# A Generalized Structured Low-Rank Matrix Completion Algorithm for MR Image Recovery

arXiv: https://arxiv.org/abs/1811.10778 · HTML: https://mlsanity.com/paper/1811.10778/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2018.

## Built on by (most cited first, 3 of 3)

- [One-dimensional Deep Low-rank and Sparse Network for Accelerated MRI](https://mlsanity.com/paper/2112.04721/) (2021)
- [Rank-1 Matrix Completion with Gradient Descent and Small Random Initialization](https://mlsanity.com/paper/2212.09396/) (2022)
- [Negative Binomial Matrix Completion](https://mlsanity.com/paper/2408.16113/) (2024)

## Abstract (arXiv)

Recent theory of mapping an image into a structured low-rank Toeplitz or Hankel matrix has become an effective method to restore images. In this paper, we introduce a generalized structured low-rank algorithm to recover images from their undersampled Fourier coefficients using infimal convolution regularizations. The image is modeled as the superposition of a piecewise constant component and a piecewise linear component. The Fourier coefficients of each component satisfy an annihilation relation, which results in a structured Toeplitz matrix, respectively. We exploit the low-rank property of the matrices to formulate a combined regularized optimization problem. In order to solve the problem efficiently and to avoid the high memory demand resulting from the large-scale Toeplitz matrices, we introduce a fast and memory efficient algorithm based on the half-circulant approximation of the Toeplitz matrix. We demonstrate our algorithm in the context of single and multi-channel MR images recovery. Numerical experiments indicate that the proposed algorithm provides improved recovery performance over the state-of-the-art approaches.