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.

This chapter sets the stage for the study of algorithms used in the practical solution of the phase retrieval problem. It defines the types of maps used and many of the algorithms that will be considered in Chapters 7–10. Many of the concepts that arise in these chapters are introduced here as well.