## 2 Theoretical Basis and Related Work
### Greedy Reconstruction Algorithms  
Various iterative greedy algorithms have been developed for reconstruction with the basic idea that signal support entries are iteratively obtained by correlating the measurement vector with columns of the sensing matrix. According to whether the signal sparsity level is known, greedy algorithms can be classified into two categories. The first category requires signal sparsity level as prior information to control iterations, such as OMP [14]. While the second one does not require any prior knowledge on the sparsity level and can perform reconstruction through adjustment of step sizes, such as SAMP [19] and AStMP [20]. [PERSON] first proposed matching pursuit (MP), i.e., one atom can be obtained from the support set in each iteration, but there is no guarantee that the residual is orthogonal to atoms. Orthogonal matching pursuit (OMP) does not repeat selection of the same atom, but selects only one atom in each iteration, which results in low reconstruction efficiency [14]. To select multiple atoms on one iteration, stagewise orthogonal matching pursuit (StOMP) with threshold principle [15] and regularized orthogonal matching pursuit (ROMP) with regularization strategy [16] were proposed. The computational complexity of the two methods is considerably lower than that of BP. However, these methods suffer from large reconstruction errors, and excellent reconstruction performance can be acquired only when there are plentiful measurements. Therefore, compressive sampling matching pursuit (CoSaMP) and subspace pursuit (SP) including pruning were introduced [17]. The expanded subspace pursuit achieves blind sparse reconstruction by incorporating a simple backtracking technique based on SP [18]. The innovation of those approaches lies in pruning selected atoms in support set to exclude the wrong atoms. Unfortunately, a constant selection for all iterations easily leads to redundant atoms required. In addition, those aforementioned methods depend heavily on prior information regarding the sparsity level. Generally speaking, signal sparsity level is unknown. Hence algorithms in the second category are widely applicable, and have been proved to be effective. These algorithms divide reconstruction into several phases. In each phase, cardinality \(W\) of the signal support set \(F\) does not change, and multiple iterations are conducted. When \(F\) cannot meet a terminating criterion, we can increase a step size \(S\) to enter next phase. As \(W\) increases, actual sparsity level \(K\) can be obtained approximatively without prior information of the signal sparsity. Researchers proposed sparsity adaptive matching pursuit (SAMP), which does not rely on signal sparsity and achieves reconstruction through gradual approximation of the actual sparsity level by multiplying fixed step size [19]. However, the fixed step size results in either low reconstruction efficiency or low reconstruction accuracy. Further, several adaptive matching pursuit algorithms were proposed on the basis of SAMP. Adaptive step-size matching pursuit (AStMP) precisely estimates signal sparsity level by combining subspace pursuit with variable step-size in each iteration [20]. Adaptive regularized compressive sampling matching pursuit (ARCoSaMP) chooses the support set adaptively and exploits the regularization process to achieve the accuracy of reconstruction [21]. An improved reconstruction method utilizes the fuzzy threshold method and a double threshold iterative method to improve the precision [22]. A modified sparsity adaptive matching pursuit (MSAMP) achieves adaptive sparsity realization and index set backtracking optimization [23]. More recently, a novel fast global matching pursuit (FGMP) solves the \(l_{0}\) minimization with global matching pursuit strategies [24]. To address the issue of step size selection, a constrained backtracking matching pursuit (CBMP) uses two kinds of composite strategies to estimate the true support set [25]. However, those algorithms have some obvious disadvantages.