to the convexity of a nonempty domain $D$ in $\\mathbb{R}^k$.  Let $\\partial D$ denotes the boundary of $D$ and  $\\pi(x)$ the projection of $x\\in\\mathbb{R}^k$ on $D$. We have the following properties:\n\\begin{eqnarray}\n(x'-x)^*(x-\\pi(x))\\leq 0,  ~~ \\forall x\\in\\mathbb{R}^d, ~ \\forall x'\\in \\bar{D}\\label{prop1}\n\\end{eqnarray}\n\\begin{eqnarray}\n(x'-x)^*(x-\\pi(x))\\leq (x'-\\pi(x'))^*(x-\\pi(x)),  ~~ \\forall x, x'\\in\\mathbb{R}^k\\label{prop2}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\exists a \\in D, \\gamma > 0,\\, \\mbox{such that}\\,\\, (x-a)^*((x-\\pi(x))\\geq \\gamma|x-\\pi(x)|, ~~ \\forall x\\in\\mathbb{R}^k. \\label{prop3}\n\\end{eqnarray}\nFor $x\\in\\partial D$, we denote by $n(x)$ the set of outward normal unit vectors at the point $x$.\\\\\n\n\n\\noindent Since $D$ is not regular, we define a sequence of regular convex domains which approximate uniformly $D$. Indeed, the function $h(x)= d(x,D)$ is convex and uniformly continuous in $\\mathbb{R}^k$. If we denote $(g_\\delta)_{0\\leq \\delta\\leq\\delta_0}$ the approximation identity with compact supports, then $h_\\delta=g_\\delta*h$ is a sequence of regular convex functions which tends uniformly to $h$ as $\\delta\\rightarrow 0$. For a fixed $\\eta >0$, $\\{x, h_\\delta(x) <\\eta\\}$ are regular convex domains that converge uniformly to $\\{x, d(x,D) <\\eta\\}$ when $\\delta$ tends to $0$.  Letting $\\eta\\rightarrow 0$, we conclude that for all $\\varepsilon >0$ there exists a regular convex domain $D_\\varepsilon$ such that\n\\begin{eqnarray}\n\\underset{x\\in D}{\\displaystyle\\sup}\\, d(x,D_\\varepsilon) <\\varepsilon \\quad \\mbox{and} \\quad \\underset{x\\in D_\\varepsilon}{\\displaystyle\\sup}\\, d(x,D) <\\varepsilon\n\\label{propappro}\n\\end{eqnarray}\nOne can find all these results in Menaldi \\cite{M83}, page 737.\n\\subsubsection{Functional spaces and  assumptions}\nHereafter, let us define the spaces and the norms which will be needed for the formulation of the BDSDE in a domain.\\\\\n\\begin{description}\n\\item[-] $\\mathbf{L}^p_k({\\mathcal F}_T)$ the space of $k$-dimensional\n${\\mathcal F}_T$-measurable random variables $\\xi$ such that\n$$\\begin{array}{ll}\n          \\|\\xi\\|_{L^p}^p:=\\mathbb{E}(|\\xi|^p)<+\\infty;\n  \\end{array}$$\n\n\\item[-] ${\\mathcal H}^2_{k\\times d}([0,T])$ the space of $\\mathbb{R}^{k\\times\nd}$-valued ${\\cal F}_t$-measurable process $Z=(Z_t)_{t\\leq T}$\nsuch that\n$$\\begin{array}{ll}\n          \\|Z\\|_{{\\mathcal H}^2}^2:= \\mathbb{E}[\\displaystyle\\int_{0}^{T}|Z_t|^2dt]<+\\infty;\n  \\end{array}$$\n\n\\item[-] ${\\mathcal S}^2_k([0,T])$ the space of $\\mathbb{R}^{k}$-valued ${\\cal F}_t$-adapted processes $Y=(Y_t)_{t\\leq T}$, with continuous paths such that\n$$\\|Y\\|_{{\\mathcal S}^2}^2:= \\mathbb{E}[\\,\\underset{t\\leq T}{\\displaystyle\\sup}\\,|Y_t|^2]<+\\infty;$$\n\n\\item[-] ${\\mathcal A}^2_k([0,T])$ the space of $\\mathbb{R}^{k}$-valued ${\\cal F}_t$-adapted processes $K=(K_t)_{t\\leq T}$, with continuous  and bounded variation paths such that $K_0=0$ and \n$$\\|K\\|_{{\\mathcal A}^2}^2:= \\mathbb{E}[\\,\\underset{t\\leq T}{\\displaystyle\\sup}\\,|K_t|^2]<+\\infty.$$\n\\end{description}\nWe next state our main assumptions on the terminal condition $\\xi$ and the functions $f$ and $h$:\n\\begin{Assumption}\\label{Ass1} \n $\\xi\\in\\mathbf{L}^2_k({{\\cal F}}_T)$ and $\\xi\\in\\bar{D}$ a.s..\n \\end{Assumption}\n \\begin{Assumption}\\label{Ass2}\n $f:\\Omega\\times [0,T]\\times\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}\\rightarrow\\mathbb{R}^k ~,~ h:\\Omega\\times [0,T]\\times\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}\\rightarrow\\mathbb{R}^{k\\times l}$ are two random functions verifying:\n\\begin{itemize}\n\n\\item[\\rm{(i)}] For all $(y,z)\\in\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}$, $f_t(\\omega,y,z)$ and $h_t(\\omega,y,z)$ are ${\\cal F}_t$ - measurable.\n\n \\item [\\rm{(ii)}]$\\mathbb{E}\\big[\\displaystyle\\int_0^T |f_t(0,0)|^2dt\\big] < +\\infty$ \\quad,\\quad $\\mathbb{E}\\big[\\displaystyle\\int_0^T\\|h_t(0,0)\\|^2dt\\big] < +\\infty.$\n \\item [\\rm{(iii)}] There exist constants $c>0$ and $0<\\alpha<1$ such that for any $(\\omega,t)\\in\\Omega\\times[0,T]~;~\n(y_1,z_1),(y_2,z_2)\\in\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}$\n\\b*\n|f_t(y_1,z_1)-f_t(y_2,z_2)|^2 &\\leq & c\\big(|y_1-y_2|^2+\\|z_1-z_2\\|^2\\big)\\\\\n\\|h_t(y_1,z_1)-h_t(y_2,z_2)\\|^2 &\\leq & c|y_1-y_2|^2+\\alpha \\|z_1-z_2\\|^2.\n\\e*\n\\end{itemize}\n\\end{Assumption}\nWe denote by $f_t^0:=f_t(\\omega,0,0)$ and $h_t^0:=h_t(\\omega,0,0)$.\\\\\n\n\\noindent We add the following  further assumption:\n\\begin{Assumption}\\label{Ass3}\n\\begin{itemize}\n\\item[\\rm{(i)}] $\\xi\\in \\mathbf{L}^4_k({{\\cal F}}_T).$\n\\item[\\rm{(ii)}] There exist $c>0$ and $0\\leq \\beta < 1$ such that for all $(t,y,z)\\in [0,T]\\times\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}$\n$$h_t \\, h_t^* (y,z)\\leq c (Id_{\\mathbb{R}^k}+ yy^*)+ \\beta  \\, zz^*.$$\n\\item[\\rm{(iii)}] $f$ and $h$ are uniformly bounded in $(y,z)$.\n\\end{itemize}\n\\end{Assumption}\n\\begin{Remark}\n\\begin{enumerate}\n\\item The Assumption \\ref{Ass3} \\rm{(i)} and \\rm{(ii)} are needed to prove the uniform $L^4$-estimate for $(Y^n,Z^n)$ solution of BDSDE \\eqref{BDSDEpen} (see estimate \\eqref{estimateL4} in the Appendix \\ref{Proof of Lemma}). This is crucial for our proof of the fundamental lemma \\ref{fundamental:lemma}.\n\\item  The Assumption \\ref{Ass3} \\rm{(iii)} is only added for simplicity and it can be removed by standard technics of BSDEs. The natural condition instead of \\rm{(iii)} is $f^0$ and $h^0$ in $\\mathbf{L}^4(\\Omega, {\\cal F} ,\\mathbb{P}).$\n\\end{enumerate}\n\n\\end{Remark}\n\\begin{Remark}\\item We assume for the sake of simplicity that $D$ is a convex set with class $C^2$ boundary. If not we can approximate our convex domain $D$ by regular convex domains as mentioned in section \\ref{convexity:subsubsection}.\n\\end{Remark}\n\n\\noindent Now we introduce the definition of the solution of BDSDEs in a domain.\n\\begin{Definition}\\label{definition:RBSDE}\nThe triplet of processes $(Y_t,Z_t,K_t)_{\\{0\\leq t\\leq T\\}} $ is a solution of the backward doubly stochastic differential equation in a convex domain $D$, with\nterminal condition $\\xi$ and coefficients $f$ and $h$, if the following hold:\n\\begin{description}\n \\item[(i)]  $Y\\in{{\\cal S}}^2_k([0,T]) ~,~ Z\\in{{\\cal H}}^2_{k\\times d}([0,T])$ and $K \\in {\\mathcal A}^2_k([0,T])$,\n \\item[(ii)] \\begin{equation} \n\\label{RBDSDE} Y_t  = \\xi +\\displaystyle\\int_t^T f_s(Y_s,Z_s)ds +\\displaystyle\\int_t^T h_s(Y_s,Z_s)d\\overleftarrow{W}_s -\\displaystyle\\int_t^T Z_sdB_s +K_T-K_t ~,~ 0\\leq t\\leq T \\,\\,a.s.\n\\end{equation}\n \\item[(iii)] $Y_t\\in\\bar{D}~ ,~ 0\\leq t\\leq T,~ a.s.$\n \\item[(iv)] for any continuous progressively measurable process $ (z_t)_{0 \\leq t \\leq T}$ valued in $\\bar{D}$, \n \\begin{equation}\n \\label{skorohod1}\n \\displaystyle\\int_0^T (Y_t-z_t)^* dK_t \\leq 0, \\;  a.s.\n \\end{equation}\n\\end{description}\n\nThe triplet $(Y_t,Z_t,K_t)_{\\{0\\leq t\\leq T\\}}$ is called a solution of RBDSDE with data $(\\xi,f,h)$.\n\\end{Definition}\n\n\\vspace{0.2cm}\n\\begin{Remark}  From Lemma 2.1 in \\cite{GPP95}, the condition \\eqref{skorohod1} implies that the bounded variation process $K$ acts only when $Y$ reaches the boundary of the convex  domain $D$ and the so-called Skorohod condition  is satisfied:\n\\begin{equation}\n\\label{skorohod2}\n \\int_0^T {\\bf 1}_{ \\{Y_t \\in D\\} } dK_t =0.\n \\end{equation}\nMoreover there exits a $ \\mathcal F_t$-measurable process $ (\\alpha_t)_{0 \\leq t \\leq T}$ valued in $ \\mathbb R^k$ such that $$ K_t = \\displaystyle\\int_0^t \\alpha_s d\\|K_s\\|_{VT} \\quad \\mbox{and} \\; \n - \\alpha_s \\in n(Y_s).$$\n\\end{Remark}\n\\vspace{0.1cm}\n\\noindent In the following, $C$ will denote a positive constant which doesn't depend  on $n$ and can vary from line to line.\n\\subsection{Existence and uniqueness of the solution}\n\\label{existenceBDSDE:section}\nIn this section we establish existence  and uniqueness results for RBDSDE \\eqref{RBDSDE}.\n\n\n\\begin{Theorem}\\label{existence:RBDSDE}\nLet Asumptions \\ref{Ass1}, \\ref{Ass2} and \\ref{Ass3} hold. Then, the RBDSDE \\eqref{RBDSDE} has a unique solution $(Y,Z,K) \\in {\\mathcal S}^2_{k}([0,T]) \\times {\\mathcal H}^2_{k\\times d}([0,T]) \\times {\\mathcal A}^2_{k}([0,T])$.