Designing and using prior knowledge for phase retrieval

TL;DR

This paper introduces a new algorithm for phase retrieval that leverages known parts of the signal, applicable in microscopy, and explains phenomena related to energy concentration in known regions.

Contribution

The paper presents a novel phase retrieval algorithm that does not require the known part to be on the boundary, expanding applicability beyond previous boundary-based methods.

Findings

Reconstruction speed improves with increased energy in the known part.

The new algorithm is effective even when the known region is not on the boundary.

The phenomena observed in prior work are explained through phase retrieval theory.

Abstract

In this work we develop an algorithm for signal reconstruction from the magnitude of its Fourier transform in a situation where some (non-zero) parts of the sought signal are known. Although our method does not assume that the known part comprises the boundary of the sought signal, this is often the case in microscopy: a specimen is placed inside a known mask, which can be thought of as a known light source that surrounds the unknown signal. Therefore, in the past, several algorithms were suggested that solve the phase retrieval problem assuming known boundary values. Unlike our method, these methods do rely on the fact that the known part is on the boundary. Besides the reconstruction method we give an explanation of the phenomena observed in previous work: the reconstruction is much faster when there is more energy concentrated in the known part. Quite surprisingly, this can be explained using our previous results on phase retrieval with approximately known Fourier phase.