Orthonormal Expansion l1-Minimization Algorithms for Compressed Sensing

TL;DR

This paper introduces orthonormal expansion-based $ ext{l}_1$-minimization algorithms for compressed sensing, offering accurate, fast, and practical solutions for sparse signal reconstruction in noiseless and noisy scenarios.

Contribution

It presents a novel reformulation of the basis pursuit problem using orthonormal expansion and proposes two convex optimization algorithms, including a relaxed iterative soft thresholding method.

Findings

The relaxed algorithm is accurate and fast in noise-free conditions.

The methods are competitive with state-of-the-art algorithms.

Practical application extends to approximately sparse signals with noise.

Abstract

Compressed sensing aims at reconstructing sparse signals from significantly reduced number of samples, and a popular reconstruction approach is $\ell_1$-norm minimization. In this correspondence, a method called orthonormal expansion is presented to reformulate the basis pursuit problem for noiseless compressed sensing. Two algorithms are proposed based on convex optimization: one exactly solves the problem and the other is a relaxed version of the first one. The latter can be considered as a modified iterative soft thresholding algorithm and is easy to implement. Numerical simulation shows that, in dealing with noise-free measurements of sparse signals, the relaxed version is accurate, fast and competitive to the recent state-of-the-art algorithms. Its practical application is demonstrated in a more general case where signals of interest are approximately sparse and measurements are contaminated with noise.