This chapter summarizes the results of the book and describes some directions for future research.

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.

The Introduction defines the classical, phase retrieval problem, i.e., the use of auxiliary information to recover the unmeasured phase of the Fourier transform from samples of its magnitude, and the discrete, classical, phase retrieval problem, which is the model that is studied in this book. It reviews well-known facts about the phase retrieval problem, including the Hayes' uniqueness theorem, and the idea of trivial associates. Hayes' theorem states that the phase retrieval problem generically has a unique solution, up to trivial associates. It summarizes the main results in the book, including the description of the tangent and normal bundles of a magnitude torus, the various types of auxiliary data commonly employed in phase retrieval (the support constraint, the nonnegativity constraint), the problem of ill conditioning, and defines the standard algorithms used to address the phase retrieval problem in practical applications. It establishes notational conventions used throughout the text, and the approach taken to numerical experiments. It closes with appendices on the factorization of polynomials in more than one variable, and the concept of conditioning.

In the earlier chapters of the book, we show that reconstruction algorithms often stagnate at a substantial distance from an exact reconstruction. In this chapter we study the statistical properties of the set of images that result from running such an algorithm with many choices of random starting points. These collections display interesting statistical features: the distribution of errors is multimodal, reflecting the different ways in which an algorithm can stagnate. Sometimes the observable data error is well correlated with the unobservable exact reconstruction error and sometimes it is not. The empirical variances in the approximate phases can be determined on a frequency-by-frequency basis, and provide a good predictor of the accuracy of the mean value of these approximate phases. Algorithms provide a much better estimate of the phases of Fourier coefficients of large magnitude, than for those of small magnitude. As a reflection of the multi-modal character of the data-error distribution, averaging reconstructed images with the smallest data error can improve the accuracy of the reconstruction, but averaging most or all reconstructions does not.

A mathematical problem is well posed if, for all appropriate data, it has a unique solution, and this solution depends continuously, in a useful sense, on that data. The phase retrieval problem does not usually have a unique solution, but the set of solutions generically consists of trivial associates, which are, for practical purposes, equivalent. This chapter addresses various ways in which the phase retrieval is not well posed. It begins with a theorem demonstrating that the solution to the phase retrieval problem, using support as the auxiliary data, is locally defined, near a given solution, by a Lipschitz map if and only if the intersection is transversal. In the previous chapter, we have shown that this is rarely the case. Near a nontransversal intersection this map is, at best, Holder-1/2, and so the phase retrieval problem is not well posed. We then consider the question of the uniqueness of the solution, in finite precision arithmetic, showing several distinct ways in which this can fail.

Another constraint often used in the phase retrieval problem to get an essentially unique solution, is the assumption that the unknown is image is real valued, and nonnegative. This assumption alone does not guarantee a unique solution, even up to trivial associates. In this chapter we prove that, if the image is nonnegative, then the phase retrieval problem does generically have a unique solution, up to trivial associates, provided that the autocorrelation image has sufficiently small support. This condition is verifiable from Fourier magnitude data alone. We study the geometry near intersection points of a magnitude torus A and the nonnegative orthant, B+. This naturally leads to a study of the L1-norm on the tangent space of A at the point of intersection, and a criterion for such an intersection to be transversal. The chapter closes with numerical examples examining the failure of transversality.

In this chapter, we consider several ideas that might lead to improved reconstructions in coherent diffraction imaging. To avoid the conditions that render the inverse problem ill conditioned, and the usual iterative algorithms nonconvergent, we consider approaches that entail (1) a modification of the sample preparation, (2) a new experimental modality, which uses a noniterative holographic Hilbert transform method to reconstruct the image, or (3) a geometric Newton-type algorithm. The final section describes the detailed implementation of the holographic Hilbert transform method. All of these proposals are illustrated with numerical experiments.