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.

The phase retrieval problem is the problem of finding the intersections between a high-dimensional magnitude torus, A, in a Euclidean space and a second set, B, defined by the auxiliary data. The problem is difficult because the set A is not convex. In this chapter we give a very explicit description of the tangent and normal bundles of the torus in both the image-space and Fourier representations. Using this description, we show that, for practical support constraints, the intersections between A and B are not usually transversal. The chapter concludes with numerical examples demonstrating this phenomenon, with various types of images and support constraints. The chapter closes with appendices on the tangent and normal bundles of submanifolds of Euclidean spaces, and a fast algorithm for finding the orthogonal projections onto the tangent and normal bundles of a magnitude torus.

This chapter repeats much of the analysis of the previous chapter but using the assumption that the image is nonnegative, and its autocorrelation image has sufficiently small support rather than an estimate for the support of the image itself. In this case the auxiliary set is B+, the set consisting of nonnegative images, and the intersections of A and B+ lie on the boundary of B+, which is convex but not smooth. This complicates the notion of transversality, as was discussed in Chapter 4. We then present many numerical examples exploring the behavior of algorithms using nonnegativty and also the assumption that the unknown image has a given l1-norm. The chapter concludes with an appendix describing an efficient algorithm for finding the l2-nearest point on the boundary of an l1-ball.