The Alan Turing Institute

# Sparse non-negative super-resolution - simplified and stabilised

## Abstract

The convolution of a discrete measure, x=∑ki=1aiδti, with a local window function, φ(s−t), is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources {ai,ti}ki=1 with an accuracy beyond the essential support of φ(s−t), typically from m samples y(sj)=∑ki=1aiφ(sj−ti)+ηj, where ηj indicates an inexactness in the sample value. We consider the setting of x being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that x is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. ηj=0, m≥2k+1 samples are available, and φ(s−t) generates a Chebyshev system. This is independent of how close the sample locations are and {\em does not rely on any regulariser beyond non-negativity}; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions x̂  consistent with the samples within the bound ∑mj=1η2j≤δ2. Any such non-negative measure is within (δ1/7) of the discrete measure x generating the samples in the generalised Wasserstein distance, converging to one another as δ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of φ(s−t) being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.

## Citation information

A. Eftekhari, J. Tanner, A. Thompson, B. Toader, H. Tyagi, Sparse non-negative super-resolution: Simplified and stabilised, arXiv preprint arXiv:1804.01490, 2018.