[Paper Review] On the absence of the RIP in real-world applications of compressed sensing and the RIP in levels

Alexander Bastounis, Anders C. Hansen|arXiv (Cornell University)|Nov 17, 2014

Sparse and Compressive Sensing Techniques44 references20 citations

TL;DR

This paper demonstrates that the Restricted Isometry Property (RIP) does not hold in real-world compressed sensing applications such as MRI, CT, and electron microscopy due to the structure of level-based reconstruction bases. To address this, the authors introduce the 'Restricted Isometry Property in Levels' (RIP-L), a generalized framework that enables uniform recovery within each sparsity level, providing a more accurate theoretical foundation for practical compressed sensing success.

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.

Motivation & Objective

• To challenge the assumption that the Restricted Isometry Property (RIP) underpins the success of compressed sensing in real-world applications.
• To demonstrate that RIP fails for many practical compressed sensing matrices, especially those using level-based reconstruction bases like wavelets.
• To develop a new theoretical framework—RIP in Levels—that better captures the structure of real-world compressed sensing problems.
• To show that uniform recovery is feasible within each sparsity level under RIP-L, even when global uniform recovery is infeasible.
• To provide theoretical and numerical evidence for the optimality and practical relevance of the RIP-L framework.

Proposed method

• Introduces the 'flip test' as a practical method to disprove RIP for specific matrices in real applications.
• Proposes the Restricted Isometry Property in Levels (RIP-L) as a generalization of classical RIP, where recovery is uniform only within predefined levels of sparsity.
• Defines RIP-L using a block-structured sparsity pattern with parameters $\mathbf{s}$ and $\mathbf{M}$, allowing for structured recovery within each level.
• Analyzes the robust nullspace property under RIP-L to establish conditions for stable and robust $\ell^1$-minimization recovery.
• Constructs counterexamples using matrices dependent on parameters $C$, $\rho$, and $\tau$ to demonstrate failure of classical RIP and validity of RIP-L.
• Employs Cauchy-Schwarz and $\ell^1$-$\ell^2$ norm inequalities to derive bounds on recovery error and verify the nullspace property under RIP-L.

Experimental results

Research questions

• RQ1 Why does the classical Restricted Isometry Property (RIP) fail to hold in real-world compressed sensing applications such as MRI and CT?
• RQ2 To what extent does the structure of level-based reconstruction bases (e.g., wavelets) prevent the RIP from being satisfied for reasonable sparsity levels?
• RQ3 Can a generalized RIP framework be constructed that better models the actual recovery behavior observed in practical compressed sensing?
• RQ4 Under what conditions does the Restricted Isometry Property in Levels (RIP-L) enable uniform recovery within each sparsity level?
• RQ5 How do the theoretical guarantees of RIP-L compare to classical RIP in terms of measurement complexity and recovery stability?

Key findings

• The classical RIP does not hold for compressed sensing matrices in MRI, CT, electron microscopy, and other real-world applications due to the structure of level-based reconstruction bases.
• For practical matrices with wavelet-based sparsity, the number of measurements required for uniform recovery of all $s$-sparse signals under classical RIP is excessive and unrealistic.
• The Restricted Isometry Property in Levels (RIP-L) is introduced as a generalized framework that allows for uniform recovery within each sparsity level, avoiding the infeasibility of global uniform recovery.
• Under certain conditions on RIP-L, stable and robust $\ell^1$-minimization recovery is possible within each level, even when classical RIP fails.
• Theoretical counterexamples demonstrate that classical RIP fails for matrices with sparsity patterns involving $C^2$ and $\omega(\rho,C)$, while RIP-L remains valid and effective.
• The paper shows that $\|z - z^1\|_2 = 1 > \frac{\sigma_{\mathbf{s,M}}(z^1)_1}{\sqrt{\tilde{s}}} f(\eta_{\mathbf{s,M}})$ for large $\eta_{\mathbf{s,M}}$, confirming the failure of classical RIP and supporting the need for RIP-L.