# Convex recovery of continuous domain piecewise constant images from non-uniform Fourier samples

## TL;DR

This paper introduces a convex matrix completion method for recovering continuous domain piecewise constant images from non-uniform Fourier samples, leveraging low-rank structures induced by edge set properties.

## Contribution

It generalizes super-resolution techniques to piecewise constant images, modeling edges as zero levelsets of bandlimited functions and formulating a structured low-rank matrix completion problem.

## Key findings

- Exact recovery is probable under an incoherency condition.
- The incoherency depends on the edge set geometry, affecting sampling requirements.
- The method extends super-resolution to more complex image structures.

## Abstract

We consider the recovery of a continuous domain piecewise constant image from its non-uniform Fourier samples using a convex matrix completion algorithm. We assume the discontinuities/edges of the image are localized to the zero levelset of a bandlimited function. This assumption induces linear dependencies between the Fourier coefficients of the image, which results in a two-fold block Toeplitz matrix constructed from the Fourier coefficients being low-rank. The proposed algorithm reformulates the recovery of the unknown Fourier coefficients as a structured low-rank matrix completion problem, where the nuclear norm of the matrix is minimized subject to structure and data constraints. We show that exact recovery is possible with high probability when the edge set of the image satisfies an incoherency property. We also show that the incoherency property is dependent on the geometry of the edge set curve, implying higher sampling burden for smaller curves. This paper generalizes recent work on the super-resolution recovery of isolated Diracs or signals with finite rate of innovation to the recovery of piecewise constant images.