Stable super-resolution of images: theoretical study

Author:

Eftekhari Armin1,Bendory Tamir2,Tang Gongguo3

Affiliation:

1. Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden

2. Program in Applied and Computational Mathematics, Princeton University, Princeton, US

3. Department of Electrical Engineering, Colorado School of Mines, Golden, Colorado, US

Abstract

Abstract We study the ubiquitous super-resolution problem, in which one aims at localizing positive point sources in an image, blurred by the point spread function of the imaging device. To recover the point sources, we propose to solve a convex feasibility program, which simply finds a non-negative Borel measure that agrees with the observations collected by the imaging device. In the absence of imaging noise, we show that solving this convex program uniquely retrieves the point sources, provided that the imaging device collects enough observations. This result holds true if the point spread function of the imaging device can be decomposed into horizontal and vertical components and if the translations of these components form a Chebyshev system, i.e., a system of continuous functions that loosely behave like algebraic polynomials. Building upon the recent results for one-dimensional signals, we prove that this super-resolution algorithm is stable, in the generalized Wasserstein metric, to model mismatch (i.e., when the image is not sparse) and to additive imaging noise. In particular, the recovery error depends on the noise level and how well the image can be approximated with well-separated point sources. As an example, we verify these claims for the important case of a Gaussian point spread function. The proofs rely on the construction of novel interpolating polynomials—which are the main technical contribution of this paper—and partially resolve the question raised in Schiebinger et al. (2017, Inf. Inference, 7, 1–30) about the extension of the standard machinery to higher dimensions.

Funder

Alan Turing Institute

Turing Seed Funding

NSF

DARPA Lagrange Program

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

Reference73 articles.

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1. Super-resolution of positive near-colliding point sources;Information and Inference: A Journal of the IMA;2023-09-18

2. Multivariate super-resolution without separation;Information and Inference: A Journal of the IMA;2023-04-27

3. On the uniqueness of solutions for the basis pursuit in the continuum;Inverse Problems;2022-10-28

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