The goal of this chapter is to examine the behavior of algorithms defined by hybrid iterative maps in the phase retrieval problem, per se. We begin by considering these maps in a variety of simple geometric situations, which demonstrate both the range of behaviors for iterates of these maps, and also how they are related to the local geometry near to the point of intersection. When these maps converge, they converge to points on a set called the center manifold. After consideration of the model problems, we turn to an analysis of the linearization of a hybrid map near to points on the center manifold. In a numerical study, we show that, even at an attractive fixed point, the linearized map may fail to be a contraction. Its eigenvalues are complex numbers with modulus less than one, but the basis of eigenvectors is very far from orthogonal. The chapter concludes with extensive numerical experiments exploring the complexities of hybrid iterative maps in realistic phase retrieval problems utilizing the support constraint.

This chapter summarizes the results of the previous three chapters on the theoretical foundations for the study of the phase retrieval problem.

We close this part of the book with a chapter examining the behavior of hybrid iterative maps after large numbers of iterates. The content of this chapter is rather speculative, consisting mostly of examples that illustrate various experimental phenomena. It is motivated by the observation that, except under very specific circumstances, the iterates of hybrid iterative maps do not converge. Rather, stagnation seems to occur with very high probability. The discussion in this chapter is not intended to suggest new algorithms, but rather to illustrate the extraordinary range, and beauty, of the dynamics that underlie stagnation.

This book consistently uses a variety of notational conventions that are intended to make the text more readable. As some are not entirely standard, or self-explanatory, we review them here.

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.

This chapter describes the contents of Part III of the book, which covers statistical properties of hybrid iterative maps, and a range of proposals for improving the outcome of phase retrieval experiments. These include suggestions for different experimental procedures, and different reconstruction algorithms, as well as methods of postprocessing collections of approximate reconstructions.