# Error Resilience for Block Compressed Sensing with Multiple-Channel Transmission
## 2 Fundamental Concepts of Block Compressed Sensing  
The field of compressed sensing aims to look for new sampling schemes that go against conventional sampling theorems or the well-known Nyquist-Shannon theorem. With compressed sensing a rate much smaller than twice the maximal bandwidth can be achieved to meet perfect reconstruction recovery. Compressed sensing, based on the representations in [1; 2], is composed of the _sparsity_ principle, and the _incoherence_ principle. Based on the concepts of compressed sensing depicted in Figure 1, we divided the original image \(\mathbf{X}\) with the size of \(M\times N\) into a set of small blocks \(\mathbf{X}_{k}\). Each block \(\mathbf{X}_{k}\) has the size \(B\times B\), and the subscript \(k\) denoted the index of the block corresponding to the original image, \(1\leq k\leq\frac{M}{B}\times\frac{M}{B}\). With the partition of \(\mathbf{X}\) into \(\mathbf{X}_{k}\), we performed the below operations block by block to turn compressed sensing into BCS [10]. * The _sparsity_ principle implies the information rate in data compression. In BCS this was expected to use a much smaller sampling rate than conventionally required, and it can be represented via \(\Psi\), \(\Psi\in\mathbb{C}^{B\times B^{2}}\), where \(C\) denotes the complex number in the \(B^{2}\times B^{2}\) matrix. \(\Psi\) was the basis to reach sparsity with a \(k\)-sparse coefficient vector \(\mathbf{X}_{k}\), \(\mathbf{X}_{k}\in\mathbb{C}^{B^{2}\times 1}\), with the condition that \[\mathbf{f}_{k}=\Psi\mathbf{X}_{k}\] (1) where \(\mathbf{f}_{k}\) denotes the reconstruction corresponding to the original signal, \(\mathbf{X}_{k}\). * The _incoherence_ principle extends the duality between time and frequency. The measurement basis \(\Phi\), \(\Phi\in\mathbb{C}^{m\times B^{2}}\), which acts like noiselet, was employed to sense the signal \(\mathbf{f}_{k}\), with the condition that \[\mathbf{Y}_{k}=\Phi\mathbf{f}_{k}\] (2) where \(\mathbf{Y}_{k}\) denotes the measurement vector, as depicted in Figure 1a. We noted that Equation (2) was an underdetermined system. For the reconstruction from BCS signals at the decoder, several methods can be employed. Considering Equations (1) and (2), by minimizing the L1-norm of \(\mathbf{X}_{k}\), i.e., \(\min\lVert\mathbf{X}_{k}\rVert_{1}\) subject to \(\mathbf{Y}_{k}=\Phi\Psi\mathbf{X}_{k}\), compressed sensing guarantees perfect recovery with a probability close to 1.0. Both the 'equality constraint' and the 'inequality quadratic constraint' are widely employed conditions for minimization [20; 21]. The equality constraint means that  
\[\min\lVert\mathbf{X}_{k}\rVert_{1}\text{ subject to }\mathbf{Y}_{k}=\Phi\Psi \mathbf{X}_{k} \tag{3}\]  
where, \(\lVert\cdot\rVert_{1}\) denotes the L1-norm. The inequality quadratic constraint implies that  
\[\lVert\min\mathbf{X}_{k}\rVert_{1}\text{ subject to }\lVert\Phi\Psi\mathbf{X}_{k }-\mathbf{Y}_{k}\rVert_{2}\leq\varepsilon \tag{4}\]where, \(\|\cdot\|_{2}\) denotes the L2-norm. Thus, the inequality quadratic constraint leaves some tolerance for minimization. When an image **X** is represented by a BCS scheme, it focuses on the local characteristics of the image. **X** was divided into a set of blocks, \(\textbf{X}_{k}\). Therefore, it might be inefficient to assign the same number of measurement dimension to each sampled vector corresponding to the different image block. Due to the local characteristics, one block in the image had significantly different sparsity than the other. With adaptive sampling in BCS, the entropy of a block may be used to evaluate the included information. It was expected to have better reconstruction quality with error-free transmission from adaptive sampling. Regarding adaptive sampling (AS) [18], the normalized DCT coefficient \(c_{k}^{\prime}\) can be calculated by  
\[c_{k}^{\prime}=\frac{c_{k}-c_{k,\min}}{c_{k,\max}-c_{k,\min}} \tag{5}\]  
where \(c_{k,\max}\) and \(c_{k,\min}\) denote the maximal and minimal DCT coefficients in block \(\textbf{X}_{k}\).