\n\\end{Theorem}\n\\begin{proof}\\\\\n{\\bf{a) Uniqueness}}: Let $(Y^1,Z^1,K^1)$ and $(Y^2,Z^2,K^2)$ be two solutions of  the RBDSDE  \\eqref{RBDSDE}. Applying generalized It\\^o'e formula (Lemma 1.3 in \\cite{pp1994} p.213) yields\n\\begin{eqnarray} \\label{itouni}\n|Y_t^1-Y_t^2|^2&+&\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds  =  2\\displaystyle\\int_t^T (Y_s^1-Y_s^2)^*(f_s(Y_s^1,Z_s^1)-f_s(Y_s^2,Z_s^2))ds\\nonumber\\\\\n& + & 2\\displaystyle\\int_t^T (Y_s^1-Y_s^2)^*(h_s(Y_s^1,Z_s^1)-h_s(Y_s^2,Z_s^2))d\\overleftarrow{W}_s -2\\displaystyle\\int_t^T (Y_s^1-Y_s^2)(Z_s^1-Z_s^2)dB_s\\nonumber\\\\\n&+& 2\\displaystyle\\int_t^T (Y_s^1-Y_s^2)^*(dK_s^1-dK_s^2) + \\displaystyle\\int_t^T \\|h_s(Y_s^1,Z_s^1)-h_s(Y_s^2,Z_s^2)\\|^2ds. \n\\end{eqnarray}\nTherefore, under the minimality condition (iv) we have \n\\begin{eqnarray}\\label{minest}\n\\displaystyle\\int_t^T (Y_s^1-Y_s^2)^*(dK_s^1-dK_s^2)\\leq 0 ,\\quad \\text{for all}~~ t\\in[0,T].\n\\end{eqnarray}\nThen, plugging (\\ref{minest}) in (\\ref{itouni}) and taking expectation we obtain\n\\end{proof}\n\\b* \n\\mathbb{E}[|Y_t^1-Y_t^2|^2]&+&\\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds] \\leq  2\\mathbb{E}[\\displaystyle\\int_t^T (Y_s^1-Y_s^2)^*(f_s(Y_s^1,Z_s^1)-f_s(Y_s^2,Z_s^2))ds]\\nonumber\\\\\n&+& \\mathbb{E}[\\displaystyle\\int_t^T \\|h_s(Y_s^1,Z_s^1)-h_s(Y_s^2,Z_s^2)\\|^2ds]. \n\\e*\nHence from the Lipschitz Assumption on $h$ and the inequality $2ab\\leq \\epsilon a^2+\\epsilon^{-1}b^2$, for all $\\epsilon>0$, it follows that \n\\b* \n\\mathbb{E}[|Y_t^1-Y_t^2|^2]&+&\\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds]\\leq  (c+\\epsilon) \\mathbb{E}[\\displaystyle\\int_t^T |Y_s^1-Y_s^2|^2ds]\\\\\n&+&\\epsilon^{-1}\\mathbb{E}[\\displaystyle\\int_t^T |f_s(Y_s^1,Z_s^1)-f_s(Y_s^2,Z_s^2)|^2ds]\n+\\alpha \\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds],\n\\e*\nwhere $0<\\alpha <1$. Choosing $\\epsilon=\\displaystyle\\frac{2 c}{1-\\alpha}$ and using the Lipschitz Assumption on $f$, we get\n\\b* \n\\mathbb{E}[|Y_t^1-Y_t^2|^2]&+&\\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds]\\leq  (c+\\displaystyle\\frac{2 c}{1-\\alpha}+\\displaystyle\\frac{1-\\alpha}{2}) \\mathbb{E}[\\displaystyle\\int_t^T |Y_s^1-Y_s^2|^2ds]\\\\\n&+&\\displaystyle\\frac{1-\\alpha}{2}\\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds]\n+\\alpha \\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds].\n\\e*\nConsequently\n$$\\mathbb{E}[|Y_t^1-Y_t^2|^2]+(\\displaystyle\\frac{1-\\alpha}{2})\\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^1-Z_s^2\\|^2ds]\\leq  (c+\\displaystyle\\frac{2 c}{1-\\alpha}+\\displaystyle\\frac{1-\\alpha}{2}) \\mathbb{E}[\\displaystyle\\int_t^T |Y_s^1-Y_s^2|^2ds].$$\nFrom Gronwall's lemma, $\\mathbb{E}[|Y_t^1-Y_t^2|^2] = 0,~ 0\\leq t\\leq T$, and $\\mathbb{E}[\\displaystyle\\int_0^T \\|Z_s^1-Z_s^2\\|^2ds] =0$.\\\\[0.3cm]\n{\\bf{b) Existence}}: The existence of a solution will be proved by penalisation method. For $n\\in \\mathbb{N}$, we consider for\nall $t\\in [0,T]$,\n\\begin{align}\nY_{t}^{n}=\\xi+\\displaystyle\\int_{t}^{T}f_s(Y_s^n,Z_s^n)ds+\\displaystyle\\int_{t}^{T}h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s-n\n\\displaystyle\\int_{t}^{T}(Y_{s}^{n}- \\pi(Y_s^n))ds\n-\\int_{t}^{T}Z_{s}^{n}dB_{s}.\n\\label{BDSDEpen}\n\\end{align}\nDenote by $K^n_t:=-n\\displaystyle\\int_0^t(Y_{s}^{n}- \\pi(Y_s^n))ds$. In order to prove the convergence of the sequence $(Y^n,Z^n,K^n)$ to the solution of our RBDSDE \\eqref{existence:RBDSDE}, we need several lemmas.\\\\\n\n\\noindent We start with the following lemma:  \n\\begin{Lemma} \\label{lem}There exists a constant $C>0$ such that\n\\begin{eqnarray} \n\\forall n\\in\\mathbb{N} \\qquad  \\mathbb{E}[\\displaystyle\\int_0^T d^2(Y_s^n,D)ds]\\leq C\\big(\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{n^2}\\big).\n\\end{eqnarray}\n\\end{Lemma}\n\\begin{proof}\nWe apply generalized It\\^o's formula (Lemma 1.3 in \\cite{pp1994} p.213) to $\\rho(Y_t^n)=d^2(Y_t^n,D)=|Y_t^n-\\pi(Y_t^n)|^2$ to obtain\n\\begin{eqnarray}\\begin{split} \n\\rho(Y_t^n)&+ \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\rho(Y_s^n)]ds = \\rho(\\xi)\n+ \\displaystyle\\int_t^T (\\nabla \\rho(Y_s^n))^* f_s(Y_s^n,Z_s^n)ds\\\\&- \\displaystyle\\int_t^T (\\nabla \\rho(Y_s^n))^* Z_s^n dB_s + \\displaystyle\\int_t^T (\\nabla \\rho(Y_s^n))^* h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s\\\\\n& + \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[(h_sh_s^*)(Y_s^n,Z_s^n)Hess \\rho(Y_s^n)]ds - 2n \\displaystyle\\int_t^T (Y_s^n- \\pi(Y_s^n))^*(Y_s^n-\\pi(Y_s^n))ds.\n\\label{itorho}\n\\end{split}\n\\end{eqnarray} \nSince $\\xi\\in \\bar{D} ~a.s.$, we have that $\\rho(\\xi)=0$. \nWe get from the boundedness of $h$ and the Hessian of $\\rho$\n\\begin{eqnarray}\n\\begin{split}\n\\rho(Y_t^n)&+ \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\rho(Y_s^n)]ds  + 2 n\\displaystyle\\int_t^T d^2(Y_s^n,D)ds\\\\\n&\\leq 2 \\displaystyle\\int_t^T (\\rho(Y_s^n))^{1\/2} |f_s(Y_s^n,Z_s^n)|ds - 2 \\displaystyle\\int_t^T (Y_s^n- \\pi(Y_s^n))^* Z_s^n dB_s\\\\\n&+2 \\displaystyle\\int_t^T (Y_s^n- \\pi(Y_s^n))^* h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s  +C.\n\\end{split}\n\\end{eqnarray}\nNow the inequality $2ab\\leq a^2+b^2$ with $a=\\sqrt{\\displaystyle\\frac{n}{2}\\rho(Y_s^n)}$ yields \n\\b*\n(\\rho(Y_s^n))^{1\/2} |f_s(Y_s^n,Z_s^n)|&\\leq &\\displaystyle\\frac{n}{4}\\rho(Y_s^n)+\\displaystyle\\frac{1}{n} |f_s(Y_s^n,Z_s^n)|^2.\n\\e*\nThen it follows that, \n\\begin{eqnarray}\n\\begin{split}\n\\rho(Y_t^n)&+ \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\rho(Y_s^n)]ds  + \\displaystyle\\frac{3n}{2}\\displaystyle\\int_t^T d^2(Y_s^n,D)ds\\\\\n&\\leq  2\\displaystyle\\int_t^T\\displaystyle\\frac{1}{n}|f_s(Y_s^n,Z_s^n)|^2ds - 2 \\displaystyle\\int_t^T (Y_s^n- \\pi(Y_s^n))^* Z_s^n dB_s \\\\\n&+2 \\displaystyle\\int_t^T (Y_s^n- \\pi(Y_s^n))^* h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s\n+ C.\n\\label{estY}\n\\end{split}\n\\end{eqnarray}\nBy taking expectation and using the boundedness of $f$, we have \n\\begin{eqnarray}\n\\mathbb{E}[\\rho(Y_t^n)]+\\displaystyle\\frac{1}{2}\\mathbb{E}[\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\rho(Y_s^n)]ds] + \\displaystyle\\frac{3n}{2}\\mathbb{E}[\\displaystyle\\int_t^T d^2(Y_s^n,D)ds]\\leq  C\\big(1+\\displaystyle\\frac{1}{n}\\big).\n\\label{estdist}\n\\end{eqnarray}\nHence, the required result is obtained.\n\\hbox{ }\\hfill$\\Box$ \n\\end{proof} \n\\vspace{0.5cm}\n\\noindent The next lemma plays a crucial role to prove the strong convergence of $(Y^n,Z^n,K^n)$.  \n\\begin{Lemma}\n\\label{fundamental:lemma}\n\\begin{eqnarray} \n\\mathbb{E}\\Big[\\underset{0\\leq t \\leq T}{\\displaystyle\\sup}(d(Y_t^n,D))^4\\Big]\\underset{n\\rightarrow +\\infty}{\\longrightarrow} 0.\n\\label{dist}\n\\end{eqnarray}\n\\end{Lemma}\n\\begin{proof} \nWe denote by $\\rho(x)=d^2(x,D)$ and $\\varphi(x)= \\rho^2(x)$.\nBy applying It\\^o's formula to $\\varphi(Y_t^n)=d^4(Y_t^n,D)$, we obtain that\n\\begin{eqnarray}\\begin{split} \n\\varphi(Y_t^n)&+ \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\varphi(Y_s^n)]ds = \\varphi(\\xi)\n+ \\displaystyle\\int_t^T (\\nabla \\varphi(Y_s^n))^* f_s(Y_s^n,Z_s^n)ds\\\\&- \\displaystyle\\int_t^T (\\nabla \\varphi(Y_s^n))^* Z_s^n dB_s + \\displaystyle\\int_t^T (\\nabla \\varphi(Y_s^n))^* h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s\\\\\n& + \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[(h_sh_s^*)(Y_s^n,Z_s^n)Hess \\varphi_(Y_s^n)]ds - n \\displaystyle\\int_t^T (\\nabla \\varphi(Y_s^n))^*(Y_s^n-\\pi(Y_s^n))ds.\n\\label{ito}\n\\end{split}\n\\end{eqnarray} \nSince $\\xi\\in \\bar{D} ~a.s.$, we have that $\\varphi(\\xi)=0$ and the chain rule of differentiation gives that\n \\begin{eqnarray}\n\\nabla \\varphi(x)&=&2\\rho(x)\\nabla\\rho(x)=4\\rho(x)(x-\\pi(x)) \\label{gradphi} \\\\\nHess \\varphi(x)&=&2\\nabla\\rho(x)(\\nabla\\rho(x))^*+2\\rho(x) Hess \\rho(x)\\label{hessphi}.\n\\end{eqnarray}\nThen it follows that\n\\begin{eqnarray}\\label{Ito}\n\\begin{split} \n\\varphi(Y_t^n)&+ \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\varphi(Y_s^n)]ds = 4\\displaystyle\\int_t^T (\\rho(Y_s^n)(Y_s^n-\\pi(Y_s^n))^* f_s(Y_s^n,Z_s^n)ds\\\\&- 4\\displaystyle\\int_t^T (\\rho(Y_s^n)(Y_s^n-\\pi(Y_s^n))^* Z_s^n dB_s + 4\\displaystyle\\int_t^T (\\rho(Y_s^n)(Y_s^n-\\pi(Y_s^n))^* h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s\\\\\n& + \\displaystyle\\frac{1}{2}\\displaystyle\\int_t^T trace[(h_sh_s^*)(Y_s^n,Z_s^n)Hess \\varphi(Y_s^n)]ds - 4n \\displaystyle\\int_t^T \\rho^2(Y_s^n)ds.\n\\end{split}\n\\end{eqnarray} \nBy taking expectation we have \n\\begin{eqnarray}\n\\begin{split}\\label{estvarphi} \n\\mathbb{E}[\\varphi(Y_t^n)]&+ \\displaystyle\\frac{1}{2}\\mathbb{E}\\big[\\displaystyle\\int_t^T trace[Z_s^nZ_s^{n *}Hess \\varphi(Y_s^n)]ds\\big]+4n \\mathbb{E}\\big[\\displaystyle\\int_t^T \\varphi(Y_s^n)ds\\big]\\\\\n& = 4\\mathbb{E}\\big[\\displaystyle\\int_t^T (\\rho(Y_s^n)(Y_s^n-\\pi(Y_s^n))^* f_s(Y_s^n,Z_s^n)ds]\\\\\n &+ \\displaystyle\\frac{1}{2}\\mathbb{E}\\Big[\\displaystyle\\int_t^T trace[(h_sh_s^*)(Y_s^n,Z_s^n)Hess \\varphi(Y_s^n)]ds].