# Recovery of damped exponentials using structured low rank matrix completion

## TL;DR

This paper presents a structured low rank matrix completion method that efficiently recovers images from under-sampled measurements by exploiting exponential signal behavior and spatial smoothness, with applications in MR parameter mapping.

## Contribution

The paper introduces a novel IRLS-based algorithm with FFT approximations to reduce computational complexity for large-scale 3D problems in low rank matrix recovery.

## Key findings

- Improved image recovery accuracy over existing methods
- Reduced computational complexity enabling large-scale 3D applications
- Effective exploitation of exponential and spatial smoothness properties

## Abstract

We introduce a structured low rank matrix completion algorithm to recover a series of images from their under-sampled measurements, where the signal along the parameter dimension at every pixel is described by a linear combination of exponentials. We exploit the exponential behavior of the signal at every pixel, along with the spatial smoothness of the exponential parameters to derive an annihilation relation in the Fourier domain. This relation translates to a low-rank property on a structured matrix constructed from the Fourier samples. We enforce the low rank property of the structured matrix as a regularization prior to recover the images. Since the direct use of current low rank matrix recovery schemes to this problem is associated with high computational complexity and memory demand, we adopt an iterative re-weighted least squares (IRLS) algorithm, which facilitates the exploitation of the convolutional structure of the matrix. Novel approximations involving two dimensional Fast Fourier Transforms (FFT) are introduced to drastically reduce the memory demand and computational complexity, which facilitates the extension of structured low rank methods to large scale three dimensional problems. We demonstrate our algorithm in the MR parameter mapping setting and show improvement over the state-of-the-art methods.