## 4 CReSL0 Algorithm
### Selection of Optimization Method
From Figure 2b, the least time-consuming reconstruction algorithm is the ReSL0 algorithm, but the proposed CReSL0 algorithm still takes less time than the WReSL0 algorithm. Additionally, the reconstruction time of the three algorithms is not very different. Secondly, the reconstruction performance of different signal lengths (150-500, where the interval is 50) is studied in the presence of noise. The reconstruction signal still chooses the one-dimensional Gauss signal. For each signal length, the measured value and sparsity are one-half and one-tenth of the signal length, respectively. Each experiment will also measure 100 times, and the final average PSNR will be taken as the final PSNR. The final experimental PSNR values are shown in Figure 3a, and the average reconstruction times of the corresponding algorithms are shown in Figure 3b. Figure 2: (**a**) Peak reconstruction signal-to-noise ratio (PSNR) values of each algorithm for different sparsities; (**b**) running time of each algorithm for different sparsities. As can be seen clearly from Figure 3a, the reconstruction PSNR of the three algorithms for different length signals tends to be stable. This shows that the reconstructed PSNR of these three algorithms is basically not affected by the length, and the PSNR of CReSL0 algorithm is still the highest for the reconstructed signals with different lengths. In Figure 3b, we see that the reconstruction time of these three algorithms increases with different length signals. However, the reconstruction time of the WReSL0 algorithm increases the fastest, followed by the CReSL0 algorithm and ReSL0 algorithm. ### Reconstruction of Two-Dimensional Image Signal

Since the image itself is not sparse, the discrete wavelet transform (DWT) basis is selected as \(\Psi\) in Equation (1) to sparse the image. In addition, the image is a two-dimensional signal, which needs to be undersampled from two dimensions. Thus, Equation (1) is termed as below for the image reconstruction:

\[x=DWT*s_{image}*DWT^{H} \tag{24}\]

where \(DWT^{H}\) is the conjugated transposed matrix of \(DWT\). Next, the two-dimensional reconstruction of the image here treats each column of the image as a one-dimensional signal and then carries out further image reconstruction. Firstly, five images in Figure 4 are selected as experimental reconstructed images. The dimension of the images is 256*256. The selected measurement matrix is still the Gauss random matrix, and the dimension of the measurement matrix is 190*256. The reconstructed PSNR of each algorithm and the running time are shown in Table 1.

From Table 1, it can be clearly seen that the PSNR of the CReSL0 algorithm is superior to the PSNR obtained by the other two algorithms for all different reconstructed images. CReSL0 takes more time than the ReSL0 algorithm, but less time than the WReSL0 algorithm. For different images, the PSNR obtained by each algorithm and the running time of each algorithm vary slightly, and the overall

Figure 4: Selected reconstruction images. Figure 3: (**a**) PSNR of each algorithm for different signal lengths; (**b**) running time of each algorithm for different signal lengths. As shown in Figure 5, with the increment of the compression ratio, the reconstruction effect of each algorithm improves. Visually, we cannot judge which algorithm has the better reconstruction effect. However, from the data in Table 2, we can clearly see that the PNSR of the CReSL0 algorithm proposed in this paper is the highest for each compression ratio. The reconstruction of the CReSL0 algorithm takes a little more time than that of the ReSL0 algorithm but a little less time than that of the WReSL0 algorithm. The ratio of PSNR and time is included, and the ratio of PSNR and time of the CReSL0 algorithm is smaller than that of ReSL0 but bigger than that of WReSL0. Generally speaking, the performance of the algorithm proposed in this paper is the best. ## 6 Conclusions

This paper makes improvements on the shortcomings of the ReSL0 algorithm.