\n\\end{split}\n\\end{eqnarray}\nFor the last term, we get from the boundedness of $h$ and $Hess\\rho$ \n\\begin{eqnarray} \\label{esth}\n\\begin{split}\n\\mathbb{E}\\Big[\\displaystyle\\int_t^T trace[(h_sh_s^*)(Y_s^n,Z_s^n)&Hess \\varphi(Y_s^n)]ds]\\leq 2\\mathbb{E}\\Big[\\displaystyle\\int_t^T\\langle h_s(Y_s^n,Z_s^n),\\nabla\\rho(Y_s^n)\\rangle^2 ds\\Big]\\\\\n& + \\mathbb{E}\\Big[\\displaystyle\\int_t^T 2\\rho(Y_s^n)trace[(h_sh_s^*)(Y_s^n,Z_s^n)Hess \\rho(Y_s^n)]ds\\Big]\\\\\n&\\leq C\\mathbb{E}\\Big[\\displaystyle\\int_t^T|\\nabla\\rho(Y_s^n)|^2 ds\\Big] + C\\mathbb{E}\\Big[\\displaystyle\\int_t^T \\rho(Y_s^n)ds\\Big]\\\\\n&\\leq C  \\mathbb{E}\\Big[\\displaystyle\\int_0^T(d(Y_s^n,D))^{2}ds\\Big].\n\\end{split}\n\\end{eqnarray} \nNow the inequality $2ab\\leq a^2+b^2$ with $a=(d(Y_s^n,D))^{2}$ and the boundedness of $f$ yield \n\\begin{eqnarray}\n\\begin{split}\\label{estf}\n4(d(Y_s^n,D))^{3} |f_s(Y_s^n,Z_s^n)|&\\leq 2 (d(Y_s^n,D))^{4}+ 2 (d(Y_s^n,D))^{2}|f_s(Y_s^n,Z_s^n)|^2\\\\\n&\\leq 2 \\varphi(Y_s^n) + 2C(d(Y_s^n,D))^{2} .\n\\end{split}\n\\end{eqnarray}\nBy plugging the estimate (\\ref{estf}) and \\eqref{esth} in (\\ref{estvarphi}), we obtain thanks to lemma \\ref{lem}\n\\begin{eqnarray}\\label{estphi}\n\\begin{split} \n\\mathbb{E}[\\varphi(Y_t^n)]+ \\displaystyle\\frac{1}{2}\\mathbb{E}\\big[\\displaystyle\\int_t^T trace [Z_s^nZ_s^{n *}& Hess \\varphi(Y_s^n)]ds\\big]+(4n-2) \\mathbb{E}\\big[\\displaystyle\\int_t^T \\varphi(Y_s^n)ds\\big]\\\\\n& \\leq C\\mathbb{E}\\big[\\displaystyle\\int_0^T (d(Y_s^n,D))^{2}ds\\big]\\leq C\\big(\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{n^2}\\big).\n\\end{split}\n\\end{eqnarray}\nNotice also that  Hessian of $ \\varphi(Y_s^n)$ is a positive semidefinite matrix since $ \\varphi$ is a convex function, so we get that \n\\begin{eqnarray}\\label{unifY}\n\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}\\mathbb{E}[\\varphi(Y_t^n)]\\leq C\\big(\\displaystyle\\frac{1}{n}+\\displaystyle\\frac{1}{n^2}\\big).\n\\end{eqnarray}\nMoreover, we can deduce from \\eqref{estphi} that, for every $t\\in[0,T]$\n\\begin{eqnarray}\\label{estZ}\n \\mathbb{E}\\big[\\displaystyle\\int_t^T trace [Z_s^nZ_s^{n *} Hess \\varphi(Y_s^n)]ds\\big]\\longrightarrow 0, \\, \\text{as}\\, n\\rightarrow \\infty.\n\\end{eqnarray}\nOn the other hand, taking the supremum over $t$ in the equation \\eqref{Ito}, by Burkholder-Davis-Gundy's inequlity and the previous calculations it follows that\n\\begin{eqnarray}\\label{uniformestimate}\n\\begin{split}\n\\mathbb{E}[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}\\varphi(Y_t^n)]&\\leq C \\mathbb{E}[\\displaystyle\\int_0^T\\varphi(Y_s^n) ds]+C\\mathbb{E}\\Big[\\displaystyle\\int_0^T (d(Y_s^n,D))^{2}ds\\Big]\\\\\n&+C\\mathbb{E}\\Big[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}\\displaystyle\\int_t^T (\\rho(Y_s^n)\\nabla\\rho(Y_s^n))^* Z_s^n dB_s\\Big]\\\\\n&+ C \\mathbb{E}\\Big[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}\\displaystyle\\int_t^T (\\rho(Y_s^n)\\nabla\\rho(Y_s^n))^* h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s\\Big]\\\\\n&\\leq C \\mathbb{E}[\\displaystyle\\int_0^T\\varphi(Y_s^n) ds]+C\\mathbb{E}\\Big[\\displaystyle\\int_0^T (d(Y_s^n,D))^{2}ds\\Big]\\\\&+C\\mathbb{E}\\Big[\\Big(\\displaystyle\\int_0^T (\\rho(Y_s^n))^2\\langle\\nabla\\rho(Y_s^n), Z_s^n\\rangle^2 ds\\Big)^{1\/2}\\Big]\\\\\n&+ C\\mathbb{E}\\Big[\\Big(\\displaystyle\\int_0^T (\\rho(Y_s^n))^2\\langle\\nabla\\rho(Y_s^n), h_s(Y_s^n,Z_s^n)\\rangle^2 ds\\Big)^{1\/2}\\Big].\n\\end{split}\n\\end{eqnarray}\nFrom the boundedness of $h$ and the fact that $|\\nabla\\rho(x)|^2=4\\rho(x)$, we have\n\\begin{eqnarray} \\label{estimate1}\n\\begin{split}\n\\mathbb{E}\\Big[\\Big(\\displaystyle\\int_0^T (\\rho(Y_s^n))^2&\\langle\\nabla\\rho(Y_s^n)), h_s(Y_s^n,Z_s^n)\\rangle^2 ds\\Big)^{1\/2}\\Big]\\leq C \\mathbb{E}\\Big[\\Big(\\displaystyle\\int_0^T (\\rho(Y_s^n))^2\\rho(Y_s^n) ds\\Big)^{1\/2}\\Big]\\\\\n&\\leq C \\mathbb{E}\\Big[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}\\big(\\varphi(Y_s^n)\\big) ^{1\/2}\\Big(\\displaystyle\\int_0^T \\rho(Y_s^n) ds\\Big)^{1\/2}\\Big]\\\\\n&\\leq  \\displaystyle\\frac{1}{4}\\mathbb{E}\\Big[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}\\varphi(Y_s^n)\\Big]+C^2 \\mathbb{E}\\Big[\\displaystyle\\int_0^T (d(Y_s^n,D))^2ds\\Big].\n\\end{split}\n\\end{eqnarray}\nBy the Holder's inequality, we obtain\n\\begin{eqnarray} \\label{estimate2}\n\\begin{split}\n\\mathbb{E}\\Big[\\Big(\\displaystyle\\int_0^T (\\rho(Y_s^n))^2\\langle\\nabla\\rho(Y_s^n)), Z_s^n\\rangle^2 ds\\Big)^{1\/2}\\Big]&\\leq C \\mathbb{E}\\Big[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}\\big(\\varphi(Y_s^n)\\big) ^{1\/2}\\Big(\\displaystyle\\int_0^T\\langle\\nabla\\rho(Y_s^n)), Z_s^n\\rangle^2 ds\\Big)^{1\/2}\\Big]\\\\\n&\\leq  \\displaystyle\\frac{1}{4}\\mathbb{E}\\Big[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}\\varphi(Y_s^n)\\Big]+C^2 \\mathbb{E}\\Big[\\displaystyle\\int_0^T\\langle\\nabla\\rho(Y_s^n), Z_s^n\\rangle^2 ds\\Big].\n\\end{split}\n\\end{eqnarray}\nSubstituting \\eqref{estimate1} and \\eqref{estimate2} in \\eqref{uniformestimate} leads to\n\\begin{eqnarray} \\label{estimate3}\n\\begin{split}\n\\mathbb{E}[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}\\varphi(Y_t^n)]&\\leq C \\mathbb{E}[\\displaystyle\\int_0^T\\varphi(Y_s^n) ds]+C\\mathbb{E}\\Big[\\displaystyle\\int_0^T (d(Y_s^n,D))^{2}ds\\Big]\\\\\n&+C^2 \\mathbb{E}\\Big[\\displaystyle\\int_0^T\\langle\\nabla\\rho(Y_s^n), Z_s^n\\rangle^2 ds\\Big].\n\\end{split}\n\\end{eqnarray}\nSince each term of \\eqref{hessphi} is positive semidefinite and from \\eqref{estZ}, we get \n\\b*\n\\mathbb{E}\\Big[\\displaystyle\\int_0^T\\langle\\nabla\\rho(Y_s^n), Z_s^n\\rangle^2 ds\\Big]\\longrightarrow 0 \\, \\text{as}\\, n\\rightarrow\\infty.\n\\e*\n\\vspace{2cm}\nFinally, by using \\eqref{unifY}, \\eqref{estimate3} and Lemma \\ref{lem}, we get the desired result.\n\\hbox{ }\\hfill$\\Box$\n\\end{proof}\n\\begin{Lemma} \\label{conv}\nThe sequence $(Y^n,Z^n)$ is a Cauchy sequence in ${\\mathcal S}^2_{k}([0,T]) \\times {\\mathcal H}^2_{k\\times d}([0,T])$, i.e.\n$$\\mathbb{E}[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}|Y_t^n- Y_t^m|^2 + \\displaystyle\\int_0^T \\|Z_t^n-Z_t^m\\|^2dt]\\longrightarrow 0 \\,\\text{as}\\,\\, n,m \\rightarrow +\\infty.$$\n\\end{Lemma}\n\\begin{proof}\nFor all $n,m\\geq 0$, we apply It\\^o formula to $|Y_t^n-Y_t^m|^2$\n\\begin{eqnarray} \\label{itoexis}\n\\begin{split}\n|Y_t^n-Y_t^m|^2&+\\displaystyle\\int_t^T \\|Z_s^n-Z_s^m\\|^2ds  =  2\\displaystyle\\int_t^T (Y_s^n-Y_s^m)^*(f_s(Y_s^n,Z_s^n)-f_s(Y_s^m,Z_s^m))ds\\\\\n& +2\\displaystyle\\int_t^T (Y_s^n-Y_s^m)^*(h_s(Y_s^n,Z_s^n)-h_s(Y_s^m,Z_s^m))d\\overleftarrow{W}_s - 2\\displaystyle\\int_t^T (Y_s^n-Y_s^m)(Z_s^n-Z_s^m)dB_s\\\\\n& +\\displaystyle\\int_t^T \\|h_s(Y_s^n,Z_s^n)-h_s(Y_s^m,Z_s^m)\\|^2ds- 2n\\displaystyle\\int_t^T (Y_s^n-Y_s^m)^*(Y_s^n-\\pi(Y_s^n))ds\\\\\n&+2m\\displaystyle\\int_t^T (Y_s^n-Y_s^m)^*(Y_s^m-\\pi(Y_s^m))ds.\n\\end{split}\n\\end{eqnarray}\nBy the property (\\ref{prop2}), we have \n\\begin{eqnarray}\n\\begin{split}\n- 2n\\displaystyle\\int_t^T (Y_s^n-Y_s^m)^*(Y_s^n-\\pi(Y_s^n))ds &+ 2m\\displaystyle\\int_t^T (Y_s^n-Y_s^m)^*(Y_s^m-\\pi(Y_s^m))ds\\\\\n&\\leq 2(n+m)\\displaystyle\\int_t^T (Y_s^n-\\pi(Y_s^n))^*(Y_s^m-\\pi(Y_s^m))ds.\\\\\n&\n\\end{split}\n\\end{eqnarray}\nHence, from the Lipschitz continuity on $f$ and $h$, and taking expectation yields\n\\begin{eqnarray}\\label{itoestimate}\n\\begin{split}\n\\mathbb{E}[|Y_t^n-Y_t^m|^2] + \\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^n-Z_s^m\\|^2ds]&\\leq  2\\mathbb{E}[\\displaystyle\\int_t^T C(|Y_s^n-Y_s^m|^2+ |Y_s^n-Y_s^m|\\|Z_s^n-Z_s^m\\|)ds]\\\\\n&+\\mathbb{E}[\\displaystyle\\int_t^T C(|Y_s^n-Y_s^m|^2+ \\alpha\\|Z_s^n-Z_s^m\\|)ds]\\\\\n&+ 2(n+m)\\mathbb{E}[\\displaystyle\\int_t^T (Y_s^n-\\pi(Y_s^n))^*(Y_s^m-\\pi(Y_s^m))ds].\n\\end{split}\n\\end{eqnarray}\nFor the last term, we need the following lemma whose proof is postponed in the Appendix.\n\\begin{Lemma}\\label{extraestimate}\nThere exists a constant $C>0$ such that, for each $n\\geq 0$,\n\\begin{eqnarray} \n \\mathbb{E}\\big[\\Big(n\\displaystyle\\int_0^T d(Y_s^n,D) ds\\Big)^2\\big]\\leq C\n \\label{boundvariation}\n\\end{eqnarray}\n\\end{Lemma}\n\\vspace{0.5cm}\n\\noindent Now we can deduce from the H\\\"older inequality and Lemma \\ref{extraestimate} that \n\\begin{eqnarray} \\label{estdistance}\n\\begin{split} \nn\\mathbb{E}[\\displaystyle\\int_t^T &(Y_s^n-\\pi(Y_s^n))^*(Y_s^m-\\pi(Y_s^m))ds] \\leq  n\\mathbb{E}[\\displaystyle\\int_t^T  d(Y_s^n,D)d(Y_s^m,D))ds]\\\\\n&\\leq n\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d(Y_s^m,D)\\displaystyle\\int_t^T  d(Y_s^n,D)ds)]\\\\\n &\\leq \\Big(\\mathbb{E}\\big[\\Big(n\\displaystyle\\int_0^T d(Y_s^n,D) ds\\Big)^2\\big]\\Big)^{1\/2} \\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}\\\\\n&\\leq C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}.\n\\end{split}\n\\end{eqnarray}\nSubstituting (\\ref{estdistance}) in the previous inequality \\eqref{itoestimate}, we have\n\\b*\n\\begin{split}\n\\mathbb{E}[|Y_t^n-Y_t^m|^2] &+ (1-\\alpha-C\\gamma)\\mathbb{E}[\\displaystyle\\int_t^T \\|Z_s^n-Z_s^m\\|^2ds]\\leq  C(1+\\displaystyle\\frac{1}{\\gamma})\\mathbb{E}[\\displaystyle\\int_t^T |Y_s^n-Y_s^m|^2ds]\\\\\n&+ C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^n,D)]\\Big)^{1\/2}+ C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}.