# Error Resilience for Block Compressed Sensing with Multiple-Channel Transmission
## 3 Proposed Algorithm  
For the effective delivery of compressed sensing signals, and considering the robust transmission depicted in [14], we employed the use of transmission over multiple mutually independent lossy channels. Figure 1 describes the block diagram of our system. Figure 1: Block diagrams for transmission with block compressed sensing (BCS) and multiple description coding (MDC): (**a**) BCS encoder and protection with MDC; (**b**) Lossy transmission over two channels with \(p_{\text{e,1}}\) and \(p_{\text{e,2}}\); (**c**) MDC and BCS decoder for protection and reconstruction. In Figure 1a, the input image \(\mathbf{X}\) is divided into \(\mathbf{X}_{k}\). It is then compressed with BCS and the compressed sensing signal is denoted by \(\mathbf{Y}_{k}\). With the notations described in Section 2, we denoted \(\mathbf{Y}_{k}=\{Y_{k,1},\ Y_{k,2},\ \cdots,Y_{k,m}\}\) and set \(m\) to an even number for application with multiple description coding. To enable easy separation of BCS coefficients, we chose the odd numbered indices to form \(\mathbf{C}_{k1}=\{Y_{k,1},\ Y_{k,3},\ \cdots,Y_{k,m-1}\}\), and the even numbered ones to form \(\mathbf{C}_{k2}=\{Y_{k,2},\ Y_{k,4},\ \cdots,Y_{k,m}\}\). For ease of representation, we rearranged the notations to be  
\[\mathbf{C}_{k1}=\left\{Y_{k,1},\ Y_{k,3},\ \cdots,Y_{k,m-1}\right\}=\left\{ C_{k1,1},\ C_{k1,2},\ \cdots,C_{k1,\frac{m}{2}}\right\} \tag{7}\]  
and  
\[\mathbf{C}_{k2}=\left\{Y_{k,2},\ Y_{k,4},\ \cdots,Y_{k,m}\right\}=\left\{ C_{k2,1},\ C_{k2,2},\ \cdots,C_{k2,\frac{m}{2}}\right\} \tag{8}\]  
After that, we employed multiple description transform coding (MDTC) [15] in MDC to form the elements in the two descriptions of \(\mathbf{D}_{k1}\) and \(\mathbf{D}_{k2}\) in Equation (9):  
\[\left[ \begin{array}{c}D_{k1,i}\\ D_{k2,i}\end{array} \right]=\left[ \begin{array}{cc}r_{2}\cos\theta_{2}&-r_{2} \sin\theta_{2}\\ -r_{1}\cos\theta_{1}&r_{1}\sin\theta_{1}\end{array} \right]\left[ \begin{array}[ ]{c}C_{k1,i}\\ C_{k2,i}\end{array} \right], \tag{9}\]  
where \(i=1,\ 2,\ \cdots,\frac{m}{2}\). The \(2\times 2\) matrix in Equation (9) has the condition that \(r_{1}r_{2}\sin(\theta_{1}-\theta_{2})=1\), leading to the determinant of one. The resulting elements in Equation (9) form the two descriptions \(\mathbf{D}_{k1}=\left\{D_{k1,1},\ D_{k1,2},\ \cdots,D_{k1,\frac{m}{2}}\right\}\) and \(\mathbf{D}_{k2}=\left\{D_{k2,1},\ D_{k2,2},\ \cdots,D_{k2,\frac{m}{2}}\right\}\). Next, descriptions \(\mathbf{D}_{k1}\) and \(\mathbf{D}_{k2}\) were transmitted over two independent lossy channels, with the loss probability of \(p_{e,1}\) and \(p_{e,1}\) for Channels 1 and 2, as depicted in Figure 1b. The received descriptions may become due to the possibility of induced errors. Note that \(\mathbf{D}^{\prime}_{k1}\) and \(\mathbf{D}^{\prime}_{k2}\) may not be identical to their counterparts \(\mathbf{D}_{k1}\) and \(\mathbf{D}_{k2}\), respectively. At the decoder, as shown in Figure 1c by employing [16], compensation should be applied between received descriptions. Compensated descriptions \(\mathbf{D}^{\prime\prime}_{k1}=\left\{D^{\prime\prime}_{k1,1},\ D^{\prime \prime}_{k1,2},\ \cdots,D^{\prime\prime}_{k1,\frac{m}{2}}\right\}\) and \(\mathbf{D}^{\prime\prime}_{k2}=\left\{D^{\prime\prime}_{k2,1},\ D^{\prime \prime}_{k2,2},\ \cdots,D^{\prime\prime}_{k2,\frac{m}{2}}\right\}\) was calculated first.