The stability of a direct method for superresolution

Abstract

A direct method for superresolution recently proposed by Walsh and Delaney is further analyzed from the point of view of numerical stability. The method is based on a set of linear equations Ax = b, where A is m × n, and b is a subset (of cardinal n) of the Fourier transform of the object (which has a total of N samples). We give exact and best possible approximate expressions for the determinant of A, when m = n. As a corollary, it is shown that the smallest eigenvalue of A in absolute value satisfies |λ min | ≤ k(n)N −(n−1)/2 , where k(n) (which is independent of N ) is explicitly given. The magnitude of the smallest eigenvalue of A becomes increasingly small as N grows, even when the number of unknowns n remains constant. When m > n the singular values of A are studied, and related to the eigenvalues of the matrix of two other direct methods. As a result, the connection between the method and the other direct methods is clarified.

Scaling properties of superoscillations and the extension to periodic signals

“…In hindsight, examples of superoscillatory behavior can be seen already in the works of Slepian et al in the 1960s on the prolate spheroidal wave functions, a sequence of bandlimited functions which become superoscillatory [3,4]. For other early work related to superoscillations, see, e.g., [5][6][7][8].…”