A Novel Optimization Method for Bipolar Chaotic Toeplitz Measurement Matrix in Compressed Sensing

Rui Zhang,Qiang Wang,Chen Meng,Cheng Wang,Mu Zhou

doi:10.1155/2021/4024737

Abstract

In this paper, a bipolar chaotic Toeplitz measurement matrix optimization algorithm for alternating optimization is presented. The construction of measurement matrices is one of the key techniques for compressive sensing from theory to engineering applications. Recent studies have shown that bipolar chaotic Toeplitz matrices, constructed by combining the intrinsic determinism of bipolar chaotic sequences with the advantages of Toeplitz matrices, have significant advantages over other measurement matrices in terms of memory overhead, computational complexity, and hard implementation. However, problems such as strong correlation and large interdependence coefficients between measurement matrices and sparse dictionaries may still exist in practical applications. To address this problem, we propose a new bipolar chaotic Toeplitz measurement matrix alternating optimization algorithm. Firstly, by introducing the structure matrix, the optimization problem of the measurement matrix is transformed into the optimization problem of the generating sequence, thus ensuring that the optimization process does not destroy the structural properties of the matrix; then, constraints are added to the values of the generating sequence during the optimization process, so that the optimized measurement matrix still maintains the bipolar properties. Finally, the effectiveness of the optimization algorithm in this paper is verified by simulation experiments. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix can effectively reduce the reconstruction error and improve the reconstruction probability.

Highlights

Compressed sensing (CS) [1, 2] is a new framework for signal sampling
Based on the above two factors, the bipolar chaotic sequence is used to construct Toeplitz matrix as a measurement matrix for compressed sensing—called the bipolar chaotic Toeplitz measurement matrix—in the literature [10]. This measurement matrix is simple to generate and has few free elements, which greatly reduces the difficulty of hardware implementation and, at the same time, supports fast algorithms that can solve numerous problems related to convolutional operations
The introduction of the structure matrix Φj ensures that the structural properties of the measurement matrix are not destroyed during the alternating optimization process, but it is not guaranteed that the elements of the optimized matrix maintain their original bipolar properties

Summary

Introduction

Compressed sensing (CS) [1, 2] is a new framework for signal sampling. It enables to sample sparse signals at a sampling rate much lower than Nyquist’s and to achieve accurate recovery of the original signal with high probability. Based on the above two factors, the bipolar chaotic sequence is used to construct Toeplitz matrix as a measurement matrix for compressed sensing—called the bipolar chaotic Toeplitz measurement matrix—in the literature [10] This measurement matrix is simple to generate and has few free elements, which greatly reduces the difficulty of hardware implementation and, at the same time, supports fast algorithms that can solve numerous problems related to convolutional operations. Based on the idea of alternating optimization, a weighted measurement matrix optimization objective function was proposed in the literature [16] to improve the robustness of the compressive sampling system under the condition of considering both signal adaptation and the matrix’s own characteristics. The literature [18] proposed a new joint optimization algorithm of the measurement matrix and sparse dictionary to improve the signal compression-aware reconstruction effect by constructing a new objective function. The experimental results show that the optimized bipolar chaotic Toeplitz measurement matrix compression-aware reconstruction error is reduced and the reconstruction probability is significantly improved

Description of the Problem

Bipolar Chaotic Toeplitz Matrix Alternating Optimization Algorithm

Experiments and Analysis

Objective function

Conclusions