Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

TL;DR

This paper presents deterministic, finite geometry-inspired binary measurement matrices for compressed sensing, with improved theoretical guarantees and practical advantages like sparsity and cyclic structure, outperforming some random matrices.

Contribution

It introduces new finite geometry-based binary measurement matrices with enhanced spark bounds and practical structures for compressed sensing.

Findings

Matrices perform comparably or better than Gaussian matrices.

Theoretical bounds on spark are improved.

Matrices are sparse, binary, and cyclic, aiding hardware implementation.

Abstract

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as {provably} good measurement matrices for compressed sensing under $\ell_1$-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of $\ell_0$-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.