Practical Phase Retrieval Based On Theoretical Models For Multi-Dimensional Band-Limited Signals

Abstract

The analytic properties of two-dimensional band-limited functions are discussed. In practice, only a limited number of intensity samples are available, and so we choose to model the spectrum as a finite degree polynomial. The set of reducible finite degree multi-variate polynomials is of measure zero and unique recovery from noise free Fourier magnitude is expected in almost all cases. We pursue a new algorithm based on finding the complex zeros of 1-D lines of the data set which requires only that the intensity is sampled at twice the Nyquist rate or greater. All solutions compatible with the Fourier magnitude samples are generated, including ambiguities should they exist. An exact solution to the phase retrieval problem, given a polynomial model, may be regarded as factorization. We discuss the relationship of this approach to factorization and iterative procedures and describe problems arising from data truncation and the presence of noise.