A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

TL;DR

This paper introduces GIRAF, a fast and memory-efficient algorithm for convolutional structured low-rank matrix recovery in Fourier domain image reconstruction, significantly reducing computational complexity and enabling larger-scale problems.

Contribution

The paper presents GIRAF, a novel algorithm that exploits convolutional structure to perform low-rank matrix recovery efficiently without lifting data, improving speed and scalability.

Findings

GIRAF is significantly faster than previous algorithms.

It can handle larger problem sizes.

It effectively recovers images from undersampled Fourier data.

Abstract

Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation and wavelet regularization. These priors specify that a convolutional structured matrix, i.e., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank. The main challenge in applying these schemes to large-scale problems is the computational complexity and memory demand resulting from lifting the image data to a large scale matrix. We introduce a fast and memory efficient approach called the Generic Iterative Reweighted Annihilation Filter (GIRAF) algorithm that exploits the convolutional structure of the lifted matrix to work in the original un-lifted domain, thus considerably reducing the complexity. Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger problem sizes than previously studied.