Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices

TL;DR

This paper presents a deterministic method for constructing binary, bipolar, and ternary compressed sensing matrices with RIP properties, leveraging BCH codes and FFT algorithms for efficient sparse signal recovery.

Contribution

It introduces a novel deterministic construction of compressed sensing matrices using BCH codes and establishes their RIP properties, enabling efficient reconstruction algorithms.

Findings

Constructed RIP matrices with size bounds involving logarithmic factors

Demonstrated the use of FFT for faster reconstruction

Combined binary and bipolar matrices to create ternary sensing matrices

Abstract

In this paper we establish the connection between the Orthogonal Optical Codes (OOC) and binary compressed sensing matrices. We also introduce deterministic bipolar $m\times n$ RIP fulfilling $\pm 1$ matrices of order $k$ such that $m\leq\mathcal{O}\big(k (\log_2 n)^{\frac{\log_2 k}{\ln \log_2 k}}\big)$. The columns of these matrices are binary BCH code vectors where the zeros are replaced by -1. Since the RIP is established by means of coherence, the simple greedy algorithms such as Matching Pursuit are able to recover the sparse solution from the noiseless samples. Due to the cyclic property of the BCH codes, we show that the FFT algorithm can be employed in the reconstruction methods to considerably reduce the computational complexity. In addition, we combine the binary and bipolar matrices to form ternary sensing matrices ($\{0,1,-1\}$ elements) that satisfy the RIP condition.