We consider an instance of the phase-retrieval problem, where one wishes to recover a signal (viewed as a vector) from the noisy magnitudes of its inner products with locally supported vectors. Such measurements arise, for example, in ptychography, which is an imaging technique used in lense-less X-ray microscopes and in optical microscopes with increased fields of view.

Starting with the setup where the signal is one-dimensional, we present theoretical and numerical results on an approach that has two important properties. First, it allows deterministic measurement constructions, which we give examples of. Second, it uses a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by simply calculating an eigenvector. We also present a natural extension to the two-dimensional setting.

This is joint work with M. Iwen, B. Preskitts, and A. Viswanathan. Speaker(s): Rayan Saab (University of California, San Diego)