\n\\end{split}\n\\e*\nChoosing $1-\\alpha-C\\gamma>0$, by Gronwall's lemma, we obtain\n\\begin{eqnarray}\n\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}\\mathbb{E}[|Y_t^n-Y_t^m|^2]\\leq C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}+ C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^n,D)]\\Big)^{1\/2}.\n\\end{eqnarray}\nWe deduce similarly\n\\begin{eqnarray}\n \\mathbb{E}[\\displaystyle\\int_0^T \\|Z_s^n-Z_s^m\\|^2ds]\\leq C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}+ C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^n,D)]\\Big)^{1\/2}.\n\\label{estunifZ}\n\\end{eqnarray}\nNext, by \\eqref{itoexis}, the Burkholder-Davis-Gundy and the Cauchy-Schwarz inequalities we get \n\\begin{align*}\n&\\mathbb{E}[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}|Y_t^n-Y_t^m|^2]\\leq C\\mathbb{E}[\\displaystyle\\int_0^T | Y_s^n-Y_s^m||f(s,Y_s^n,Z_s^n)-f(s,Y_s^m,Z_s^m)|ds]\\\\\n&+C\\mathbb{E}\\big(\\displaystyle\\int_0^T | Y_s^n-Y_s^m|^2\\|h_s(Y_s^n,Z_s^n)-h_s(Y_s^m,Z_s^m)\\|^2ds\\big)^{1\/2}+C\\mathbb{E}\\big(\\displaystyle\\int_0^T | Y_s^n-Y_s^m|^2\\| Z_s^n-Z_s^m\\|^2ds\\big)^{1\/2}\\\\&+\\mathbb{E}[\\displaystyle\\int_0^T C(|Y_s^n-Y_s^m|^2+ \\alpha\\|Z_s^n-Z_s^m\\|^2)ds]+  2(n+m)\\mathbb{E}[\\displaystyle\\int_0^T (Y_s^n-\\pi(Y_s^n))^*(Y_s^m-\\pi(Y_s^m))ds].\n\\end{align*}\nThen, it follows by the Lipschitz Assumption \\ref{Ass2} on $f$ and $h$ and  (\\ref{estdistance}) that for any $n,m\\geq 0$\n\\b*\n\\begin{split}\n\\mathbb{E}[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}|Y_t^n-Y_t^m|^2]&\\leq  C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}+C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^n,D)]\\Big)^{1\/2}\\\\\n&+ C\\mathbb{E}(\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}|Y_t^n-Y_t^m|^2\\displaystyle\\int_0^T \\|Z_s^n-Z_s^m\\|^2ds)^{1\/2}\\\\\n&\\leq  C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}+C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^n,D)]\\Big)^{1\/2}\\\\\n&+ C\\varepsilon\\mathbb{E}(\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}|Y_t^n-Y_t^m|^2)+ C\\varepsilon^{-1}\\mathbb{E}(\\displaystyle\\int_0^T \\|Z_s^n-Z_s^m\\|^2ds).\n\\end{split}\n\\e*\nChoosing $1-C\\varepsilon>0$ and from the inequality (\\ref{estunifZ}) we conclude that\n\\b*\n\\begin{split}\n\\mathbb{E}[\\underset{0\\leq t\\leq T}{\\displaystyle\\sup}|Y_t^n-Y_t^m|^2] &\\leq C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^m,D)]\\Big)^{1\/2}+C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^2(Y_s^n,D)]\\Big)^{1\/2}\\\\\n&\\leq C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^4(Y_s^m,D)]\\Big)^{1\/4}+C\\Big(\\mathbb{E}[\\underset{0\\leq s\\leq T}{\\displaystyle\\sup}d^4(Y_s^n,D)]\\Big)^{1\/4}\\longrightarrow 0,\n\\end{split}\n\\e*\nas $n,m\\rightarrow \\infty$, where Lemma \\ref{fundamental:lemma} has been used.\\hbox{ }\\hfill$\\Box$\n\\end{proof}\n\\vspace{0.6cm}\n\\noindent Finally, we conclude that $(Y^n,Z^n)$ is a Cauchy sequence in ${{\\cal S}}^2_k([0,T])\\times {{\\cal H}}^2_{k\\times d}([0,T])$ and therefore there exists a unique pair $(Y_t,Z_t)$  of ${\\cal F}_t$- measurable processes which valued in $\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}$, satisfying\n\\begin{eqnarray}\n\\mathbb{E}(\\underset{0\\leq t\\leq T}{\\displaystyle\\sup} |Y_t^n-Y_t|^2 + \\displaystyle\\int_0^T |Z_t^n-Z_t|^2 dt) \\rightarrow 0 \\quad \\mbox{as}~~ n\\rightarrow\\infty.\n\\label{converg}\n\\end{eqnarray}\nConsequently, since for any $n\\geq 0$ and $0\\leq t\\leq T$,\n\\begin{eqnarray}\n\\begin{split}\nK_t^n-K_t^m &= Y_0^n-Y_0^m-Y_t^n-Y_t^m-\\displaystyle\\int_0^t (f_s(Y_s^n,Z_s^n)-f_s(Y_s^m,Z_s^m))ds\\\\\n&-\\displaystyle\\int_0^t (h_s(Y_s^n,Z_s^n)-h_s(Y_s^m,Z_s^m))d\\overleftarrow{W}_s\n +\\displaystyle\\int_0^t (Z_s^n-Z_s^m) dB_s.\n\\end{split}\n\\end{eqnarray}\nwe obtain from (\\ref{converg}) and Burkholder-Davis-Gundy inequality,\n\\begin{eqnarray}\n\\mathbb{E}(\\underset{0\\leq t\\leq T}{\\displaystyle\\sup} |K_t^n-K_t^m|^2)\\rightarrow 0 \\quad \\mbox{as}~~ n, m\\rightarrow\\infty.\n\\end{eqnarray}\nHence, there exists a ${\\cal F}_t$- adapted continuous process $(K_t)_{0\\leq t\\leq T}$ ( with $K_0=0$) such that $$\\mathbb{E}(\\underset{0\\leq t\\leq T}{\\displaystyle\\sup} |K_t-K_t^n|^2 )\\rightarrow 0 \\quad \\mbox{as}~~ n\\rightarrow\\infty.$$\nFurthermore, \\eqref{boundvariation} shows that the total variation of $K^n$ is uniformly bounded. Thus, $K$ is also of uniformly bounded variation.\nPassing to the limit in \\eqref{BDSDEpen}, the processes $(Y_t,Z_t,K_t)_{0\\leq t\\leq T}$ satisfy \n$$Y_t  = \\xi +\\displaystyle\\int_t^T f_s(Y_s,Z_s)ds +\\displaystyle\\int_t^T h_s(Y_s,Z_s)d\\overleftarrow{W}_s -\\displaystyle\\int_t^T Z_sdB_s +K_T-K_t ~,~ 0\\leq t\\leq T.$$\nSince we have from Lemma \\ref{fundamental:lemma} that $Y_t$ is in $\\bar{D}$, it remains to check the minimality property for $ (K_t)$, namely i.e., for any continuous progressively measurable process $(z_t)$ valued in $\\bar{D}$, $$\\displaystyle\\int_0^T (Y_t-z_t)^* dK_t \\leq 0.$$\nWe note that (\\ref{prop1}) gives us \n$$\\displaystyle\\int_0^T (Y_t^n-z_t)^*dK_t^n= -n\\displaystyle\\int_0^T (Y_t^n-z_t)^*(Y_t^n-\\pi(Y_t^n))dt \\leq 0.$$\nTherefore, we will show that we can extract a subsequence such that $\\displaystyle\\int_0^T (Y_t^n-z_t)^* dK_t^n$ converge a.s. to $\\displaystyle\\int_0^T (Y_t-z_t)^* dK_t.$ Following the proof of Lemma \\ref{estapriori} in Appendix, we have\n\\begin{eqnarray}\n2\\gamma \\|K^n\\|_{VT}&\\leq & |\\xi -a|^2 +2\\displaystyle\\int_0^T (Y_s^n-a)f_s(Y_s^n,Z_s^n)ds + 2\\displaystyle\\int_0^T  (Y_s^n-a) h_s(Y_s^n,Z_s^n)d\\overleftarrow{W}_s\\nonumber\\\\\n&+& \\displaystyle\\int_0^T  \\|h_s(Y_s^n,Z_s^n)\\|^2ds - 2\\displaystyle\\int_0^T(Y_s^n-a)Z_s^n dB_s.\n\\end{eqnarray}\nNotice that the right hand side tends in probability as $n$ goes to infinity  to $$|\\xi -a|^2 +2\\displaystyle\\int_0^T (Y_s-a)f_s(Y_s,Z_s)ds + 2\\displaystyle\\int_0^T  (Y_s-a) h_s(Y_s,Z_s)d\\overleftarrow{W}_s + \\displaystyle\\int_0^T \\|h_s(Y_s,Z_s)\\|^2ds - 2\\displaystyle\\int_0^T(Y_s-a)Z_s dB_s.$$ \nThus, there exists a subsequence $(\\phi(n))_{n\\geq 0}$ such that the convergence is almost surely and $\\|K^{\\phi(n)}\\|_{VT}$ is bounded. Moreover, due to the convergence in $\\mathbb{L}^2$ of $\\underset{0\\leq t\\leq T}{\\displaystyle\\sup} |Y_t^n-Y_t|^2$ to $0$, we can extract a subsequence  from $(\\phi(n))_{n\\geq 0}$ such that $Y^{\\phi(\\psi(n))}$ converges uniformly to $Y$. Hence, we apply  Lemma 5.8 in \\cite{GPP95} and we obtain\n$$\\displaystyle\\int_0^T(Y_t^{\\phi(\\psi(n))}-z_t)^*dK_t^{\\phi(\\psi(n)))}\\longrightarrow\\displaystyle\\int_0^T(Y_t -z_t)^*dK_t\\quad a.s. ~~\\mbox{as}~~ n\\rightarrow\\infty$$\nwhich is  the required result.\n\\hbox{ }\\hfill$\\Box$\n\\section{Weak solution of semilinear SPDE in a convex domain}\n\\label{section:SPDE}\nThe aim of this section is to give a Feynman-Kac's formula for the weak solution of a  semilinear reflected SPDEs  \\eqref{OSPDE1}  in a given convex domain $D$\nvia Markovian class of RBDSDEs  studied in the last section.  As explained in the introduction, the solution  of such SPDEs  is expressed as a pair  $(u,\\nu) $ where \n$u$ is a predictable continuous process which takes values in a  Sobolev space and $\\nu$ is a signed Radon regular measure. The bounded variation processes $K $ component of the  solution of the reflected BDSDE  controls the set when   $ u$ reaches the boundary of $D$. In fact, this bounded variation process determines the measure $\\nu$ from a particular relation by using the inverse of the flow associated to the  diffusion operator. \n\\subsection{Notations and Hypothesis}\nLet us first introduce some notations:\\\\\n- $C^n_{l,b}(\\mathbb{R}^p,\\mathbb{R}^q)$ the set of $C^n$-functions which grow at most linearly at infinity and whose partial derivatives of order less than or equal to $n$ are bounded.\\\\\n- $\\mathbf{L}_{\\rho}^2\\left( \\mathbb{{R}}^d\\right) $ will be a Hilbert with the inner product,\n$$ \\left( u,v\\right)_{\\rho} =\\int_{\\mathbb{R}^d}u\\left( x\\right) v\\left(\nx\\right) \\rho (x) dx,\\;\\left\\| u\\right\\| _2=\\left(\n\\int_{\\mathbb{R}^d}u^2\\left( x\\right) \\rho (x) dx\\right) ^{\\frac\n12}. $$ \n\\vspace{0.1cm}\n\\noindent\n\\begin{Assumption}\\label{assweight}\nWe assume that $ \\rho$ is the weight function that satisfy the following conditions:\n\\begin{itemize}\n\\item $\\rho$ is a continuous positive function.\n\\item $\\rho$ is integrable and $ \\displaystyle\\frac{1}{\\rho}$ is locally integrable.