Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples

TL;DR

This paper presents a novel method for recovering piecewise constant images from limited Fourier samples by exploiting the structure of the image edges, with applications to super-resolution MRI.

Contribution

It introduces a new theoretical framework and a practical algorithm for exact image recovery from few Fourier samples, leveraging edge set localization and annihilation relations.

Findings

Successful super-resolution MRI reconstruction from limited Fourier data.

The proposed method outperforms standard super-resolution techniques.

Robustness to noise and model mismatch demonstrated.

Abstract

We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric polynomial, we show the Fourier coefficients of the partial derivatives of the image satisfy a linear annihilation relation. We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation. We also propose a practical two-stage recovery algorithm which is robust to model-mismatch and noise. In the first stage we estimate a continuous domain representation of the edge set of the image. In the second stage we perform an extrapolation in Fourier domain by a least squares two-dimensional linear prediction, which recovers the exact Fourier coefficients of the underlying image. We demonstrate our algorithm on the super-resolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image super-resolution MRI.