An Extended Reweighted ℓ1 Minimization Algorithm for Image Restoration

This paper proposes an effective extended reweighted ℓ1 minimization algorithm (ERMA) to solve the basis pursuit problem minu∈Rnu1:Au=f in compressed sensing, where A∈Rm×n, m≪n. The fast algorithm is based on linearized Bregman iteration with soft thresholding operator and generalized inverse iteration. At the same time, it also combines the iterative reweighted strategy that is used to solve minu∈Rnupp:Au=f problem, with the weight ωiu,p=ε+ui2p/2−1. Numerical experiments show that this l1 minimization persistently performs better than other methods. Especially when p=0, the restored signal by the algorithm has the highest signal to noise ratio. Additionally, this approach has no effect on workload or calculation time when matrix A is ill-conditioned.

Application of compressed sensing for image compression based on optimized Toeplitz sensing matrices

In compressed sensing, the Toeplitz sensing matrices are generated by randomly drawn entries and further optimizes them with suitable optimization methods. However, during an optimization process, state-of-the-art optimization methods tend to lose control over the structure of measurement matrices. In this paper, we proposed the novel approach for optimization of Toeplitz sensing matrices based on evolutionary algorithms such as Genetic Algorithm (GA), Simulated Annealing (SA), and Particle Swarm Optimization (PSO) for compression of an image signal. Furthermore, we investigated the performance of Basis Pursuit (BP) and Orthogonal Matching Pursuit (OMP) algorithms for the reconstruction of the images. The proposed optimized Toeplitz sensing matrices based on evolutionary algorithms such as GA, SA, and PSO exhibit a significant reduction in the mutual coherence (μ) and thus improved the recovery performance of 2D images compared to state-of-the-art non-optimized Toeplitz sensing matrices. The result reveals that the optimized Toeplitz sensing matrices with Basis Pursuit (BP) achieved more accurate results with a robust and uniform reconstruction guarantee compared to the OMP algorithm. However, BP shows the slow reconstruction performance of the image signal. On the other hand, an optimized Toeplitz sensing matrix with OMP shows a fast reconstruction guarantee, but at the cost of a reduction in the PSNR. Furthermore, the proposed approach retains the structure of Toeplitz sensing matrices and improves the image recovery performance of compressed sensing. Finally, the experimental results validate the effectiveness of the proposed method based on evolutionary algorithms for image compression.

A Compressed Sensing Recovery Algorithm Based on Support Set Selection

The theory of compressed sensing (CS) has shown tremendous potential in many fields, especially in the signal processing area, due to its utility in recovering unknown signals with far lower sampling rates than the Nyquist frequency. In this paper, we present a novel, optimized recovery algorithm named supp-BPDN. The proposed algorithm executes a step of selecting and recording the support set of original signals before using the traditional recovery algorithm mostly used in signal processing called basis pursuit denoising (BPDN). We proved mathematically that even in a noise-affected CS system, the probability of selecting the support set of signals still approaches 1, which means supp-BPDN can maintain good performance in systems in which noise exists. Recovery results are demonstrated to verify the effectiveness and superiority of supp-BPDN. Besides, we set up a photonic-enabled CS system realizing the reconstruction of a two-tone signal with a peak frequency of 350 MHz through a 200 MHz analog-to-digital converter (ADC) and a signal with a peak frequency of 1 GHz by a 500 MHz ADC. Similarly, supp-BPDN showed better reconstruction results than BPDN.