\n\\end{itemize}\n\\end{Assumption}\n\\vspace{0.2cm}\n\\noindent In general, we shall use for the usual $L^2$-scalar product\n$$(u,v)=\\displaystyle\\int_{\\mathbb{R}^d} u(x)v(x)\\, dx,$$\nwhere $u$, $v$ are measurable functions defined in $\\mathbb{R }^d$\nand $uv \\in \\mathbf{L}^1 (\\mathbb{R}^d )$.\\\\\nOur evolution problem will be considered over a fixed time interval\n$[0,T]$ and the norm for an element of $\\mathbf{L}_{\\rho}^2\\left(\n[0,T] \\times \\mathbb{{R}}^d\\right) $ will be denoted by\n$$\\left\\| u\\right\\| _{2,2}=\\left(\\displaystyle\\int_0^T  \\int_{\\mathbb{R}^d} |u (t,x)|^2 \\rho(x)dx dt \\right)^{\\frac 12}. $$\nWe assume the following hypotheses :\n\\begin{Assumption}\\label{assxi}\n$\\Phi:\\mathbb{R}^d\\rightarrow\\mathbb{R}^k$ belongs to $\\mathbf{L}_{\\rho}^2(\\mathbb{R}^d)$ and $\\Phi(x)\\in\\bar{D}~~ a.e. ~\\forall x\\in\\mathbb{R}^d$;\n\\end{Assumption}\n\\begin{Assumption} \\label{assgener}\n\\begin{itemize} \n\\item[\\rm{(i)}] $f:[0,T]\\times \\mathbb{R}^d\\times \\mathbb{R}^k\\times \\mathbb{R}^{k\\times\nd}\\rightarrow\\mathbb{R}^n$ and $h:[0,T]\\times \\mathbb{R}^d\\times \\mathbb{R}^k\\times \\mathbb{R}^{k\\times\nd}\\rightarrow\\mathbb{R}^{k\\times l}$ are measurable in $(t,x,y,z)$ and\nsatisfy $ f^0, h^0 \\in \\mathbf{L}_{\\rho}^2\\left( [0,T] \\times\n\\mathbb{{R}}^d\\right) $ where $f_t^0 (x) := f (t,x,0, 0)$, $h_t^0 := h (t,\n,x,0, 0)$.\n\\item[\\rm{(ii)}] There exist constants $c>0$ and $0<\\alpha<1$ such that for any $(t,x)\\in[0,T]\\times\\mathbb{R}^d~;~\n(y_1,z_1),(y_2,z_2)\\in\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}$\n\\b*\n|f_t(x,y_1,z_1)-f_t(x,y_2,z_2)|^2 &\\leq & c\\big(|y_1-y_2|^2+\\|z_1-z_2\\|^2\\big)\\\\\n\\|h_t(x,y_1,z_1)-h_t(x,y_2,z_2)\\|^2 &\\leq & c|y_1-y_2|^2+\\alpha \\|z_1-z_2\\|^2.\n\\e*\n\\end{itemize}\n\\end{Assumption}\n\\begin{Assumption}\\label{assdiff}\nThe coefficients $b$ and $\\sigma$ of the second order differential operator ${\\cal L}$ (\\ref{operator conv}) satisfy $b\\in C^2_{l,b}(\\mathbb{R}^d;\\mathbb{R}^d)$, $\\sigma\\in C^3_{l,b}(\\mathbb{R}^d;\\mathbb{R}^{d\\times d}).$\n\\end{Assumption}\n\\begin{Assumption}\\label{assint}\n\\begin{itemize}\n\\item[\\rm{(i)}] $\\Phi\\in \\mathbf{L}_{\\rho}^4(\\mathbb{R}^d).$\n\\item[\\rm{(ii)}] There exist $c>0$ and  $0\\leq \\beta < 1$ such that for all $(t,x,y,z)\\in [0,T]\\times\\mathbb{R}^k\\times\\mathbb{R}^{k\\times d}$\n$$h_t \\, h_t^* (x,y,z)\\leq c (Id_{\\mathbb{R}^k}+ yy^*)+ \\beta  \\, zz^*.$$\n\\item[\\rm{(iii)}] $f$ and $h$ are uniformly bounded in $(x,y,z)$.\n\\end{itemize}\n\\end{Assumption}\n\\subsection{Weak formulation for a solution of Stochastic  PDEs}\n\\label{definition:solution}\nThe space of test functions which we employ in the definition of\nweak solutions of the evolution equations  \\eqref{SPDE1} is $\n\\mathcal{D}_T  = \\mathcal{C}^{\\infty} (\\left[0,T]\\right) \\otimes\n\\mathcal{C}_c^{\\infty} \\left(\\mathbb{R}^d\\right)$, where\n\\begin{itemize}\n\\item $\\mathcal{C}^{\\infty} \\left([0,T]\\right)$ denotes the space of real\nfunctions which can be extended as infinitely differentiable functions\nin the neighborhood of $[0,T]$, \n\\item $\\mathcal{C}_c^{\\infty}\\left(\\mathbb{R}^d\\right)$ is the space of\ninfinite differentiable functions with compact supports in\n$\\mathbb{R}^d$.\n\\end{itemize}\nWe denote by  $ {\\mathcal H}_T$ the space of  $ {\\cal F}_{t,T}^W$-progressively measurable  processes  $(u_t ) $ with values  in the weighted Sobolev space $ H_{\\rho} ^1 (\\mathbb{R}^d)$ where \n$$ H_{\\rho} ^1 (\\mathbb{R}^d):=\\{v \\in \\mathbf{L}_{\\rho}^2(\\mathbb{R}^d) \\; \\big|\\; \\nabla v\\sigma\\in \\mathbf{L}_{\\rho}^2(\\mathbb{R}^d)\\} $$\nendowed with the norm\n$$\\begin{array}{ll}\n\\|u\\|_{{\\mathcal H}_T}^2=\n \\mathbb{E} \\,  \\big[\\underset{ 0 \\leq t \\leq T}{\\displaystyle\\sup} \\|u_s \\|_2^2 +   \\displaystyle\\int_{ \\mathbb{R}^d} \\displaystyle\\int_0^T  |\\nabla\nu_s (x)\\sigma(x)|^2 ds\\rho(x)dx \\big],\n\\end{array}\n$$\nwhere we denote the gradient by $\\nabla u (t,x) = \\big(\\partial_1 u\n(t,x), \\cdot \\cdot \\cdot, \\partial_d u (t,x) \\big)$. Here, the derivative is defined in the weak sense (Sobolev sense).\n\\begin{Definition}[{\\textbf{Weak solution of regular SPDE}}]\nWe say that $ u \\in \\mathcal{H}_T $ is a Sobolev solution of SPDE  $\\left(\n\\ref{SPDE1}\\right) $ if the following\nrelation holds, for each $\\varphi \\in \\mathcal{D}_T ,$\n\\begin{equation}\\label{wspde1}\n\\begin{array}{ll}\n\\displaystyle\\int_t^T(u(s,x),\\partial_s\\varphi(s,x))ds+(u(t,x),\\varphi(t,x))-(\\Phi(x),\\varphi(T,x))\n-\\int_t^T( u(s,x),\\mathcal L^\\ast \\varphi(s,x))ds\n\\\\=\\displaystyle\\int_t^T(f_s(x,u(s,x),\\nabla u(s,x)\\sigma(x)),\\varphi(s,x))ds +\\displaystyle\\int_t^T(h_s(x,u(s,x),\\nabla u(s,x)\\sigma(x)),\\varphi(s,x))d\\overleftarrow{W}_s.\n\\end{array}\n\\end{equation}\nwhere ${\\mathcal L}^\\ast$ is the adjoint operator of ${\\mathcal L}$.\nWe denote by $ u:=\\mathcal{U }(\\Phi, f,h)$ the solution of SPDEs with data $(\\Phi,f,h)$.\n\\end{Definition}\n\\vspace{0.2cm}\n\\noindent The existence and uniqueness  of weak solution  for SPDEs \\eqref{wspde1} is ensured by Theorem 3.1 in Bally and Matoussi \\cite{BM01} or Denis and Stoica \\cite{DS04}.\n\\subsection{Stochastic flow of diffeomorphisms and random test functions}\n\\label{Flow} We are concerned in this part with solving our problem by developing a stochastic flow method which was first introduced in Kunita \\cite{K84}, \\cite{K90} and Bally, Matoussi\n\\cite{BM01}. We recall that  $\\{X_{t,s}(x), t\\leq s\\leq T\\}$ is the diffusion process starting from $x$ at time $t$ and is the strong solution  of the equation:\n \\begin{equation}\\label{sde}\n  X_{t,s}(x)=x+\\displaystyle\\int_{t}^{s}b(X_{t,r}(x))dr+\\displaystyle\\int_{t}^{s}\\sigma(X_{t,r}(x))dB_r.\n\\end{equation}\nThe existence and uniqueness of this solution was proved in Kunita \\cite{K84}. Moreover, we have the following properties:\n\\begin{Proposition}\\label{estimatesde}\nFor each $t>0$, there exists a version of $\\{X_{t,s}(x);\\,x\\in\n\\mathbb{R}^d,\\,s\\geq t\\}$ such that $X_{t,s}(\\cdot)$ is a $C^2(\\mathbb{R}^d)$-valued\ncontinuous process which satisfy the flow property: $X_{t,r}(x)=X_{s,r} \\circ X_{t,s} (x)$, $0\\leq t<s<r$.\nFurthermore,\nfor all $p\\geq 2$, there exists $M_p$ such that for all $0\\leq t<s$,\n$x,x'\\in\\mathbb{R}^d$, $h,h'\\in\\mathbb{R}\\backslash{\\{0\\}}$,\n$$\\begin{array}{ll}\n\\mathbb{E}(\\underset{t\\leq r\\leq s}{\\displaystyle\\sup}|X_{t,r}(x)-x|^p)\\leq\nM_p(s-t)(1+|x|^p),\\\\\n\\mathbb{E}(\\underset{t\\leq r\\leq s}{\\displaystyle\\sup}|X_{t,r}(x)-X_{t,r}(x')-(x-x')|^p)\\leq\nM_p(s-t)(|x-x'|^p),\\\\\n\\mathbb{E}(\\underset{t\\leq r\\leq s}{\\displaystyle\\sup}|\\Delta_h^i[X_{t,r}(x)-x]|^p)\\leq\nM_p(s-t),\\\\\n\\mathbb{E}(\\underset{t\\leq r\\leq s}{\\displaystyle\\sup}|\\Delta_h^i X_{t,r}(x)-\\Delta_{h'}^i\nX_{t,r}(x')|^p)\\leq M_p(s-t)(|x-x'|^p+|h-h'|^p),\n\\end{array}$$\nwhere $\\Delta_h^ig(x)=\\frac{1}{h}(g(x+he_i)-g(x))$, and\n$(e_1,\\cdots,e_d)$ is an orthonormal basis of $\\mathbb{R}^d$.\n\\end{Proposition}\nUnder regular conditions (Assumption \\ref{assdiff}) on the diffusion, it is known that the stochastic flow  solution of a continuous SDE satisfies the homeomorphic property (see Bismut \\cite{b}, Kunita \\cite{K84}, \\cite{K90}). We have  the following result\nwhose proof can be found in  \\cite{K84}.\n\\begin{Proposition}\\label{flow}\nLet Assumption \\ref{assdiff} holds. Then\n$\\{X_{t,s}(x); x \\in \\mathbb{R}^d \\}$ is a $C^2$-diffeomorphism a.s.\nstochastic flow. Moreover the inverse of the flow satisfies the\nfollowing backward SDE\n\\begin{equation}\\label{inverse:flow}\n\\begin{split}\nX_{t,s}^{-1}(y) &  = y - \\int_t^s \\widehat{b}(X_{r,s}^{-1}(y)) dr  -\n\\int_t^s \\sigma (X_{r,s}^{-1} (y)) d\\overleftarrow{B}_r .\n\\end{split}\n\\end{equation}\nfor any  $t<s$,  where\n\\begin{equation}\n\\label{drift:backward}\n\\begin{split}\n\\widehat{b}(x) =  b(x) -  \\sum_{i,j}\\frac{\\partial \\sigma^j (x)\n}{\\partial x_i}  \\sigma^{ij} (x) .\n\\end{split}\n\\end{equation}\n\\end{Proposition}\n\\vspace{0.2cm}\n\\noindent We denote by $J(X_{t,s}^{-1}(x))$ the determinant of the Jacobian\nmatrix of $X_{t,s}^{-1}(x)$, which is positive and\n$J(X_{t,t}^{-1}(x))=1$. For $\\varphi\\in C_c^{\\infty}(\\mathbb{R}^d)$, we define\na process $\\varphi_t:\\,\\Omega\\times [t,T]\\times \\mathbb{R}^d\\rightarrow \\mathbb{R}$ by\n\\begin{equation}\n\\label{random:testfunction conv}\n\\varphi_t(s,x):=\\varphi(X_{t,s}^{-1}(x))J(X_{t,s}^{-1}(x)).\n\\end{equation}\nWe know that for $v\\in \\mathbf{L}^2(\\mathbb{R}^d)$, the composition of $v$\nwith the stochastic flow is\n$$(v\\circ X_{t,s} (\\cdot), \\varphi):=(v,\\varphi_t(s,\\cdot)).$$\nIn fact, by a change of variable, we have (see Kunita \\cite{K94a}, \\cite{K94b}, Bally and Matoussi \\cite{BM01}) $$(v\\circ X_{t,s} (\\cdot),\n\\varphi)=\\displaystyle\\int_{\\mathbb{R}^d}v(X_{t,s}\n(x))\\varphi(x)dx=\\displaystyle\\int_{\\mathbb{R}^d}v(y)\\varphi(X_{t,s}^{-1}(y))J(X_{t,s}^{-1}(y))dy\n=(v,\\varphi_t(s,\\cdot)).