Algorithms for Reconstruction of Partially Known, Band Limited Fourier Transform Pairs from Noisy Data

摘要

This paper is a summary of more detailed mathematical work by the author on recovery of partially know 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 (due to existence of multiple exact solutions) and local ill-conditioning (due to existence of large neighborhoods of the true solution, all of whose members are indistinguishable from the true solution if the data is 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.