RIP sensing matrices construction for sparsifying dictionaries with application to MRI imaging

TL;DR

This paper introduces a new method for constructing RIP measurement matrices tailored for redundant dictionaries, enhancing MRI imaging by enabling reliable sparse recovery with physical measurement constraints.

Contribution

It presents a simple construction for RIP matrices compatible with specific dictionaries and measurement mechanisms, with applications to MRI imaging.

Findings

Constructed RIP matrices with high probability in the $k ext{log}(n/k)$ regime.

Demonstrated recovery of wavelet coefficients in MRI using the proposed method.

Validated the approach on fastMRI dataset examples.

Abstract

Practical applications of compressed sensing often restrict the choice of its two main ingredients. They may (i) prescribe using particular redundant dictionaries for certain classes of signals to become sparsely represented, or (ii) dictate specific measurement mechanisms which exploit certain physical principles. On the problem of RIP measurement matrix design in compressed sensing with redundant dictionaries, we give a simple construction to derive sensing matrices whose compositions with a prescribed dictionary have a high probability of the RIP in the $k \log(n/k)$ regime. Our construction thus provides recovery guarantees usually only attainable for sensing matrices from random ensembles with sparsifying orthonormal bases. Moreover, we use the dictionary factorization idea that our construction rests on in the application of magnetic resonance imaging, in which also the sensing matrix is prescribed by quantum mechanical principles. We propose a recovery algorithm based on transforming the acquired measurements such that the compressed sensing theory for RIP embeddings can be utilized to recover wavelet coefficients of the target image, and show its performance on examples from the fastMRI dataset.