$$ Since  $ (\\varphi_t(s,x))_{ t\\leq s}$ is a\nprocess,  we may not use it directly as a test function because \n$\\displaystyle\\int_t^T(u(s,\\cdot),\\partial_s\\varphi_t(s,\\cdot))$ has no sense. However\n$\\varphi_t(s,x)$ is a semimartingale and we have the following\ndecomposition of $\\varphi_t(s,x)$\n\\begin{Lemma}\\label{decomposition conv}\nFor every function $\\varphi\\in C_c^{\\infty}(\\mathbb{R}^d),$\n\\begin{equation}\\label{decomp conv}\\begin{array}{ll}\n\\varphi_t(s,x)&=\\varphi(x)+\\displaystyle\\displaystyle\\int_t^s{\\mathcal\nL}^\\ast\\varphi_t(r,x)dr-\\sum_{j=1}^{d}\\displaystyle\\int_t^s\\left(\\sum_{i=1}^{d}\\frac{\\partial}{\\partial\nx_i}(\\sigma^{ij}(x)\\varphi_t(r,x))\\right)dB_r^j,\n \\end{array} \\end{equation}\nwhere ${\\mathcal L}^\\ast$ is the adjoint operator of ${\\mathcal L}$.\n\\end{Lemma}\n\\vspace{0.2cm}\n\\noindent We also  need  equivalence of norms result  which plays an important role in the proof of the existence of the solution for SPDE as a connection between the functional norms and random norms.\nFor continuous SDEs, this result  was first proved by Barles and Lesigne \\cite{BL97} by\nusing an analytic method and Bally, Matoussi \\cite{BM01} by a probabilistic method. \n\\begin{Proposition}\\label{equivalence:normes conv}\nThere exists two constants $c>0$ and $C>0$ such that for every\n$t\\leq s\\leq T$ and $\\varphi\\in L^1(\\mathbb{R}^d,dx)$,\n\\begin{equation}\\label{equi1 conv} c\\displaystyle\\int_{\\mathbb{R}^d}|\\varphi(x)|\\rho(x)dx\\leq\n\\displaystyle\\int_{\\mathbb{R}^d}\\mathbb{E}(|\\varphi(X_{t,s}(x))|)\\rho(x)dx\\leq\nC\\displaystyle\\int_{\\mathbb{R}^d}|\\varphi(x)|\\rho(x)dx. \n\\end{equation} \nMoreover, for\nevery $\\Psi\\in L^1([0,T]\\times\\mathbb{R}^d,dt\\otimes dx)$,\n\\begin{equation} \\label{equi2 conv}\nc\\displaystyle\\int_{\\mathbb{R}^d}\\displaystyle\\int_t^T|\\Psi(s,x)|ds\\rho(x)dx \\leq\n\\displaystyle\\int_{\\mathbb{R}^d}\\int_t^T\\mathbb{E}(|\\Psi(s,X_{t,s}(x))|)ds\\rho(x)dx\\leq\nC\\displaystyle\\int_{\\mathbb{R}^d}\\displaystyle\\int_t^T|\\Psi(s,x)|ds\\rho(x)dx.\n\\end{equation}\n \\end{Proposition}\n\\subsection{Existence and uniqueness of solutions for the reflected SPDE}\n\\label{subsection:SPDE}\nIn order  to provide a probabilistic representation to the solution of the RSPDEs \\eqref{OSPDE1}, we introduce the following  Markovian RBDSDE:\n\\begin{equation}\n\\label{rbsde1}\n \\left\\lbrace\n\\begin{aligned}\n&(i)~ Y_{s}^{t,x}\n =\\Phi(X_{t,T}(x))+\n\\displaystyle\\int_{s}^{T}f_r(X_{t,r}(x),Y_{r}^{t,x},Z_{r}^{t,x})dr+\\displaystyle\\int_{s}^{T}h_r(X_{t,r}(x),Y_{r}^{t,x},Z_{r}^{t,x})d\\overleftarrow{W}_r+K_{T}^{t,x}-K_{s}^{t,x}\\\\\n&\\hspace{2.5cm}\n -\\displaystyle\\int_{s}^{T}Z_{r}^{t,x}dB_{r},\\;\n\\mathbb{P}\\text{-}a.s. , \\; \\forall \\,  s \\in [t,T]  \\\\\n& (ii)~ Y_{s}^{t,x} \\in \\bar{D} \\, \\, \\quad \\mathbb{P}\\text{-}a.s.\\\\\n& (iii) \\displaystyle\\int_0^T (Y_{s}^{t,x}-v_s(X_{t,s}(x)))^* dK_{s}^{t,x}\\leq 0., \\, \\, \\mathbb{P}\\text{-}a.s., \\\\\n&~ \\text{for any continuous }\\, {\\cal F}_t -\\text{random function}\\, v \\, : \\,[0,T] \\times \\Omega \\times \\mathbb R^d \\longrightarrow \\,  \\bar{D}.\n\\end{aligned}\n\\right.\n\\end{equation}\n\n\\noindent Moreover, using Assumptions \\ref{assxi} and \\ref{assgener} and  the equivalence of norm results  (\\ref{equi1 conv}) and (\\ref{equi2 conv}), we get\n\\b*\n\\begin{split}\n&\\Phi(X_{t,T}(x)) \\in \\mathbf{L}^2({\\cal F}_T) \\, \\,  \\mbox{and} \\, \\,    \\Phi(X_{t,T}(x)) \\in \\bar{D},\\\\\n &f_s^0(X_{t,s}(x)) \\in \\mathcal {H}_{k}^2(t,T)\\,\\mbox{ and }\\,h_s^0(X_{t,s}(x)) \\in \\mathcal {H}_{k\\times d}^2(t,T).\\\\\n\\end{split}\n\\e*\nTherefore  under Assumption \\ref{assxi}-\\ref{assint} and according to Theorem \\ref{existence:RBDSDE},  there exists a unique triplet \n$ (Y^{t,x},Z^{t,x},K^{t,x}) $ solution of the RBDSDE \\eqref{rbsde1} associated to $ (\\Phi, f, h)$.\\\\\n\n\\noindent We  now consider  the  following definition  of weak solutions for the reflected SPDE (\\ref{OSPDE1}):\n\\begin{Definition}[{\\textbf{Weak solution of RSPDE}}]\n\\label{o-pde}We say that $(u,\\nu ):= (u^i,\\nu^i )_{1\\leq i\\leq k}$ is the weak solution of the reflected SPDE (\\ref{OSPDE1}) associated to $(\\Phi,f,h)$, if for each $1\\leq i\\leq k$ \n\\begin{itemize}\n\\item[(i)]$\\left\\| u\\right\\|_{{\\mathcal H}_T} <\\infty $, $u_t(x)\\in \\bar{D}, dx\\otimes dt\\otimes d\\mathbb{P}~a.e.$, and  $u(T,x)=\\Phi(x)$.\n\\item[(ii)] $\\nu^i $ is a signed \\textit{Random measure} on $[0,T]\\times\\mathbb{R}^d$ such that:\n\\begin{itemize}\n\\item[a)] $\\nu^i $ is adapted in the sense that for any measurable function $\\psi:[0,T]\\times \\mathbb{R}^d\\longrightarrow\\mathbb{R}^d$ and for each $s\\in[t,T]$,$\\displaystyle\\int_{s}^{T}\\!\\displaystyle\\int_{\\mathbb{R}^{d}}\\!\\psi (r,x)\\nu^i(dr,dx)$ is ${\\cal F}_{s,T}^W$-measurable.\n\\item[b)] $\\mathbb{E}\\big[\\displaystyle\\int_{0}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}\\rho (x)|\\nu^i| (dt,dx)\\big]<\\infty.$\n\n\\end{itemize} \n\\item[(iii)] for every $\\varphi \\in \\mathcal D_T$%\n\\begin{align}\\label{OPDE}\n\\nonumber &\\displaystyle\\int_{t}^{T}\\!\\!\\displaystyle\\int_{\\mathbb{R}^{d}}\\!u^i(s,x)\\partial _{s}\\varphi(s,x)dxds+\\displaystyle\\int_{\\mathbb{R}^{d}}\\!\\!(u^i(t,x )\\varphi (t,x\n)-\\Phi^i(x )\\varphi (T,x))dx-\\displaystyle\\int_{t}^{T}\\!\\!\\displaystyle\\int_{\\mathbb{R}^{d}}\\!u^i(s,x)\\mathcal{L}^*\\varphi(s,x)dxds\\\\\n\\nonumber &=\\displaystyle\\int_{t}^{T}\\!\\!\\displaystyle\\int_{\\mathbb{R}^{d}}\\!\\!f_s(x ,u(s,x),\\nabla u(s,x)\\sigma(x))\\varphi(s,x)dxds+\\displaystyle\\int_{t}^{T}\\!\\!\\displaystyle\\int_{\\mathbb{R}^{d}}\\!\\!h_s(x ,u(s,x),\\nabla u(s,x)\\sigma(x))\\varphi(s,x)dxd\\overleftarrow{W}_s\\\\\n& +\\displaystyle\\int_{t}^{T}\\!\\!\\displaystyle\\int_{\\mathbb{R}^{d}}\\!\\!\\varphi (s,x)1_{\\{u\\in \\partial D\\}}(s,x)\\nu^i(ds,dx). \n\\end{align}\n\\end{itemize}\nFor the sake of simplicity we will omit in the sequel the subscript $i$.\n\\end{Definition}\n\\vspace{0.2cm}\n\\noindent The main result of this section is the following:\n\\begin{Theorem}\n\\label{existence:RSPDE}\nLet  Assumptions \\ref{assxi}-\\ref{assint}   hold and  $\\rho (x)=(1+\\left| x\\right|)^{-p}$ with $p >  d+1 $. Then\nthere exists a weak solution $(u,\\nu\n)$ of  the reflected SPDE (\\ref{OSPDE1}) associated to $(\\Phi,f,h)$ such that, $ u (t, x) := Y_t^{t,x}$,   $dt\\otimes\nd\\mathbb{P}\\otimes\\rho(x)dx-a.e.$,  and  \\begin{eqnarray}\\label{con-pre}\n Y_{s}^{t,x}=u(s,X_{t,s}(x)), \\quad \\quad Z_{s}^{t,x}=(\\nabla\nu\\sigma)(s,X_{t,s}(x)), \\quad ds\\otimes\nd\\mathbb{P}\\otimes\\rho(x)dx-a.e.. \n\\end{eqnarray}\nMoreover, $\\nu^i $ is a \\textit{regular measure} in the following sense: for every measurable bounded and positive functions $\\varphi $ and $\\psi $,\n\\begin{align}\n\\nonumber &\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{t}^{T}\\varphi (s,X^{-1}_{t,s}(x))J(X^{-1}\n_{t,s}(x))\\psi (s,x)1_{\\{u\\in \\partial D\\}}(s,x)\\nu^i (ds,dx)\\\\\n&=\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{t}^{T}\\varphi (s,x)\\psi (s,X_{t,s}(x))dK_{s}^{t,x,i}dx\\text{, a.s..}\n\\label{con-k}\n\\end{align}\nwhere $(Y_{s}^{t,x},Z_{s}^{t,x},K_{s}^{t,x})_{t\\leq s\\leq T}$ is the\nsolution of RBDSDE (\\ref{rbsde1}).and satisfying the probabilistic interpretation \\eqref{con-k}.\\\\\nIf $(\\overline{u},\\overline{\\nu })$ is another solution of the reflected SPDE (\\ref{OSPDE1}) such that $\\overline{\\nu }$ satisfies (\\ref{con-k}) with some $%\n\\overline{K}$ instead of $K$, where $\\overline{K}$ is a continuous process, then $\\overline{u}=u$ and $\\overline{\\nu }=\\nu $.\\\\\nIn other words, there is a unique Randon regular measure with support $\\{u\\in\\partial D\\}$ which satisfies (\\ref{con-k}).\n\\end{Theorem}\n\\begin{Remark}\nThe expression (\\ref{con-k}) gives us the probabilistic\ninterpretation (Feymamn-Kac's formula) for the measure $\\nu $ via\nthe nondecreasing process $K^{t,x}$ of the RBDSDE. This formula was\nfirst introduced in Bally et al. \\cite{BCEF} (see also \\cite{MX08}) in the context of obstacle problem  for PDEs. Here we adapt this notion to the case of SPDEs in a convex domain.\n\\end{Remark}\n\\vspace{0.2cm}\n\\noindent We give now the following result which allows us to link in a natural way the solution of RSPDE with the associated RBDSDE.\n Roughly speaking, if we choose in the variational formulation \\eqref{OPDE} the random test functions $\\varphi_t(\\cdot,\\cdot)$, then we obtain the  associated RBDSDE.