Algorithms for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data

Abstract

This paper is a summary of more detailed mathematical work by the authors on recovery of partially known Fourier transforms. These problems of inversion of the finite Fourier transform and of phase retrieval are known to be ill posed. We draw a distinction in the resultant ill conditioning of the problems between global ill conditioning (which is due to the existence of multiple exact solutions) and local ill conditioning (which is due to the existence of large neighborhoods of the true solution, all of whose members are indistinguishable from the true solution if the data are noisy). We then develop extensions of known algorithms that attempt to reduce at least the effects of local ill conditioning on numerical solutions by using the idea of filtered singular-value decomposition and present some numerical examples of the use of those algorithms in the context of optical-diffraction theory.