## 3 The Proposed Algorithm and Its Applications
### Description of FRSVssMP  
After description of the initial estimation approach for sparsity level, we draw a sketch of FRSVssMP. After employing initial sparsity level estimation to establish initialization for iterations, it mainly consists of two components, iterations and filtering. In iterations part, leveraging the strategies of regularization and sparsity variable step-size adaptive, FRSVssMP iteratively identifies the support set of the sparse signal by appropriately correlating the measurement or residual with the columns of the sensing matrix. More specifically, during each iteration, residuals acquired in contiguous phases are compared and a relative threshold is applied for adjusting step size. Through several rapid approximations with a large step size and precise approximations with a small step size, a superior support set of the sparsity signal is attained. Then, the sparse signal can be obtained based upon the confirmed support set by least square method. In contrast with existing methods, sparsity variable step-size adaptive performed in our algorithm considers both residuals and the relative threshold. Discriminant criterions are designed according to comparison of old and new residuals. If the relative error obtained by old and new residuals in contiguous phases is larger than the relative threshold given, cardinality of the estimated support set should be increased by an integral step size to approximate the true signal sparsity level. This means that, the algorithm can accomplish a rapid reconstruction by reducing time with a large step size. If the relative error obtained is smaller than the relative threshold given, cardinality of the estimated support set should be increased by less than an integral step size. In other words, the algorithm can implement a precise reconstruction by approximating the true signal sparsity with a small step size. Furthermore, in filtering part, the obtained result is filtered to exclude atoms that exerts minimal influence on reconstruction until the residual cannot be further diminished. Therefore, we can further prevent incorrect atoms from degrading the performance and acquire a more appropriate sparsity signal. According to the process described above, we present the pseudocode of the proposed FRSVssMP algorithm in details with the same symbols in Section 2. Let \(S\) be the initial step size, \(L\) be the estimated sparsity level, \(k_{0}\) be the estimated initial sparsity level, \(\mathbf{r_{\mathrm{t}}}\) be the residual, and \(t\) be number of iterations. \(\mathcal{O}\) represents an empty set, and \(\Lambda_{t}\) is nonzero indices (number of columns) set with \(L\) elements for the \(t\)-th iteration. \(\mathbf{a_{\mathrm{j}}}\) denotes the \(j\)-th column of \(\mathbf{A}\). Put \(\mathbf{A_{\mathrm{t}}}\) to be the set of columns of \(\mathbf{A}\) selected according to the indices set (the number of columns is \(L_{t}\)). The regularization function can be denoted as \(regularize(\mathbf{u},l)\), where \(\mathbf{u}\) is the set of atoms selected, and \(l\) is the number of candidate atoms in \(\mathbf{u}\) to be regularized. Function \(max(\mathbf{v},i)\) is defined to select the largest \(i\) elements in \(\mathbf{v}\). Steps of the proposed Algorithm 1 are illustrated as follows.