Exploring the Irreducible Support

Abstract

The phase problem confronts one with the task obtaining a function having a specified autocorrelation (or Fourier modulus) which is also consistent with a limited number of object constraints. In many case of interest it is known that the object has compact support; indeed, in two dimensions this is often sufficient to uniquely define it [1]. Nonetheless, one can usually find an autocorrelation such that some object defined on the same compact support is but one of a multitude of functions having that autocorrelation. However, if the object happens to be defined on Eisenstein's support, it has been shown that the object is always uniquely described by its autocorrelation [2]. We shall call supports of the Eisenstein-type irreducible supports. Unfortunately, variations on Eisenstein's support are the only ones to have been shown to be irreducible, and the method of proof for Eisenstein's support does not lend itself to ready generalization. Herein the irreducible support will be defined in a wider sense, and a method of testing an arbitrary convex support for irreducibility will be presented.