On the absence of the RIP in real-world applications of compressed sensing and the RIP in levels

TL;DR

This paper reveals that the traditional RIP does not hold in many practical compressed sensing applications and introduces a new 'RIP in Levels' framework that better explains successful structured recovery.

Contribution

The paper introduces the 'Restricted Isometry Property in Levels', a new framework that accounts for level-based structures in compressed sensing, addressing limitations of the traditional RIP.

Findings

RIP does not hold in many real-world applications

Uniform recovery of all sparse signals is often unrealistic

Structured recovery within levels is achievable under new RIP in Levels

Abstract

The purpose of this paper is twofold. The first is to point out that the Restricted Isometry Property (RIP) does not hold in many applications where compressed sensing is successfully used. This includes fields like Magnetic Resonance Imaging (MRI), Computerized Tomography, Electron Microscopy, Radio Interferometry and Fluorescence Microscopy. We demonstrate that for natural compressed sensing matrices involving a level based reconstruction basis (e.g. wavelets), the number of measurements required to recover all $s$-sparse signals for reasonable $s$ is excessive. In particular, uniform recovery of all $s$-sparse signals is quite unrealistic. This realisation shows that the RIP is insufficient for explaining the success of compressed sensing in various practical applications. The second purpose of the paper is to introduce a new framework based on a generalised RIP-like definition that fits the applications where compressed sensing is used. We show that the shortcomings that show that uniform recovery is unreasonable no longer apply if we instead ask for structured recovery that is uniform only within each of the levels. To examine this phenomenon, a new tool, termed the 'Restricted Isometry Property in Levels' is described and analysed. Furthermore, we show that with certain conditions on the Restricted Isometry Property in Levels, a form of uniform recovery within each level is possible. Finally, we conclude the paper by providing examples that demonstrate the optimality of the results obtained.