Decision-Theoretic Treatment of Superresolution Based upon Oversampling and Finite Support

Abstract

It is well-known (at least in one dimension) that a function vanishing outside of a finite support domain has a Fourier transform that is analytic everywhere in frequency space. Consequently, if the transform is known exactly on a finite line segment in the complex frequency plane, it can by analytic continuation be determined everywhere and thus the original function can be recovered exactly. In this paper we consider realistic imaging problems in both one and two dimensions where the transform is imperfectly known from a set of noisy measurements at a discrete set of points in spatial frequency space. The true image in physical space is assumed to vanish identically outside of a specified support domain. The problem of estimating the image from the noisy measurements is approached within the well-established framework of linear Gaussian estimation theory.