A General Approach for Construction of Deterministic Compressive Sensing Matrices

TL;DR

This paper introduces a deterministic method for constructing large, low-coherence sensing matrices in compressive sensing using linear codes and column replacement, achieving near-optimal coherence and outperforming existing methods.

Contribution

It presents a novel deterministic construction technique for sensing matrices based on linear codes and column replacement, with proven coherence bounds and resizing capabilities.

Findings

Constructed matrices achieve asymptotic optimal coherence.

Resized matrices maintain low coherence and performance.

Simulation results confirm the effectiveness of the proposed method.

Abstract

In this paper, deterministic construction of measurement matrices in Compressive Sensing (CS) is considered. First, by employing the column replacement concept, a theorem for construction of large minimum distance linear codes containing all-one codewords is proposed. Then, by applying an existing theorem over these linear codes, deterministic sensing matrices are constructed. To evaluate this procedure, two examples of constructed sensing matrices are presented. The first example contains a matrix of size ${{p}^{2}}\times {{p}^{3}}$ and coherence ${1}/{p}\;$, and the second one comprises a matrix with the size $p\left( p-1 \right)\times {{p}^{3}}$ and coherence ${1}/{\left( p-1 \right)}\;$, where $p$ is a prime integer. Based on the Welch bound, both examples asymptotically achieve optimal results. Moreover, by presenting a new theorem, the column replacement is used for resizing any sensing matrix to a greater-size sensing matrix whose coherence is calculated. Then, using an example, the outperformance of the proposed method is compared to a well-known method. Simulation results show the satisfying performance of the column replacement method either in created or resized sensing matrices.