Projection methods for high numerical aperture phase retrieval

TL;DR

This paper introduces a mathematical framework for applying projection algorithms to high-NA phase retrieval, analyzing their steps, convergence, and demonstrating improved performance over existing methods through numerical results.

Contribution

It extends projection algorithms to high-NA phase retrieval using a new framework and vectorial PSF, with convergence analysis and superior numerical performance.

Findings

Projection algorithms are effective for high-NA phase retrieval.

The framework provides closed-form prox-operators for the problem.

Projection methods outperform existing approaches in numerical tests.

Abstract

We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant prox-operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.