\nThis result plays the same role as It\\^o's formula used in \\cite{pp1994} \n to relate the solution of some semilinear RSPDEs with the associated RBDSDEs:\n\\begin{Proposition}\n\\label{weak:Itoformula1 conv} Let  Assumptions \\ref{assxi}-\\ref{assint} hold and $u\\in {\\mathcal H_T}$ be a weak solution of the reflected \nSPDE (\\ref{OSPDE1}) associated to $(\\Phi,f,h)$, \nthen for $s\\in[t,T]$ and $\\varphi\\in\nC_c^{\\infty}(\\mathbb{R}^d)$, \n\\begin{equation}\\label{wspde2}\n\\begin{array}{ll}\n\\displaystyle\\int_{\\mathbb{R}^d}\\int_s^Tu(r,x)d\\varphi_t(r,x)dx+(u(s,\\cdot),\\varphi_t(s,\\cdot))-(\\Phi(\\cdot),\\varphi_t(T,\\cdot))\n-\\displaystyle\\int_{\\mathbb{R}^d}\\int_s^T u(r,x)\\mathcal L^\\ast \\varphi_t(r,x))drdx\\\\\n=\\displaystyle\\int_{\\mathbb{R}^d}\\int_s^Tf_r(x,u(r,x),\\nabla\nu(r,x)\\sigma(x))\\varphi_t(r,x)drdx+\\displaystyle\\int_{\\mathbb{R}^d}\\int_s^T h_r(x,u(r,x),\\nabla\nu(r,x)\\sigma(x))\\varphi_t(r,x)d\\overleftarrow{W}_rdx\\\\\n+\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{s}^{T}\\varphi _{t}(r,x)1_{\\{u\\in\\partial D\\}}(r,x)\\nu (dr,dx) \\quad a.s.\n\\end{array}\n\\end{equation}\nwhere  $\\displaystyle\\int_{\\mathbb{R}^d}\\displaystyle\\int_s^Tu(r,x)d\\varphi_t(r,x)dx $ is well defined in the semimartingale decomposition result (Lemma \\ref{decomposition conv}).\n\\end{Proposition}\n\n\\vspace{0.5em}\n\\noindent\\textbf{Proof of Theorem \\ref{existence:RSPDE}.}\\\\\n\\noindent\\textbf{a) Existence}: The existence of a solution\nwill be proved in two steps. For the first step, we suppose that\n$h$ doesn't depend on $y,z$, then we are able to apply the classical penalization method. \nIn the second step, we study the case when $h$ depends on $y,z$ with the result obtained in the first step.\\\\[0.1cm] \n\n\\textit{Step 1} :\nWe will use the penalization method. For $n\\in \\mathbb{N}$, we consider for\nall $s\\in [t,T]$,\n\\begin{align*}\nY_{s}^{n,t,x}=\\Phi(X_{t,T}(x))&+\\int_{s}^{T}f_r(X_{t,r}(x),Y_{r}^{n,t,x},Z_{r}^{n,t,x})dr+\\int_{s}^{T}h_r(X_{t,r}(x))d\\overleftarrow{W}_r\\\\\n&-n\n\\int_{s}^{T}(Y_{r}^{n,t,x}-\\pi(Y_{r}^{n,t,x}))dr\n-\\int_{s}^{T}Z_{r}^{n,t,x}dB_{r}.\n\\end{align*}\n\n\\noindent From Theorem 3.1 in Bally and Matoussi \\cite{BM01}, we know that $u_{n}(t,x):=Y_{t}^{n,t,x}$, is a solution of the SPDE $(\\Phi,f_{n},h)$ (\\ref{SPDE1}), with $\nf_{n}(t,x,y)=f(t,x,y,z)-n(y-\\pi(y))$, i.e. for every $\\varphi \\in\\mathcal{D}_T$\n\\begin{align}\\label{o-equa1}\n \\nonumber\\displaystyle\\int_{t}^{T}(u^{n}(s,\\cdot),\\partial\n_{s}\\varphi(s,\\cdot) )ds & +(u^{n}(t,\\cdot ),\\varphi\n(t,\\cdot ))-(\\Phi(\\cdot ),\\varphi (T,\\cdot))-\\displaystyle\\int_{t}^{T}\n(u^{n}(s,\\cdot),\\mathcal{L}^*\\varphi(s,\\cdot))ds\\\\\n\\nonumber&=\\displaystyle\\int_{t}^{T}(f_s(\\cdot,u^{n}(s,\\cdot),\\sigma ^{*}\\nabla\nu_{n}(s,\\cdot)),\\varphi(s,\\cdot))ds+\\displaystyle\\int_{t}^{T}(h_s(\\cdot),\\varphi(s,\\cdot))d\\overleftarrow{W}_s\\\\\n&-n\\int_{t}^{T}((u^{n}-\\pi(u^{n}))(s,\\cdot ),\\varphi(s,\\cdot))ds.\n\\end{align}\nMoreover from Theorem 3.1 in Bally and Matoussi \\cite{BM01}, we also have\n\\begin{align}\\label{rep1}\n\\nonumber &Y_{s}^{n,t,x}=u_{n}(s,X_{t,s}(x))\\,\\,\\,,\\,\\,\\,Z_{s}^{n,t,x}=(\\nabla\nu_{n}\\sigma)(s,X_{t,s}(x)),\\,\\, ds\\otimes\nd\\mathbb{P}\\otimes\\rho(x)\\,dx-a.e. \\\\\n\\end{align}\nSet $K_{s}^{n,t,x}=-n \\displaystyle\n\\displaystyle\\int_{t}^{s}(Y_{r}^{n,t,x}-\\pi(Y_{r}^{n,t,x}))dr$. Then by\n(\\ref{rep1}), we have that \n\\begin{eqnarray}\nK_{s}^{n,t,x}=-n \\displaystyle\n\\displaystyle\\int_{t}^{s}(u_{n}-\\pi(u_{n}))(r,X_{t,r}(x))dr.\\label{defKn}\\end{eqnarray}\nFollowing the estimates and convergence results for\n$(Y^{n,t,x},Z^{n,t,x},K^{n,t,x})$ in Section 2 and  estimate (\\ref{estunif}),  we get :\n\\begin{eqnarray}\n\\label{K-estimate}\n &\\underset{n}{\\displaystyle\\sup}\\, \\mathbb{E}\\left[\\underset{t\\leq s\\leq T}{\\displaystyle\\sup}\\left| Y_{s}^{n,t,x}\\right| ^{2}+\\int_{t}^{T}\\left\\|\nZ_{s}^{n,t,x}\\right\\|^2ds + \\|K^{n,t,x}\\|_{VT}\\right] \\leq C\\left( T,x\\right),\n\\end{eqnarray} \nwhere  $$ C (T,x):=  \\mathbb{E} \\Big[ \\left| \\Phi (X_{t,T} (x))\\right|^2 + \\displaystyle\\int_t^T  \\big( \\left| f_s^0 (X_{t,s} (x))\\right|^2 \\,  + \\left| h_s^0 (X_{t,s} (x))\\right|^2 \\, \\big)\\, ds \\Big],$$\nand\n\\begin{align*}\n& \\mathbb{E}[\\underset{t\\leq s\\leq T}{\\displaystyle\\sup}\\left| Y_{s}^{n,t,x}-Y_{s}^{m,t,x}\\right|\n^{2}]+\\mathbb{E}[\\displaystyle\\int_{t}^{T}\\left\\| Z_{s}^{n,t,x}-Z_{s}^{m,t,x}\\right\\|\n^{2}ds]\\\\\n&+\\mathbb{E}[\\underset{t\\leq s\\leq T}{\\displaystyle\\sup}\\left|\nK_{s}^{n,t,x}-K_{s}^{m,t,x}\\right| ^{2}]\\quad\\longrightarrow 0,~\\text{as}~ n,m \\longrightarrow +\\infty.\n\\end{align*}\nMoreover, the equivalence of norms results (\\ref{equi2 conv}) yield:\n\\begin{eqnarray*}\n&&\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{t}^{T}\\rho (x)(\\left|\nu_{n}(s,x)-u_{m}(s,x)\\right| ^{2}+\\left| (\\nabla u_{n}\\sigma)(s,x)-(\\nabla u_{m}\\sigma)(s,x)\\right| ^{2})dsdx \\\\\n&\\leq &\\frac{1}{k_{2}}\\displaystyle\\int_{\\mathbb{R}^{d}}\\rho (x)\\mathbb{E}\\displaystyle\\int_{t}^{T}(\\left|\nY_{s}^{n,t,x}-Y_{s}^{m,t,x}\\right| ^{2}+\\left\\|\nZ_{s}^{n,t,x}-Z_{s}^{m,t,x}\\right\\| ^{2})dsdx\\longrightarrow 0.\n\\end{eqnarray*}\nThus $(u_{n})_{n\\in\\mathbb N}$ is a Cauchy sequence in $\\mathcal{H}_T$, and the limit $%\nu=\\underset{n\\rightarrow \\infty }{\\lim}u_{n}$ belongs to  $\\mathcal{H}_T$.\nDenote $\\nu _{n}(dt,dx)= -n(u_{n}-\\pi(u_{n}))(t,x)dtdx$ and $\\pi _{n}(dt,dx)=\\rho\n(x)\\nu _{n}(dt,dx)$, then by the equivalence norm result (\\ref{equi2 conv}) we get \n\\begin{eqnarray*}\n\\mathbb{E}\\big[|\\pi _{n}|([t,T]\\times \\mathbb{R}^{d})\\big] &=&\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\mathbb{E}\\big[ n|(u_{n}-\\pi(u_{n}))(s,x)| \\big]ds\\rho\n(x)dx \\\\\n&\\leq &\\frac{1}{k_{2}}\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\mathbb{E}\\big[ n|(u_{n}-\\pi(u_{n}))(s,X_{t,s}(x))| \\big]ds\\rho\n(x)dx.\n\\end{eqnarray*}\nFinally, using \\eqref{defKn} and \\eqref{K-estimate}, we obtain\n\\begin{eqnarray*}\n\\mathbb{E}\\big[|\\pi _{n}|([0,T]\\times \\mathbb{R}^{d})\\big]&\\leq &\\frac{1}{k_{2}}\\displaystyle\\int_{\\mathbb{R}^{d}}\\rho (x)\\mathbb{E}\\left\\|\nK^{n,0,x}\\right\\|_{VT} dx\\leq C\\displaystyle\\int_{\\mathbb{R}^{d}}\\rho (x)dx<\\infty .\n\\end{eqnarray*}\nIt follows that\n\\begin{equation}\n\\underset{n}{\\displaystyle\\sup}\\,|\\pi _{n}|([0,T]\\times \\mathbb{R}^{d})<\\infty .  \\label{est-measure}\n\\end{equation}\nMoreover  by Lemma \\ref{tight} (see Appendix \\ref{appendix:tight}),  the sequence of measures $(\\pi _{n})_{n  \\in \\mathbb N}$  is tight.  Therefore, there exits a subsequence such that $(\\pi _{n})_{n  \\in \\mathbb N}$  converges  weakly to a measure $\\pi $. Define $\\nu =\\rho^{-1}\\pi $; $\\nu $ is a measure such that $\\displaystyle\\int_{0}^{T}\\displaystyle\\int_{\n\\mathbb{R}^{d}}\\rho (x)|\\nu| (dt,dx)<\\infty $, and so we have for $\\varphi \\in \\mathcal{D}_T$ with compact support in $x$,\n$$\n\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{t}^{T}\\varphi d\\nu _{n}=\\displaystyle\\int_{\\mathbb{R}%\n^{d}}\\displaystyle\\int_{t}^{T}\\frac{\\phi }{\\rho }d\\pi _{n}\\rightarrow \\displaystyle\\int_{\\mathbb{R}\n^{d}}\\displaystyle\\int_{t}^{T}\\frac{\\phi }{\\rho }d\\pi =\\displaystyle\\int_{\\mathbb{R}\n^{d}}\\displaystyle\\int_{t}^{T}\\varphi d\\nu .\n$$\nNow passing to the limit in the SPDE $(\\Phi,f_{n},h)$ (\\ref{o-equa1}), we get that  that $(u,\\nu )$\nsatisfies the reflected SPDE associated to  $(\\Phi,f,h)$, i.e. for every $\\varphi \\in\n\\mathcal{D}_T$, we have\n\\begin{eqnarray}\n&&\\displaystyle\\int_{t}^{T}(u(s,\\cdot),\\partial _{s}\\varphi(s,\\cdot) )ds+(u(t,\\cdot ),\\varphi (t,\\cdot))-(\\Phi(\\cdot),\\varphi (T,\\cdot))-\\displaystyle\\int_{t}^{T}(u(s,\\cdot),\\mathcal{L}^*\\varphi(s,\\cdot))ds\n\\nonumber \\\\\n&=&\\displaystyle\\int_{t}^{T}(f_s(\\cdot, u(s,\\cdot),\\sigma^{*}\\nabla u(s,\\cdot)),\\varphi(s,\\cdot) )ds+\\displaystyle\\int_{t}^{T}(h_s(\\cdot),\\varphi(s,\\cdot) )d\\overleftarrow{W}_s+\\displaystyle\\int_{t}^{T}\\displaystyle\\int_{\\mathbb{R}\n^{d}}\\varphi (s,x)\\nu (ds,dx).\\nonumber\\\\\n  \\label{equa1} \n\\end{eqnarray}\nThe last point is to prove that $\\nu $ satisfies the probabilistic interpretation (\\ref{con-k}). Since $K^{n,t,x}$ converges to $K^{t,x}$ uniformly in $t$, the\nmeasure $dK^{n,t,x}$ converges to $dK^{t,x}$ weakly in probability.\nFix two continuous functions $\\varphi $, $\\psi $ : $[0,T]\\times \\mathbb{R}%\n^{d}\\rightarrow \\mathbb{R}^{+}$ which have compact support in $x$ and a\ncontinuous function with compact support $\\theta :\\mathbb{R}^{d}\\rightarrow %\n\\mathbb{R}^{+}$, from Bally et al \\cite{BCEF} (The proof of Theorem 4), we have (see also Matoussi and Xu \\cite{MX08})\n\\begin{eqnarray*}\n&&\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi (s,X^{-1}_{t,s}(x))J(X\n^{-1}_{t,s}(x))\\psi (s,x)\\theta (x)\\nu (ds,dx) \\\\\n&=&\\lim_{n\\rightarrow \\infty }-\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi (s,\nX^{-1}_{t,s}(x))J(X^{-1}_{t,s}(x))\\psi (s,x)\\theta\n(x)n(u_{n}-\\pi(u_{n}))(s,x)dsdx \\\\\n&=&\\lim_{n\\rightarrow \\infty }-\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi\n(s,x)\\psi (s,X_{t,s}(x))\\theta\n(X_{t,s}(x))n(u_{n}-\\pi(u_{n}))(t,X_{t,s}(x))dtdx \\\\\n&=&\\lim_{n\\rightarrow \\infty }\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi\n(s,x)\\psi (s,X_{t,s}(x))\\theta (X_{t,s}(x))dK_{s}^{n,t,x}dx \\\\\n&=&\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi (s,x)\\psi (s,X_{t,s}(x))\\theta\n(X_{t,s}(x))dK_{s}^{t,x}dx.