From compression to compressed sensing

TL;DR

This paper explores how compression algorithms can be used for signal recovery in compressed sensing, establishing theoretical links and proposing an algorithm that guarantees accurate recovery from limited measurements.

Contribution

It introduces the compressible signal pursuit (CSP) algorithm and proves its effectiveness in recovering signals from underdetermined linear measurements, connecting compression performance to measurement requirements.

Findings

CSP accurately recovers signals with high probability.

Theoretical link between compression rate-distortion and measurement count.

Effective for infinite dimensional signals.

Abstract

Can compression algorithms be employed for recovering signals from their underdetermined set of linear measurements? Addressing this question is the first step towards applying compression algorithms for compressed sensing (CS). In this paper, we consider a family of compression algorithms $\mathcal{C}_r$, parametrized by rate $r$, for a compact class of signals $\mathcal{Q} \subset \mathds{R}^n$. The set of natural images and JPEG at different rates are examples of $\mathcal{Q}$ and $\mathcal{C}_r$, respectively. We establish a connection between the rate-distortion performance of $\mathcal{C}_r$, and the number of linear measurements required for successful recovery in CS. We then propose compressible signal pursuit (CSP) algorithm and prove that, with high probability, it accurately and robustly recovers signals from an underdetermined set of linear measurements. We also explore the performance of CSP in the recovery of infinite dimensional signals.