Non-uniform sampling, image recovery from sparse data and the discrete sampling theorem

TL;DR

This paper develops a theoretical framework for recovering band-limited discrete signals from irregular or sparse samples, with applications in image super-resolution and reconstruction.

Contribution

It introduces a discrete sampling theorem that provides conditions for unique recovery of signals from sparse data, applicable to various transforms.

Findings

Established conditions for unique image recovery from sparse samples.

Demonstrated applications in image super-resolution.

Validated the approach with examples of image reconstruction from sparse projections.

Abstract

In many applications sampled data are collected in irregular fashion or are partly lost or unavailable. In these cases it is required to convert irregularly sampled signals to regularly sampled ones or to restore missing data. In this paper, we address this problem in a framework of a discrete sampling theorem for band-limited discrete signals that have a limited number of non-zero transform coefficients in a certain transform domain. Conditions for the image unique recovery, from sparse samples, are formulated and then analyzed for various transforms. Applications are demonstrated on examples of image super-resolution and image reconstruction from sparse projections.