7 - Algorithms for Phase Retrieval

Summary

In this chapter we introduce the basic types of algorithms used in to find intersections of sets in Euclidean space. Among other things we analyze their behavior on pairs of linear subspaces. This analysis shows that, when two linear subspaces meet at a very shallow angle, the known algorithms can be expected to converge very slowly. The linear case then allows us to analyze the behavior of these algorithms on nonlinear subspaces.We begin with the classical alternating projection algorithm, and then consider algorithms based on hybrid iterative maps, which are motivated by the HIO algorithms introduced by Fienup. We also include a brief analysis of the RAAR algorithm. We introduce nonorthogonal splitting of the ambient space, which have proved very useful for analyzing algorithms of this general type. In a final section we outline a new, noniterative method for phase retrieval that uses the Hilbert transform to directly. This approach requires a holographic modification to the standard experimental protocol, which we describe. The chapter closes with an appendix relating alternating projection to gradient flows.