Deterministic Construction of Compressed Sensing Matrices using BCH Codes

TL;DR

This paper presents a deterministic method for constructing compressed sensing matrices using BCH codes, enabling efficient and exact sparse signal recovery with simple algorithms.

Contribution

Introduces a new deterministic construction of RIP matrices using BCH codes, expanding to matrices with 10,1,-1 elements for compressed sensing.

Findings

Matrices satisfy RIP with specific log-ratio properties.

Matching Pursuit can exactly reconstruct signals with these matrices.

Extension to 10,1,-1 matrices using Devore's binary matrices.

Abstract

In this paper we introduce deterministic $m\times n$ RIP fulfilling $\pm 1$ matrices of order $k$ such that $\frac{\log m}{\log k}\approx \frac{\log(\log_2 n)}{\log(\log_2 k)}$. The columns of these matrices are binary BCH code vectors that their zeros are replaced with -1 (excluding the normalization factor). The samples obtained by these matrices can be easily converted to the original sparse signal; more precisely, for the noiseless samples, the simple Matching Pursuit technique, even with less than the common computational complexity, exactly reconstructs the sparse signal. In addition, using Devore's binary matrices, we expand the binary scheme to matrices with $\{0,1,-1\}$ elements.