## 1 Introduction  
Compressed sensing, also known as compressive sampling or sparse sampling, is a technique for finding sparse solutions of underdetermined linear systems. In 2006, [PERSON] et al. proposed the concept of compressed sensing (CS), which provides a new idea for signal sampling theory [1; 2; 3]. CS theory has attracted the attention of many experts and scholars and has been widely used in many fields such as signal processing [4], radar imaging [5], wireless sensor networks [6], blind source separation [7] and so on. According to compressed sensing theory, if the signal is sparse on a dictionary basis, it can be sampled and compressed simultaneously with an observation matrix that is not related to the sparse dictionary to project the high-dimensional signal in the low-dimensional space. Then, the sparse representation vector of the signal can be obtained by solving the \(l_{0}\) norm optimization problem, and the original signal will be reconstructed. Unlike the traditional Nyquist sampling theory, compressed sensing theory requires that the signal must be compressible or sparse in a transform space. In other words, the new compressed sensing theory breaks through the limitation of the traditional Nyquist sampling theorem in signal acquisition, innovatively realizes the simultaneous sampling and compression of signals, and can accurately recover the original signals by random sampling compared with the traditional method. Accurate signal reconstruction refers to the accurate recovery of the original high-dimensional signal from a small amount of data. The essence of the reconstruction process is to find the optimal solution for a set of undetermined equations. Therefore, it can be seen that the reconstruction algorithm plays an important role in all of compressed sensing theory. The reasonable design and selection of the method is directly related to the accuracy of reconstructed data. The higher the reconstructed accuracy, the greater the practicality of the theory. Compressed sensing theory, with its unique advantages, has made rapid development and at the same time has also set off a research boom. At present, many universities and research institutes have joined in the research of compressed sensing theory and have achieved many excellent results. Research results for signal reconstruction algorithms for compressed sensing are emerging in an endless stream. There are many kinds of compressed sensing reconstruction algorithms, but the most common variants are greedy and convex relaxation algorithms. Among greedy algorithms, the earliest matching pursuit algorithm used for reconstruction is the matching pursuit algorithm (MP) [8]. Additionally, there are many variants of it, such as the orthogonal matching pursuit algorithm (OMP) [9], subspace pursuit algorithm (SP) [10], compressed sampling matching pursuit algorithm (CoSaMP) [11], sparsity adaptive matching pursuit algorithm (SAMP) [12], multi-orthogonal least squares algorithm (MOLS) [13] and so on. There are also many convex relaxation optimization algorithms, such as the basis pursuit algorithm (BP) [14], interior point algorithm [15], gradient projection for sparse reconstruction algorithm (GPSR) [16], fast iterative threshold shrinkage algorithm (FISTA) [17] and so on. In 2009, [PERSON] proposed a smooth SL0 [18] reconstruction algorithm that directly minimizes the \(l_{0}\) norm. Compared with the previous convex optimization method, in this algorithm, the discrete \(l_{0}\) norm is replaced by the continuous Gaussian function with parameters to construct a new objective function, then the steepest descent method is used to minimize the approximate continuous function, and finally, the minimal solution is projected into the solution space to satisfy the constraints. From the implementation process of the algorithm, the algorithm combines the rapidity of the greedy algorithm and the accuracy of convex optimization and is a better reconstruction algorithm. There are many improved algorithms based on the SL0 algorithm.