A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

Abstract

Fourier domain structured low-rank matrix priors are emerging as powerful alternatives to traditional image recovery methods such as total variation (TV) 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 a 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.

A general algorithm for compensation of trajectory errors: Application to radial imaging

“…However, the high memory demands of the present method would require the adaption of memory‐efficient structured low rank methods such as Refs. and for efficient implementation.…”

Wave‐LORAKS: Combining wave encoding with structured low‐rank matrix modeling for more highly accelerated 3D imaging

“…Similar autocalibration ideas have also been used in several subsequent publications . Another ingredient of our new implementation is the use of fast Fourier transforms (FFTs) for implementing matrix‐vector multiplications with high‐dimensional structured low‐rank matrices, as originally proposed by Ongie and Jacob. We have recently made both of these ingredients available to the community through a new public open‐source LORAKS software release…”

Improved MUSSELS reconstruction for high‐resolution multi‐shot diffusion weighted imaging

“…The implicit incorporation of the phase information accelerates the recovery problem. IRLS‐based implementations were previously employed to accelerate SLRMC reconstructions . We note that the exact implementation employing Hankel matrix multiplication is of comparable performance to the approximate FFT‐based implementation.…”

“…Fast algorithms proposed in the past to address the computational complexity of SLRMC include the UV factorization method and the GIRAF algorithm . While the former method eliminated the need for SVD computation, the latter employed the IRLS formulation and exploited the convolutional structure of the lifted matrix to work in the original un‐lifted domain.…”

“…We reformulate the MUSSELS reconstruction to enable a computationally efficient implementation to solve the matrix recovery problem embedded in the recovery scheme. The reformulation is based on the iterative re‐weighted least squares (IRLS) methods that were previously proposed to address the computational complexity in similar reconstruction problems involving matrix completion . It has been shown that nuclear norm minimization involving large matrices can be efficiently solved by reformulating such problems using the IRLS formulation .…”