Stable Restoration and Separation of Approximately Sparse Signals

TL;DR

This paper introduces new algorithms and theoretical insights for recovering approximately sparse signals from noisy, interfered measurements across various applications like audio and image restoration.

Contribution

It provides a unified framework for signal recovery considering general dictionaries and interference, with algorithms adaptable to different support-set knowledge levels.

Findings

Effective recovery in noisy, interfered conditions

Successful application to audio and image restoration

Theoretical recovery guarantees established

Abstract

This paper develops new theory and algorithms to recover signals that are approximately sparse in some general dictionary (i.e., a basis, frame, or over-/incomplete matrix) but corrupted by a combination of interference having a sparse representation in a second general dictionary and measurement noise. The algorithms and analytical recovery conditions consider varying degrees of signal and interference support-set knowledge. Particular applications covered by the proposed framework include the restoration of signals impaired by impulse noise, narrowband interference, or saturation/clipping, as well as image in-painting, super-resolution, and signal separation. Two application examples for audio and image restoration demonstrate the efficacy of the approach.