Stable Super-Resolution of Images
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We study the ubiquitous problem of super-resolution 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 nonnegative 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 can be decomposed into horizontal and vertical components, and the translations of these components form a Chebyshev system, namely, a system of continuous functions that loosely behave like algebraic polynomials; we argue that is often the case in practice. Building upon recent results for one-dimensional signals, we also prove that this super-resolution algorithm is stable in the generalized Wasserstein metric to model mismatch (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 is the main technical contribution of this paper.
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