Deterministic partial binary circulant compressed sensing matrices

TL;DR

This paper introduces a new deterministic, binary, partial circulant measurement matrix for compressed sensing, leveraging Legendre symbols, offering fast computations and reconstruction, with verified performance through numerical experiments.

Contribution

The paper proposes a novel deterministic measurement matrix based on Legendre symbols with a simple binary, partial circulant structure for efficient compressed sensing.

Findings

Provides a bound on sparsity levels for successful measurement and reconstruction.

Demonstrates competitive performance of the proposed matrices through numerical experiments.

Offers a fast matrix-vector multiplication and reconstruction algorithm.

Abstract

Compressed sensing (CS) is a signal acquisition paradigm to simultaneously acquire and reduce dimension of signals that admit sparse representation. This is achieved by collecting linear, non-adaptive measurements of a signal, which can be formalized as multiplying the signal with a "measurement matrix". Most of matrices used in CS are random matrices as they satisfy the restricted isometry property (RIP) in an optimal regime of number of measurements with high probability. However, these matrices have their own caveats and for this reason, deterministic measurement matrices have been proposed. While there is a wide classes of deterministic matrices in the literature, we propose a novel class of deterministic matrices using the Legendre symbol. This construction has a simple structure, it enjoys being a binary matrix, and having a partial circulant structure which provides a fast matrix-vector multiplication and a fast reconstruction algorithm. We will derive a bound on the sparsity level of signals that can be measured (and be reconstructed) with this class of matrices. We perform quantization using these matrices, and we verify the performance of these matrices (and compare with other existing constructions) numerically.