# Error Resilience for Block Compressed Sensing with Multiple-Channel Transmission
## 4 Simulation Results  
In our simulations we provided three sets of experiments based on three test images. The first was cameraman grey-level test image with a size of \(256\times 256\). The second was the ducks color image, taken by the authors, with a size of \(1024\times 1024\). The third was the Pasadena-houses color image with a size of \(1760\times 1168\)[22]. These three images were employed in the experiments. Here we start the first set of experiments with the test image cameraman. In Figure 2, we present the use of entropy for adaptive sampling. Considering the practical implementation of adaptive sampling, we applied quantization to the entropy values with a step size of 0.1. Figure 2a presents the original test image, cameraman. The relationship between the number of blocks and entropy is depicted in Figure 2b. Smaller entropy values imply smoother blocks. Figure 3 presents the error-free transmission of BCS for cameraman. Considering the practical applications, we chose \(B=8\) in BCS in Equation (1). The measurement rate was set to \(\frac{12}{64}=0.1875\), meaning that \(m=12\) compressed sensing coefficients were selected per block on the average. The reconstructed images were assessed with the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM) [23]. Figure 3a shows only the reconstruction for equality constraint (EQ). Because adaptive sampling was not applied, it is implied that every block was reconstructed with 12 BCS coefficients. Figure 3b presents the reconstruction for equality constraint (EQ) with adaptive sampling (AS) [10]. Here we use the abbreviation EQ + AS for the results in Figure 3b. Because adaptive sampling was applied, the number of BCS coefficients varied from one block to another. Blocks with larger entropies were designated a higher number of BCS coefficients, and vice versa. The average number of BCS coefficients for all the blocks was 12. Via adaptive sampling, we easily observed enhanced performances. Figure 3c,d led to the results for quadratic constraint (QC) and quadratic constraint with  
Figure 2: The test image cameraman: (**a**) original image with a size of \(256\times 256\); and (**b**) the entropy distribution. adaptive sampling (QC + AS), respectively. Again, with adaptive sampling, better performances were observed. We also compared the results of Figure 3a,c. With the quadratic constraint in Equation (4) it performed slightly better than its counterpart, equality constraint, in Equation (3). Similar phenomena can also be found by comparing Figure 3b,d. In Figure 4 we present the lossy transmission for block compressed sensing with adaptive sampling. In Figure 4a,b, when the BCS coefficients were transmitted over the two independent and lossy channels in Figure 1b, we set \(p_{e,1}=p_{e,2}=0.1\). Here we employed the equality constraint (EQ) for reconstruction. Constructing the received BCS coefficients directly led to the result in Figure 4a, meaning that some protection may be required. In Figure 4b, we applied multiple description coding from Equation (9) for protection. The parameters were chosen to be \(\theta_{1}=\frac{\pi}{3}\), \(\theta_{2}=\frac{-\pi}{4}\), and \(r_{2}=3\), and this led to \(r_{1}=\frac{1}{r_{2}\sin(\theta_{1}-\theta_{2})}=0.3451\), from calculations using Equation (9). With the compensation techniques of MDC from Equation (9), we observed that the reconstructed quality was greatly improved. In addition, for the evaluation of reconstruction under severely lost channels we set \(p_{e,1}=p_{e,2}=0.5\), with the results shown in Figure 4c,d. Severe degradation was easily visible in Figure 4c, as was the improvement of reconstructed quality after protection in Figure 4d. Figure 3: Reconstruction of BCS coefficients: (**a**) equality constraint only (EQ); (**b**) equality constraint and adaptive sampling (EQ + AS); (**c**) quadratic constraint only (QC); and (**d**) quadratic constraint and adaptive sampling (QC + AS).