# Saliency Detection with Bilateral Absorbing Markov Chain Guided by Depth Information
## 2 Methodology
### Mid-Level Saliency Maps Generation by Bilateral Absorbing Markov Chain
#### 2.3.3 Cross-Modal Multi-Graph Learning Model
\(\mathbf{W}_{c}^{(v)}=\left[w_{ij}^{c(v)}\right]_{N\times N}\) is the graph affinity matrix computed by the \(\
u-\)th color feature and \(\mathbf{W}_{d}^{(\tau)}=\left[w_{ij}^{d(\tau)}\right]_{N\times N}\) is the graph affinity matrix computed by the \(\tau-\)th depth feature. Then, the final learning affinity matrix optimization equation can be defined as  
\[\min_{\beta\mathbf{Y}}\sum_{v=1}^{m}\left(\beta_{c}{}^{(v)} \right)^{\top} \sum_{i,j=1}^{N}w_{ij}^{c(v)}\|\mathbf{y}_{i}-\mathbf{y}_{j}\|^{2}+ \sum_{\tau=1}^{n}\left(\beta_{d}{}^{(\tau)}\right)^{\top}\sum_{i,j=1}^{N}w_{ ij}^{d(\tau)}\|\mathbf{y}_{i}-\mathbf{y}_{j}\|^{2}+\mu\sum_{i=1}^{N}\|\mathbf{y}_{i}- \mathbf{i}_{i}\|^{2},\] \[s.t.\sum_{v=1}^{m}\beta_{c}{}^{(v)}+\sum_{\tau=1}^{n}\beta_{d}{ }^{(\tau)}=1,0\leq\beta_{c}{}^{(v)},\beta_{d}{}^{(\tau)}\leq 1 \tag{19}\]  
where the parameter \(\gamma\) controls the weight distribution of all affinity matrices, ensuring that different-mode features can be fully utilized. Without this parameter, in some cases, it is possible that only partial features participate in the learning of affinity matrix, which may utilize the complementarity between different features insufficiently. The parameter \(\mu\) and \(\gamma\) are set to 0.001 and 4 respectively. To facilitate the derivation, we rewrite the above objective function (19) in the form of matrix,  
\[\mathcal{J}=\sum_{v=1}^{m}\left(\beta_{c}^{(v)}\right)^{\top}\!Tr\!\left( \mathbf{Y}^{\mathrm{T}}\mathbf{L}_{c}^{(v)}\mathbf{Y}\right)+\sum_{\tau=1}^{n }\left(\beta_{d}^{(\tau)}\right)^{\top}\!Tr\!\left(\mathbf{Y}^{\mathrm{T}} \mathbf{L}_{d}^{(\tau)}\mathbf{Y}\right)+\mu\|\mathbf{Y}-\mathbf{I}\|_{F}^{2} \tag{20}\]  
where \(\mathbf{L}_{c}^{(v)}=\mathbf{D}_{c}^{(v)}-\mathbf{W}_{c}^{(v)}\) is the graph Laplacian matrix of the \(\
u-\)th color feature, \(\mathbf{D}_{c}^{(v)}\) is the degree matrix and \(d_{ii}^{(v)}=\sum\limits_{j=1}^{N}w_{ij}^{c(v)}\). Similarly, \(\mathbf{L}_{d}^{(\tau)}=\mathbf{D}_{d}^{(\tau)}-\mathbf{W}_{d}^{(\tau)}\) is the graph Laplacian matrix of the \(\tau-\)th depth feature, \(\mathbf{D}_{d}^{(\tau)}\) is the degree matrix and \(d_{ii}^{d(v)}=\sum\limits_{j=1}^{N}w_{ij}^{d(v)}\). \(Tr(\cdot)\) and \(\|\cdot\|_{F}\) compute the trace and the Frobenius norm of the matrix separately. We can see that there are two unknown items \(\beta\) and \(\mathbf{Y}\) to be solved in Equation (20), so we decompose it into two sub-problems to solve this optimization problem by iteration. Fix \(\beta\), Update \(\mathbf{Y}\):  
\[\mathbf{Y}=\mu\!\left(\sum_{v=1}^{m}\beta_{c}{}^{(v)}\mathbf{L}_{c}^{(v)}+ \sum_{\tau=1}^{n}\beta_{d}{}^{(\tau)}\mathbf{L}_{d}^{(\tau)}+\mu\mathbf{I} \right)^{-1} \tag{21}\]Fix **Y**, Update \(\beta\):  
\[\beta_{c}{}^{(v)}=\frac{\left(Tr\Big{(}\mathbf{Y}^{\mathrm{T}}\mathbf{L}_{c}^{(v )}\mathbf{Y}\Big{)}\right)^{\frac{1}{1-\gamma}}}{\sum\limits_{\
u^{\prime}=1}^{ m}\left(Tr\Big{(}\mathbf{Y}^{\mathrm{T}}\mathbf{L}_{c}^{(\
u^{\prime})} \mathbf{Y}\Big{)}\right)^{\frac{1}{1-\gamma}}+\sum\limits_{\tau^{\prime}=1}^{n} \left(Tr\Big{(}\mathbf{Y}^{\mathrm{T}}\mathbf{L}_{d}^{(\tau^{\prime})}\mathbf{ Y}\Big{)}\right)^{\frac{1}{1-\gamma}}} \tag{22}\]  
\[\beta_{d}{}^{(\tau)}=\frac{\left(Tr\Big{(}\mathbf{Y}^{\mathrm{T}}\mathbf{L}_{d} ^{(\tau)}\mathbf{Y}\Big{)}\right)^{\frac{1}{1-\gamma}}}{\sum\limits_{\
u^{ \prime}=1}^{m}\left(Tr\Big{(}\mathbf{Y}^{\mathrm{T}}\mathbf{L}_{c}^{(\
u^{ \prime})}\mathbf{Y}\Big{)}\right)^{\frac{1}{1-\gamma}}+\sum\limits_{\tau^{ \prime}=1}^{n}\left(Tr\Big{(}\mathbf{Y}^{\mathrm{T}}\mathbf{L}_{d}^{(\tau^{ \prime})}\mathbf{Y}\Big{)}\right)^{\frac{1}{1-\gamma}}} \tag{23}\]  
To get the optimal solution of sub-problems, we utilize partial derivative and Lagrange Multiplier Method. The specific derivation process can refer to [48]. With the learned affinity matrix **Y**, we can calculate the transition matrixes of absorbing Markov chain.