Inversion of Autocorrelation Functions
Wednesday, November 9, 2005 - 9:30am - 10:30am
James Fienup (University of Rochester)
Solving the phase retrieval problem, i.e. reconstructing a compact, multidimensional function from the modulus of its Fourier transform, has applications in astronomy, x-ray diffraction, optical wave-front sensing, and other areas of physics and engineering. To solve such problems, one must have constraints on the function in order to have a chance for a unique solution. For some of these problems, the most important constraint is S, the support of the function, i.e., the set of points outside of which the function has value zero. The autocorrelation of the function can computed from the Fourier modulus, and A, the support of the autocorrelation function, is the Minkowski sum of S with -S. Therefore, in order to solve the phase retrieval problem, we first want to determine S, or at least estimate the smallest upper bound on S, from A = S - S. This paper will describe methods we have discovered so far for performing this inversion with the hope that others will point us to, or discover, additional approaches.