Compressed sensing (CS) theory has attracted widespread attention in recent years and has been widely used in signal and image processing, such as underdetermined blind source separation (UBSS), magnetic resonance imaging (MRI), etc. As the main link of CS, the goal of sparse signal reconstruction is how to recover accurately and effectively the original signal from an underdetermined linear system of equations (ULSE). For this problem, we propose a new algorithm called the weighted regularized smoothed $L_0$ -norm minimization algorithm (WReSL0). Under the framework of this algorithm, we have done three things: (1) proposed a new smoothed function called the compound inverse proportional function (CIPF); (2) proposed a new weighted function; and (3) a new regularization form is derived and constructed. In this algorithm, the weighted function and the new smoothed function are combined as the sparsity-promoting object, and a new regularization form is derived and constructed to enhance de-noising performance. Performance simulation experiments on both the real signal and real images show that the proposed WReSL0 algorithm outperforms other popular approaches, such as SL0, BPDN, NSL0, and L ${}_p$ -RLSand achieves better performances when it is used for UBSS.

This paper addresses the problems of sparse signal and image recovery using compressive sensing (CS), especially in the case of Gaussian noise. The main contribution of this paper is the proposal of the regularization re-weighted Composite Sine function smoothed L ₀-norm minimization (RRCSFSL0) algorithm where the Composite Sine function (CSF), the iteratively re-weighted scheme and the regularization mechanism represent the core of an approach to the solution of the problem. Compared with other state-of-the-art functions, the CSF we proposed can better approximate the L ₀-norm and improve the reconstruction accuracy, the new re-weighted scheme we adopted can promote sparsity and speed up convergence. Moreover, the use of the regularization mechanism makes the RRCSFSL0 algorithm more robust against noise. The performance of the proposed algorithm is verified via numerical experiments in the noise environment. Furthermore, experiments and comparisons demonstrate the superiority of the RRCSFSL0 algorithm in image restoration.

Sparse-signal recovery in noisy conditions is a problem that can be solved with current compressive-sensing (CS) technology. Although current algorithms based on $L_1$ regularization can solve this problem, the $L_1$ regularization mechanism cannot promote signal sparsity under noisy conditions, resulting in low recovery accuracy. Based on this, we propose a regularized reweighted composite trigonometric smoothed $L_0$ -norm minimization (RRCTSL0) algorithm in this paper. The main contributions of this paper are as follows: (1) a new smoothed symmetric composite trigonometric (CT) function is proposed to fit the $L_0$ -norm; (2) a new reweighted function is proposed; and (3) a new $L_0$ regularization objective function framework is constructed based on the idea of $Tikhonov$ regularization. In the new objective function framework, Contributions (1) and (2) are combined as sparsity regularization terms, and errors as deviation terms. Furthermore, the conjugate-gradient (CG) method is used to optimize the objective function, so as to achieve accurate recovery of sparse signal and image under noisy conditions. The numerical experiments on both the simulated and real data verify that the proposed algorithm is superior to other state-of-the-art algorithms, and achieves advanced performance under noisy conditions.

Compressed sensing (CS) relies on the sparse priorin posed on the signal to solve the ill-posed recovery problem in an under-determined linear system (ULS). Motivated by the theory, this paper proposes a new algorithm called regularized re-weighted inverse trigonometric smoothed function approximating L 0-norm minimization (RRITSL0) algorithm, where the inverse trigonometric (IT) function, iteratively re-weighted scheme and regularization mechanism constitute the core of the proposed RRITSL0 algorithm. Compared with other state-of-the-art functions, our proposed IT function cluster can better approximate the L 0-norm, thus improving the reconstruction accuracy. And the new re-weighted scheme we adopted can promote sparsity and speed up convergence. Moreover, the regularization mechanism makes the RRITSL0 algorithm more robust against noise. The performance of the proposed algorithm is verified via numerical experiments with additive noise. Furthermore, the experiments prove the superiority of the RRITSL0 algorithm in magnetic resonance (MR) image recovery.

Compressive sensing has gained great attention due to its effectiveness in solving linear inverse problems. However, how to further improve the accuracy of compressed image inversion while ensuring or even accelerating the speed is still a major challenge. To tackle this problem, we present a novel multi-branch denoising-based approximate message passing algorithm via deep neural network, dubbed MB-DAMPNet. It mainly consists of three components, i.e. sampling subnet, initial recovery subnet, and deep recovery subnet, which are optimized jointly. The sampling subnet is constructed to obtain the compressed measurements, the initial recovery subnet is employed to generate the reconstructed image by inverse transformation, while the deep recovery subnet is designed to refine the reconstructed results obtained by the former, so as to improve the image accuracy. Moreover, the matrix multiplication in the network is all designed as matrix convolution which can be learned automatically, so that the input image of the MB-DAMPNet can be of different scales, which improves the flexibility and applicability of the network. In addition, all parameters in the network are learned end-to-end instead of fixed or hand-crafted. The numerical results validate that our method significantly outperforms other state-of-the-art methods in image reconstruction accuracy.