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

arXiv: https://arxiv.org/abs/1510.00384 · HTML: https://mlsanity.com/paper/1510.00384/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2015.

## Built on by (most cited first, 5 of 5)

- [A review and experimental evaluation of deep learning methods for MRI reconstruction](https://mlsanity.com/paper/2109.08618/) (2021)
- [K-UNN: k-Space Interpolation With Untrained Neural Network](https://mlsanity.com/paper/2208.05827/) (2022)
- [Towards a Sampling Theory for Implicit Neural Representations](https://mlsanity.com/paper/2405.18410/) (2024)
- [Wave-Encoded Model-based Deep Learning for Highly Accelerated Imaging with Joint Reconstruction](https://mlsanity.com/paper/2202.02814/) (2022)
- [Zero-shot Dynamic MRI Reconstruction with Global-to-local Diffusion Model](https://mlsanity.com/paper/2411.03723/) (2024)

## Abstract (arXiv)

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.