Coded Aperture Ptychography: Uniqueness and Reconstruction

TL;DR

This paper proves the uniqueness of solutions in coded aperture ptychography with random masks, analyzes convergence of reconstruction algorithms, and introduces a minimalist scheme with practical insights and numerical validation.

Contribution

It establishes theoretical uniqueness results, analyzes algorithm convergence, and proposes a minimalist overlapping mask scheme with empirical performance evaluation.

Findings

DR algorithm has a unique fixed point in the object domain.

A simple criterion distinguishes the true solution among fixed points.

Numerical experiments confirm the impact of mask features and measurement noise on reconstruction quality.

Abstract

Uniqueness of solution is proved for any ptychographic scheme with a random masks under a minimum overlap condition and local geometric convergence analysis is given for the alternating projection (AP) and Douglas-Rachford (DR) algorithms. DR is shown to possess a unique fixed point in the object domain and for AP a simple criterion for distinguishing the true solution among possibly many fixed points is given. A minimalist scheme is proposed where the adjacent masks overlap 50\% of area and each pixel of the object is illuminated by exactly four times during the whole measurement process. Such a scheme is conveniently parametrized by the number $q$ of shifted masks in each direction. The lower bound $1-C/q^2$ is proved for the geometric convergence rate of the minimalist scheme, predicting a poor performance with large $q$ which is confirmed by numerical experiments. Extensive numerical experiments are performed to explore what the general features of a well-performing mask are like, what the best-performing values of $q$ for a given mask are, how robust the minimalist scheme is with respect to measurement noise and what the significant factors affecting the noise stability are.