Phase Retrieval in Hardy Space

TL;DR

This paper addresses phase retrieval in Hardy spaces by transforming the problem into factorization, developing algorithms for outer and inner functions, and introducing a sparse representation method.

Contribution

It introduces a novel approach to phase retrieval in Hardy spaces using Nevanlinna factorization and develops algorithms for reconstructing functions from intensity measurements.

Findings

Successful reconstruction algorithms for Hardy space functions from intensity data

Effective methods for identifying zero points of Blaschke products

Introduction of a sparse representation via unwinding adaptive Fourier decomposition

Abstract

This paper concerns the study of reconstructing a function $f$ in the Hardy space of the unit disc $\D$ from intensity measurements $|f(z)|,\ z\in \D.$ It's known as the problem of phase retrieval. We transform it into solving the corresponding outer and inner function through the Nevanlinna factorization Theorem. The outer function will be established based on the mechanical quadrature method, while we use two different ways to find out the zero points of Blashcke product, thereby computing the inner function under the assumption that the singular inner function part is trivial. Then the concrete algorithms and illustrative experiments follow. Finally, we give a sparse representation of $f$ by introducing the unwinding adaptive Fourier decomposition.