Recovery of Sparsely Corrupted Signals

TL;DR

This paper develops deterministic recovery guarantees and algorithms for reconstructing signals with sparse representations in general dictionaries, even when corrupted by sparse noise, applicable to various real-world signal processing problems.

Contribution

It introduces a novel uncertainty relation for pairs of general dictionaries and provides practical recovery algorithms with guarantees based on sparsity and coherence parameters.

Findings

Recovery guarantees depend on sparsity levels and dictionary coherence.

Algorithms successfully recover signals under specified sparsity and noise conditions.

Applicable to image inpainting, super-resolution, and interference removal.

Abstract

We investigate the recovery of signals exhibiting a sparse representation in a general (i.e., possibly redundant or incomplete) dictionary that are corrupted by additive noise admitting a sparse representation in another general dictionary. This setup covers a wide range of applications, such as image inpainting, super-resolution, signal separation, and recovery of signals that are impaired by, e.g., clipping, impulse noise, or narrowband interference. We present deterministic recovery guarantees based on a novel uncertainty relation for pairs of general dictionaries and we provide corresponding practicable recovery algorithms. The recovery guarantees we find depend on the signal and noise sparsity levels, on the coherence parameters of the involved dictionaries, and on the amount of prior knowledge about the signal and noise support sets.