## 4 CReSL0 Algorithm
### Selection of Optimization Method
\begin{array}{l}\text{(1)}\quad\text{Let }\sigma=\sigma_{j}\text{:}\\ \text{(2)}\quad\text{Initialization:}\ x=\hat{\sigma}^{j-1}\text{:}\\ \text{-for }k=1,\dots,\beta,\dots,L\\ \text{If }k\leq\beta\text{ (a) }x\gets x+\mu\
abla F_{\sigma}(x)\text{.}\\ \text{(b) }x\gets x-A^{H}(AA^{H}I_{\text{n}}+\lambda^{-1}I_{\text{m}})^{-1}(Ax-y) \text{.}\\ \text{Else }\quad\text{(c) }x\gets x-G^{-1}\Delta Y_{\sigma}(x)\text{.}\\ \text{(d) }x\gets x-A^{H}(AA^{H}I_{\text{n}}+\lambda^{-1}I_{\text{m}})^{-1}(Ax-y) \text{.}\\ \text{(3)}\quad\text{Set }\hat{\sigma}^{j}=x\text{.}\\ \text{The estimated value is }\hat{x}=\hat{\sigma}^{j}\text{.}\end{array} \)
```

**Algorithm 2**The pseudocode of CReSL0 algorithm. ### Selection of Parameters

For the selection of \(\mu\), from Algorithm 2, we can obtain the steepest descent method part \(x\gets x+\mu\
abla F_{\sigma}(x)\) of the CReSL0 algorithm; that is, \(x\gets x+\mu 2\alpha\sigma^{2}x_{i}/\left(ax_{i}^{2}+\sigma^{2}\right)^{2}\). According to Equation (11), we know that \(x\gets x+\mu 2\alpha\sigma^{2}x_{i}/\left(ax_{i}^{2}+\sigma^{2}\right)^{2}\) must be approaching zero when \(\sigma\to 0\), so we can obtain \(\mu=\sigma^{2}/2\alpha\). For the selection of the \(\sigma\) generation mode, we do not utilize \(\rho(0<\rho<1)\) times the decreasing mode, but select the generation mode \(\sigma_{t}=\sigma_{1}\exp(-\log(\sigma_{1}/\sigma_{T})(t-1)/(T-1))\) for \(t=1,2,\dots,T\) in [15]. ## 5 Simulation and Results

In order to verify the performance of the proposed algorithm, we compare the popular WReSL0 [19] and ReSL0 [5] algorithms with the proposed CReSL0 algorithm in reconstructing a one-dimensional signal and two-dimensional image by the MATLAB simulation platform. The MATLAB simulation platform runs on a 64-bit Intel i5-4210 [EMAIL_ADDRESS] GHz processor and Windows 8 system. The noise added in each experiment is Gauss white noise, which is generated by \(i*N(0,1)\). \(i\) in all the following experiments is set as 0.01. The peak reconstruction signal-to-noise ratio (PSNR) and running time of the reconstruction algorithms are selected as the performance indicators. The running time is obtained by the tic and toc functions of MATLAB. The PSNR of the one-dimensional signal and two-dimensional signal is defined by the following two formulas, respectively:

\[PSNR=20\log(norm(x)/norm(x-\hat{x})) \tag{22}\]

\[PSNR=10\log(255^{2}mn/\sum_{i}\sum_{j}\left(\hat{x}(i,j)-x(i,j)\right)) \tag{23}\]

Each algorithm chooses the parameter settings when its performance is the best. For WReSL0, the iteration time \(T=30\), step length \(L=5\), and regularization parameter \(\lambda=0.1\); for ReSL0, the iteration step \(L=5\), constant \(\mu=2.5\), regular parameter \(\lambda=1.5\); for CReSL0, the iteration time \(T=30\), step length \(L=5\), constant \(\mu=2.5\), regular parameter \(\lambda=1.5\), \(\beta=3\). \(\sigma_{J}=0.01\) is set for all algorithms. ### Reconstruction of One-Dimensional Gauss Signal

Firstly, the one-dimensional Gauss signal is chosen as the signal to be reconstructed, and the Gauss random matrix is used as the measurement matrix. The aim of this experiment is to set the signal length to 256 and the measurement number to 128 to study the reconstruction performance of each algorithm for different sparsities in the presence of noise. For each sparsity value (15-85, where the interval is 5), we repeat the process 100 times and take the final average PSNR as the final PSNR. The final experimental PSNR values are shown in Figure 2a, and the average reconstruction time of the corresponding algorithms is shown in Figure 2b. As shown clearly in Figure 2a, the PSNR of CReSL0 algorithm is much higher than that of the other two algorithms for each sparsity value. Especially when the sparsity is 30, the PSNR of the CReSL0 algorithm proposed in this paper can reach 34, while the other two algorithms can only get to 31 and 18, respectively.