A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery
For researchers in image recovery, this work provides a faster and more scalable algorithm for using convolutional structured low-rank priors, addressing a key computational bottleneck.
The paper introduces the GIRAF algorithm for convolutional structured low-rank matrix recovery, which reduces computational complexity and memory demand by working in the original un-lifted domain. Experiments show it is considerably faster and can handle larger problem sizes than previous methods.
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.