# Resource Optimization for 3D Video SoftCast with Joint Texture/Depth Power Allocation
## 3 Minimum Distortion Optimization
### Problem Formulation  
Given a 3D video sequence, a 3D-DCT is used to decorrelate the texture and depth frames of the reference viewpoints in one GOP, which results in \(N\) chunks of DCT coefficients with variance indicated by \(\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\). Without loss of generality, assume \(\lambda_{i}\geq\lambda_{j}\) for all \(i<j\). Let \(K\) be the available bandwidth resources (e.g., frequency or time slots). Let \(k=[k_{1},k_{2},\ldots,k_{N}]\) be a binary allocation vector for each GOP. Therefore, the chunk selection is indicated as follows: chunk \(i\) is discarded when \(k_{i}=0\) and transmitted over a bandwidth resource slot when \(k_{i}=1\). The expected MSE of the \(i\)th chunk is denoted as the Euclidean norm \(\varepsilon_{i}\)=\(\|x_{i}[j]-\hat{x}_{i}[j]\|^{2}\). The optimal values of power allocation \(\rho^{*}=[\rho_{1}^{*},\rho_{2}^{*},\ldots,\rho_{N}^{*}]\) and bandwidth allocation are found by minimizing the transmission distortion under constrained power and bandwidth resources. The problem is formulated as follows [33]:  
\[MSE(k,\rho)=\min_{(k,\rho)}\mathbf{E}[\varepsilon_{i}|\sigma_{i}^{2}]\!=\! \frac{1}{N}\sum_{i=1}^{N}\!\frac{\lambda_{i}}{k_{i}\rho_{i}+1} \tag{6}\]  
\[\text{s.t.}\sum_{i=1}^{N}k_{i}<K \tag{7}\]  
\[\frac{1}{N}\sum_{i=1}^{N}k_{i}\rho_{i}\leq\frac{P}{N(h\text{x}w)\sigma_{in}^{ 2}}:=SNR_{chk} \tag{8}\]  
\[k_{i}\!\in\!\{0,1\},\quad 1\!\leq\!i\!\leq\!N \tag{9}\]  
\[\rho_{i}\geq 0,\quad\quad\quad 1\!\leq\!i\!\leq\!N \tag{10}\]  
where \(\rho_{i}=\frac{\delta_{i}^{2}\lambda_{i}}{\sigma_{in}^{2}}\) is the signal noise-to-power ratio for the \(i\)th chunk for a given power allocation, \(K\) should be in the range \([0,N]\), and \(SNR_{chk}\) is the SNR budget for each chunk. As systems run short on bandwidth, with \(K\!<\!N\), to minimize distortion the system will resort to transmitting a number \(K\) of HP chunks and discarding the remaining chunks. \[k= \begin{cases}1,\text{ if }i\leq K\\ 0,\text{ if }i>K\end{cases}  \tag{11}\]  
By re-expressing MSE as  
\[\text{MSE}=\frac{1}{N}\Big{(}\sum_{i=1}^{K}\frac{\lambda_{i}}{\rho_{i}+1}+ \sum_{i=K+1}^{N}\lambda_{i}\Big{)} \tag{12}\]the problem becomes one of finding the optimal power allocation \(\rho_{i}^{*}\) and bandwidth \(K^{*}\). Since the second-order derivative of the objective function in (12) is derived as  
\[\frac{\partial^{2}\text{MSE}}{\partial\rho_{i}^{2}}=\frac{2}{N}\sum_{i=1}^{K} \frac{\lambda_{i}}{\big{(}\rho_{i}+1\big{)}^{3}}>0,\rho_{i}>0 \tag{13}\]  
the Lagrange multiplier method is used to solve (12), after defining \(\mu>0\) and the Lagrange function \(J\) as  
\[\min_{(k,\rho)}J= \frac{1}{N}\Big{(}\sum_{i=1}^{K}\frac{\lambda_{i}}{\rho_{i}+1}+ \sum_{i=K+1}^{N}\lambda_{i}\Big{)}-\mu\left(\frac{1}{N}\sum_{i=1}^{K}\rho_{i}- SNR_{chk}\right) \tag{14}\]  
By setting \(\frac{\partial J}{\partial\rho_{i}}=0,i=1,2,\ldots,N\) and \(\frac{\partial J}{\partial\mu}=0\)  
\[\left\{ \begin{aligned} \frac{\partial J}{\partial\rho_{i}}& =\frac{1}{N}\frac{-\lambda_{i}}{\big{(}\rho_{i}+1\big{)}^{2}}+ \frac{\mu}{N}=0\\ \frac{\partial J}{\partial\mu}&=\frac{1}{N}\sum_{i= 1}^{K}\rho_{i}-SNR_{chk}=0\end{aligned} \right. \tag{15}\]  
\[SNR_{chk}=\frac{1}{N}\left(\frac{\sum_{i=1}^{K}\sqrt{\lambda_{i}}}{\frac{\sqrt {\lambda_{i}}}{(\rho_{i}+1)}}-K\right)=\frac{\sum_{i=1}^{K}\sqrt{\lambda_{i}} }{N\cdot\sqrt{\lambda_{i}}}(\rho_{i}+1)-\frac{K}{N} \tag{16}\]  
The optimal power allocation value \(\rho_{i}^{*}\) is derived as  
\[\rho_{i}^{*}=\frac{N\sqrt{\lambda_{i}}}{\sum_{i=1}^{K}\sqrt{\lambda_{i}}} \big{(}SNR_{chk}+\frac{K}{N}\big{)}-1\approx\frac{N\sqrt{\lambda_{i}}}{\sum_{ i=1}^{K}\sqrt{\lambda_{i}}}SNR_{chk} \tag{17}\]  
This confirms the chunk scaling in (2).