The stability of a direct method for superresolution

A direct method for superresolution proposed by Walsh and Nielsen-Delaney (1994) 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