\n\\end{eqnarray*}\nWe take $\\theta =\\theta _{R}$ to be the regularization of the indicator\nfunction of the ball of radius $R$ and pass to the limit with $R\\rightarrow\n\\infty $, to get that\n\\begin{equation}\\label{con-k1}\n\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi (s,X^{-1}_{t,s}(x))J(X^{-1}\n_{t,s}(x))\\psi (s,x)\\nu (ds,dx)=\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi\n(s,x)\\psi (s,X_{t,s}(x))dK_{s}^{t,x}dx.\n\\end{equation}\nFrom Section 2, it follows that $dK_{s}^{t,x}=1_{\\{u\\in\\partial D\\}}(s,X_{t,s}(x))dK_{s}^{t,x}$. In (\\ref{con-k1}), setting $\\psi =1_{\\{u\\in\\partial D\\}}$ yields\n\\begin{align*}\n&\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi (s,X^{-1}_{t,s}(x))J(X^{-1}\n_{t,s}(x))1_{\\{u\\in\\partial D\\}}(s,x)\\nu (ds,dx)\\\\\n&=\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{t}^{T}\\varphi\n(s,X^{-1}_{t,s}(x))J(X^{-1}_{t,s}(x))\\nu (ds,dx)\\text{, a.s.}\n\\end{align*}\nNote that the family of functions $A(\\omega )=\\{(s,x)\\rightarrow \\phi (s,%\nX^{-1}_{t,s}(x)):\\varphi \\in C_{c}^{\\infty }\\}$ is an algebra which\nseparates the points (because $x\\rightarrow X^{-1}_{t,s}(x)$ is a\nbijection). Given a compact set $G$, $A(\\omega )$ is dense in $C([0,T]\\times\nG)$. It follows that $J(X^{-1}_{t,s}(x))1_{\\{u\\in\\partial D\\}}(s,x)\\nu (ds,dx)=J(\nX^{-1}_{t,s}(x))\\nu (ds,dx)$ for almost every $\\omega $. While $J(\nX^{-1}_{t,s}(x))>0$ for almost every $\\omega $, we get $\\nu\n(ds,dx)=1_{\\{u\\in\\partial D\\}}(s,x)\\nu (ds,dx)$, and (\\ref{con-k}) follows.\\\\\nThen we get easily that $Y_{s}^{t,x}=u(s,X_{t,s}(x))$ and $\nZ_{s}^{t,x}=(\\nabla u\\sigma)(s,X_{t,s}(x))$,  in view of the convergence\nresults for $(Y_{s}^{n,t,x},Z_{s}^{n,t,x})$ and the equivalence of norms. So $u(s,X_{t,s}(x))=Y_{s}^{t,x}\\in \\bar{D}$. Specially for $s=t$, we\nhave $u(t,x)\\in \\bar{D}$.\\\\[0.2cm]\n\\textit{ Step 2 } : \\textit{The  nonlinear  case where $h$\ndepends on $y$ and $z$}.\\\\  \nDefine $H(s,x)\\triangleq\nh(x,Y_s^{s,x},Z_s^{s,x}).$ Due to the fact that\n$h^0\\in \\mathbf{L}^2_\\rho([0,T]\\times\\mathbb{R}^d)$ and $h$ is Lipschitz\nwith respect to $(y,z)$, then \nwe have $H_s(x)\\in \\mathbf{L}^2_\\rho([0,T]\\times\\mathbb{R}^d)$. Since $H$ is independent of $y,z$, by applying the result of Step 1 yields that there exists $(u,\\nu )$ satisfying the SPDE with obstacle $(\\Phi,f,H)$, i.e. for every $\\varphi \\in\n\\mathcal{D}_T$, we have\n\\begin{eqnarray}\n&&\\displaystyle\\int_{t}^{T}(u(s,\\cdot),\\partial _{s}\\varphi(s,\\cdot) )ds+(u(t,\\cdot ),\\varphi (t,\\cdot))-(\\Phi(\\cdot),\\varphi (T,\\cdot))-\\displaystyle\\int_{t}^{T}(u(s,\\cdot),\\mathcal{L}^*\\varphi(s,\\cdot))ds\n\\nonumber \\\\\n&=&\\displaystyle\\int_{t}^{T}(f_s(\\cdot, u(s,\\cdot),\\sigma^{*}\\nabla u(s,\\cdot)),\\varphi(s,\\cdot) )ds+\\displaystyle\\int_{t}^{T}(H_s(\\cdot),\\varphi(s,\\cdot) )d\\overleftarrow{W}_s\\nonumber\\\\\n&+&\\int_{t}^{T}\\displaystyle\\int_{\\mathbb{R}\n^{d}}\\phi (s,x)1_{\\{u\\in\\partial D\\}}(s,x)\\nu (ds,dx).  \\label{equa2}\n\\end{eqnarray}\nThen by the uniqueness of the solution to the RBDSDE ($\\Phi(X_{t,T}(x))$, $f$, $h$),  we get easily that $Y_{s}^{t,x}=u(s,X_{t,s}(x))$, $%\nZ_{s}^{t,x}=(\\nabla u\\sigma)(s,X_{t,s}(x))$, and $\\nu$ satisfies the probabilistic interpretation (\\ref{con-k}). So $u(s,X_{t,s}(x))=Y_{s}^{t,x}\\in\\bar{D}$. Specially for $s=t$, we have $u(t,x)\\in\\bar{D}$, which is the desired result.\\\\\n\n\\noindent\\textbf{b) Uniqueness } :  Set $(\\overline{u},\\overline{\\nu\n})$ to be another weak solution of the reflected SPDE (\\ref{OSPDE1}) associated to \n$(\\Phi,f,h)$; with $\\overline{\\nu }$ verifies (\\ref{con-k}) for a continuous process $\\overline{K}$. We fix $\\varphi :\\mathbb{R}^{d}\\rightarrow \\mathbb{R}^k$, a smooth function in $C_{c}^{2}(%\n\\mathbb{R}^{d})$ with compact support and denote $\\varphi _{t}(s,x)=\\varphi (X^{-1}_{t,s}(x))J(X^{-1}_{t,s}(x))$. From Proposition \\ref{weak:Itoformula1 conv}, one may use $\\varphi _{t}(s,x)$ as a test function in the SPDE $(\\Phi,f,h)$ with $\\partial _{s}\\varphi (s,x)ds$ replaced by a stochastic integral with respect to the semimartingale $\\varphi _{t}(s,x)$. Then we get, for $t\\leq s\\leq T$\n\n\\begin{align}\n&\\displaystyle\\int_{\\mathbb{R}^{d}}\\int_{s}^{T}\\overline{u}(r,x)d\\varphi _{t}(r,x)dx+\\displaystyle\\int_{\\mathbb{R}^{d}}\\overline{u}(s,x)\\varphi _{t}(s,x)dx-\\displaystyle\\int_{\\mathbb{R}^{d}}\\Phi(x)\\varphi _{t}(T,x)dx-\\displaystyle\\int_{s}^{T}\\int_{\\mathbb{R}^{d}}\\overline{u}(r,x)\\mathcal{L}^*\\varphi _{t}(r,x)drdx  \\label{o-pde-u1}\n\\nonumber \\\\\n&=\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}f_r(x,\\overline{u}(r,x),(\\nabla \\overline{u}\\sigma)(r,x))\\varphi _{t}(r,x)drdx+ \\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}h_r(x,\\overline{u}(r,x),(\\nabla \\overline{u}\\sigma)(r,x))\\varphi _{t}(r,x)dx d\\overleftarrow{W}_r\\nonumber \\\\\n& +\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{\nR}^{d}}\\varphi _{t}(r,x)1_{\\{\\overline{u}\\in\\partial D\\}}(r,x)\\overline{\\nu }(dr,dx).\n\\end{align}\nBy (\\ref{decomp conv}) in Lemma \\ref{decomposition conv}, we have\n\n\\begin{align*}\n&\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{s}^{T}\\overline{u}(r,x)d\\varphi _{t}(r,x)dx\n=\\displaystyle\\int_{s}^{T}(\\displaystyle\\int_{\\mathbb{R}^{d}}(\\nabla \\overline{u}\\sigma)(r,x)\\varphi_{t}(r,x)dx)dB_{r}+\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}\\overline{u}(r,x)\\mathcal{L}^*\\varphi _{t}(r,x)drdx .\\\\\n\\end{align*}\nSubstituting this equality in (\\ref{o-pde-u1}), we get\n\\begin{align*}\n&\\displaystyle\\int_{\\mathbb{R}^{d}}\\overline{u}(s,x)\\varphi _{t}(s,x)dx =\\displaystyle\\int_{\\mathbb{R}^{d}}\\Phi(x)\\varphi _{t}(T,x)dx-\\displaystyle\\int_{s}^{T}(\\displaystyle\\int_{\\mathbb{R}^{d}}(\\nabla \\overline{u}\\sigma)(r,x)\\varphi\n_{t}(r,x)dx)dB_{r}\\\\\n&+\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}f_r(x,\\overline{u}(r,x),(\\nabla \\overline{u}\\sigma)(r,x))\\varphi _{t}(r,x)drdx+ \\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}h_r(x,\\overline{u}(r,x),(\\nabla \\overline{u}\\sigma)(r,x))\\varphi _{t}(r,x)dx d\\overleftarrow{W}_r\\\\&+\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{\nR}^{d}}\\varphi _{t}(r,x)1_{\\{\\overline{u}\\in\\partial D\\}}(r,x)\\overline{\\nu }(dr,dx).\n\\end{align*}\nThen by changing of variable $y=X^{-1}_{t,r}(x)$ and applying (\\ref\n{con-k}) for $\\overline{\\nu }$, we obtain\n\\begin{align*}\n&\\displaystyle\\int_{\\mathbb{R}^{d}}\\overline{u}(s,X_{t,s}(y))\\varphi (y)dy =\\displaystyle\\int_{\\mathbb{R}^{d}}\\Phi(X_{t,T}(y))\\varphi (y)dy\\\\\n&+\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{s}^{T}\\varphi(y)f_r(X_{t,r}(y),\\overline{u}(r,X_{t,r}(y)),(\\nabla \\overline{u}\\sigma)(r,X_{t,r}(y)))drdy\\\\\n& +\\displaystyle\\int_{\\mathbb{R}^{d}}\\displaystyle\\int_{s}^{T}\\varphi(y)h_r(X_{t,r}(y),\\overline{u}(r,X_{t,r}(y)),(\\nabla \\overline{u}\\sigma)(r,X_{t,r}(y)))dyd\\overleftarrow{W}_r \\\\\n&+\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^{d}}\\phi (y)1_{\\{\\overline{u}\\in\\partial D\\}}(r,X_{t,s}(y))d\\overline{K}_{r}^{t,y}dy-\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}\n^{d}}\\varphi (y)(\\nabla \\overline{u}\\sigma)(r,X_{t,r}(y))dydB_{r}\n\\end{align*}\nSince $\\varphi $ is arbitrary, we can prove that for $\\rho (y)dy$ almost every $\ny$, ($\\overline{u}(s,X_{t,s}(y))$, $(\\nabla \\overline{u}\\sigma\n)(s,X_{t,s}(y))$, $\\widehat{K}_{s}^{t,y}$) solves the RBDSDE \n$(\\Phi(X_{t,T}(y)),f,h)$. Here $\\widehat{K}_{s}^{t,y}$=$\\displaystyle\\int_{t}^{s}1_{\\{%\n\\overline{u}\\in\\partial D\\}}(r,X_{t,r}(y))d\\overline{K}_{r}^{t,y}$. Then by the\nuniqueness of the solution of the RBDSDE, we know $\\overline{u}%\n(s,X_{t,s}(y))=Y_{s}^{t,y}=u(s,X_{t,s}(y))$, $(\\nabla \\overline{\nu}\\sigma)(s,X_{t,s}(y))=Z_{s}^{t,y}=(\\nabla u\\sigma)(s,X_{t,s}(y))$,  and $%\n\\widehat{K}_{s}^{t,y}=K_{s}^{t,y}$. Taking $s=t$ we deduce that $\\overline{u}%\n(t,y)=u(t,y)$, $\\rho (y)dy$-a.s. and by the probabilistic interpretation (\n\\ref{con-k}), we obtain\n$$\n\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^d} \\varphi _{t}(r,x)1_{\\{\\overline{u}\\in\\partial D\\}}(r,x)\\overline{\\nu }\n(dr,dx)=\\displaystyle\\int_{s}^{T}\\displaystyle\\int_{\\mathbb{R}^d} \\varphi _{t}(r,x)1_{\\{u\\in \\partial D\\}}(r,x)\\nu (dr,dx).\n$$\nSo $1_{\\{\\overline{u}\\in\\partial D\\}}(r,x)\\overline{\\nu\n}(dr,dx)=1_{\\{u\\in\\partial D\\}}(r,x)\\nu (dr,dx)$.\\\\\n\\hbox{ }\\hfill$\\Box$ \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}