Numerical investigation of the uniqueness of phase retrieval

J. H. Seldin, J. R. Fienup
1990 Optical Society of America. Journal A: Optics, Image Science, and Vision  
Both a new iterative grid-search technique and the iterative Fourier-transform algorithm are used to illuminate the relationships among the ambiguous images nearest a given object, error metric minima, and stagnation points of phase-retrieval algorithms. Analytic expressions for the subspace of ambiguous solutions to the phase-retrieval problem are derived for 2 X 2 and 3 X 2 objects. Monte Carlo digital experiments using a reduced-gradient search of these subspaces are used to estimate the
more » ... to estimate the probability that the worst-case nearest ambiguous image to a given object has a Fourier modulus error of less than a prescribed amount. Probability distributions for nearest ambiguities are estimated for different object-domain constraints. A 413 as a polynomial of two complex variables, z = exp(j7ru/M) and w = exp(j7rv/N). It is also equivalent to the z transform. Then the presence of ambiguity in the phase-retrieval problem is equivalent to the factorability of the polynomial. This explains the vast difference between the 1-D and 2-D cases, because polynomials (of degree 2 or greater) of a single complex variable are always factorable, whereas polynomials of two (or more) complex variables are rarely factor-
doi:10.1364/josaa.7.000412 fatcat:chl2gxuc7nb4xnaq4yrf66ty3i