Robust analysis $\ell_1$-recovery from Gaussian measurements and total variation minimization

TL;DR

This paper establishes bounds on the number of Gaussian measurements needed for successful analysis $ ext{l}_1$-recovery of signals, including total variation minimization, especially when the analysis sparsity is moderate.

Contribution

It provides new measurement bounds for analysis $ ext{l}_1$-recovery applicable to total variation and frame-based analysis operators, improving understanding of recovery conditions.

Findings

Derived bounds for Gaussian measurements in total variation minimization

Extended analysis to frame-based analysis operators

Applicable to signals with moderate analysis sparsity

Abstract

Analysis $\ell_1$-recovery refers to a technique of recovering a signal that is sparse in some transform domain from incomplete corrupted measurements. This includes total variation minimization as an important special case when the transform domain is generated by a difference operator. In the present paper we provide a bound on the number of Gaussian measurements required for successful recovery for total variation and for the case that the analysis operator is a frame. The bounds are particularly suitable when the sparsity of the analysis representation of the signal is not very small.