Structured low-rank recovery of piecewise constant signals with performance guarantees

TL;DR

This paper presents a theoretical framework and algorithm for exact recovery of piecewise constant images from limited Fourier samples, with guarantees based on low-rank matrix completion and incoherence conditions, demonstrated on MRI data.

Contribution

It introduces a structured low-rank matrix completion method with provable guarantees for recovering images from non-uniform Fourier samples, extending previous work to piecewise constant signals.

Findings

Exact recovery with high probability under certain conditions

Algorithm successfully reconstructs undersampled MRI data

Theoretical guarantees are established for the recovery process

Abstract

We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities of the image are localized to the zero level-set of a bandlimited function, which induces certain linear dependencies in Fourier domain, such that a multifold Toeplitz matrix built from the Fourier data is known to be low-rank. The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints. This work adapts results by Chen and Chi on the recovery of isolated Diracs via nuclear norm minimization of a similar multifold Hankel structure. We show that exact recovery is possible with high probability when the bandlimited function describing the edge set satisfies an incoherency property. Finally, we demonstrate the algorithm on the recovery of undersampled MRI data.