## 4 CReSL0 Algorithm
### Selection of Optimization Method  
In this paper, the combination of the steepest descent method and Newton method is used to replace the simple steepest descent method in ReSL0. From Figure 1, we can see that the steepness of the Gauss function and CIPF function is almost the same when they are very close to zero; that is, the final stage of the optimization process. In addition, the form of the Gauss function \(y_{\sigma}(x)=\exp(-x^{2}/2\sigma^{2})\) will be much simpler. Thus, we select the CIPF function to approximate zero norm in the early steepest descent method part (that is, only the first derivative is needed). We choose the Gauss function to approximate zero norm in the latter Newton method part (that is, the second derivative is needed). Only in this way can we attain a more accurate reconstruction in less time. In the next, let  
\[F_{\sigma}(x_{i})=\ \underset{\sigma\to 0}{\lim}\underset{i=1}{\sum}f_{ \sigma}(x_{i}) \tag{14}\]  
\[Y_{\sigma}(x_{i})=\ \underset{\sigma\to 0}{\lim}\underset{i=1}{\sum}y_{ \sigma}(x_{i}) \tag{15}\]  
Then, the gradients of Equations (14) and (15) are obtained as follows, respectively:  
\[\
abla F_{\sigma}(x)=[\frac{\partial f_{\sigma}(x_{1})}{\partial x_{1}}, \ldots,\frac{\partial f_{\sigma}(x_{N})}{\partial x_{N}}]^{T}, \tag{16}\]  
where \(\frac{\partial f_{\sigma}(x_{i})}{\partial x_{i}}=\frac{2\alpha\sigma^{2}x_{i }}{(ax_{i}^{2}+\sigma^{2})^{2}}\). \[\
abla Y_{\sigma}(x)=[\frac{\partial y_{\sigma}(x_{1})}{\partial x_{1}}, \ldots,\frac{\partial y_{\sigma}(x_{N})}{\partial x_{N}}]^{T}, \tag{17}\]  
where \(\frac{\partial y_{\sigma}(x_{i})}{\partial x_{i}}=-\frac{s_{i}}{\sigma^{2}}e^ {-\pi x_{i}^{2}/2\sigma^{2}}\). Further, we can get the Hessen matrix of Equation (15) as follows:  
\[\
abla^{2}Y_{\sigma}(x)=\left[ \begin{array}{cccc}\frac{\partial^{2}f_{ \sigma}(x_{1})}{\partial x_{1}{}^{2}}&0&\ldots&0\\ 0&\frac{\partial^{2}f_{\sigma}(x_{2})}{\partial x_{2}{}^{2}}&\ldots&0\\ \ldots&\ldots&&\ldots\\ 0&0&\ldots&\frac{\partial^{2}f_{\sigma}(x_{N})}{\partial x_{N}{}^{2}}\end{array} \right] \tag{18}\]where \(\frac{\partial^{2}f_{g}(x_{i})}{\partial x^{2}}=-e^{-\sigma_{i}^{2}/2\sigma^{2}}( \frac{\sigma^{2}-\hat{\sigma}^{2}}{\sigma^{4}})\). From the above Hessen matrix, we see that it cannot be guaranteed that all the elements of the diagonal elements are greater than 0. If the Hessen matrix is not more than 0, it cannot be guaranteed that the direction of the Newton method is always the direction approaching the optimal solution. Thus, we need to further modify the Hessen matrix to get a modified Newton direction to meet the requirement of the positive definite matrix. The modified method replaces the Hessen matrix with the following matrix:  
\[G=\
abla^{2}Y_{\sigma}(x)+p_{i}I \tag{19}\]  
where the modified factor chosen in this paper is \(p_{i}=e^{-\sigma_{i}^{2}/2\sigma^{2}}\frac{2\sigma_{i}^{2}}{\sigma^{4}}\). In this way, we can get the optimization iteration formulas of the front and back parts of the optimization. The optimization iteration part of the steepest descent method in this paper is as follows:  
\[x\gets x+\mu\
abla F_{\sigma}(x) \tag{20}\]  
The optimization iteration part of the Newton method in this paper is as follows:  
\[x\gets x-G^{-1}\Delta Y_{\sigma}(x) \tag{21}\]  
In summary, we can get the pseudocode of the CReSL0 algorithm as shown in Algorithm 2. ```
Initialization: (1) Set \(L\), \(\mu>0\) and \(\hat{\sigma}^{0}=A^{H}(AA^{H})^{-1}y\). (2) Set \(\sigma_{1,\dots,\sigma_{J}}\). ```
While \(\sigma_{j}>\sigma_{J}\)  
\(