Compressed Sensing with coherent tight frames via $l_q$-minimization for $0<q\leq1$

TL;DR

This paper develops new conditions for accurate signal recovery using $l_q$-minimization in compressed sensing with tight frames, improving existing results and extending to cases with coherence-independent guarantees.

Contribution

It introduces coherence-independent recovery conditions and explores $l_q$-minimization for sparse signals in tight frames, extending prior work to broader settings.

Findings

Established a coherence-independent recovery condition.

Proved existence of $q_0$ such that $l_q$-minimization approximates the true signal.

Extended results to cases where the tight frame is an identity or orthonormal basis.

Abstract

Our aim of this article is to reconstruct a signal from undersampled data in the situation that the signal is sparse in terms of a tight frame. We present a condition, which is independent of the coherence of the tight frame, to guarantee accurate recovery of signals which are sparse in the tight frame, from undersampled data with minimal $l_1$-norm of transform coefficients. This improves the result in [1]. Also, the $l_q$-minimization $(0<q<1)$ approaches are introduced. We show that under a suitable condition, there exists a value $q_0\in(0,1]$ such that for any $q\in(0,q_0)$, each solution of the $l_q$-minimization is approximately well to the true signal. In particular, when the tight frame is an identity matrix or an orthonormal basis, all results obtained in this paper appeared in [13] and [26].