Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI

TL;DR

This paper extends compressed sensing theory to practical inverse problems like deconvolution, source recovery, and MRI, providing rigorous guarantees and insights for improved reconstruction strategies.

Contribution

It generalizes the theoretical framework to multiple inverse problems, demonstrating practical recovery guarantees and optimized sampling strategies.

Findings

Successful reconstruction guarantees for deconvolution, source recovery, and MRI.

Balanced sampling strategies improve reconstruction performance.

Unified theoretical foundation for compressed sensing in inverse problems.

Abstract

This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.