Numerical Investigation of Phase Retrieval Uniqueness

The uniqueness of phase retrieval has been explored from a theoretical standpoint via the factorability of polynomials. The Fourier transform of a function (an image), f, sampled on a regular grid is a polynomial, F. Bruck and Sodin [1] have shown that if f can be written as the convolution of K non-Hermitian functions (equivalent to K irreducible polynomial factors in Fourier space), then there exist 2K-1 ambiguous solutions to the phase retrieval problem of reconstructing f from IFI. It is well known for the two-dimensional case that polynomials of two complex variables have probability zero of being factorable [1-3]. It has also been shown from a theoretical standpoint that the uniqueness condition is not sensitive to noise [4]. However, this does not answer the practical question: what is the likelihood of significant ambiguity when IFI is corrupted by a given level of noise? We approached this question by trying to determine, given an arbitrary image and its Fourier polynomial, how close is the nearest factorable polynomial, and does it have an ambiguous solution that is significantly different from the given polynomial? This paper explores this question for the case of images defined within a 3x2 support. A derivation of object-domain conditions for factorability provides a means for finding nearest factorable polynomials through a constrained minimization search over the space of 3x2 ambiguous images. These searches are implemented with different object-domain constraints in a Monte Carlo simulation to estimate the probability that the nearest factorable polynomial, with an ambiguous solution that is significantly different from a given image, is within some distance of the given polynomial.