Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers

Huibin Chang,Stefano Marchesini,Pablo Enfedaque

doi:10.1137/18m1188446

Abstract

Ptychography has risen as a reference X-ray imaging technique: it achieves resolutions of one billionth of a meter, macroscopic field of view, or the capability to retrieve chemical or magnetic contrast, among other features. A ptychographyic reconstruction is normally formulated as a blind phase retrieval problem, where both the image (sample) and the probe (illumination) have to be recovered from phaseless measured data. In this article we address a nonlinear least squares model for the blind ptychography problem with constraints on the image and the probe by maximum likelihood estimation of the Poisson noise model. We formulate a variant model that incorporates the information of phaseless measurements of the probe to eliminate possible artifacts. Next, we propose a generalized alternating direction method of multipliers designed for the proposed nonconvex models with convergence guarantee under mild conditions, where their subproblems can be solved by fast element-wise operations. Numerically, the proposed algorithm outperforms state-of-the-art algorithms in both speed and image quality.

Highlights

Ptychographic phase retrieval (Ptycho-PR) [33, 35, 26] is an increasingly popular imaging technique used in scientific fields as diverse as condensed matter physics, cell biology, and materials science, among others
In order to deal with data contaminated by different types of noise, based on the maximum likelihood estimation (MLE), more general mappings [8] to measure the distance between the recovered intensity g ∈ Rm+ and the collected noisy intensity f ∈ Rm+ have been extensively studied for the PR problem: (2.1)
The R-factors of alternating direction method of multipliers (ADMM) are smaller than those obtained by proximal alternating linearized minimization (PALM) and DR, and the signal-to-noise ratio (SNR) of the recovery results of ADMM are higher than those obtained by PALM, which is consistent with the results shown in Figures 7 and 8

Summary

Introduction

Ptychographic phase retrieval (Ptycho-PR) [33, 35, 26] is an increasingly popular imaging technique used in scientific fields as diverse as condensed matter physics, cell biology, and materials science, among others. Instead of directly solving the quadratic multidimensional systems in (1.1), following [34] for nonblind Ptycho-PR, a nonlinear least squares model for BP-PR can be given as below min ω∈Cm ,u∈Cn. In order to deal with data contaminated by different types of noise, based on the MLE, more general mappings [8] to measure the distance between the recovered intensity g ∈ Rm+ and the collected noisy intensity f ∈ Rm+ have been extensively studied for the PR problem:. The generalized ADMM will be adopted to solve the proposed models, which permits bigger stepsizes by avoiding directly calculating the gradient of the objective functional such that fast convergence speed is gained. A natural scheme to solve the above saddle point problem is to split them, which consists of four-step iterations for the generalized ADMM (only the subproblems w.r.t. ω or u have proximal terms), as follows:. 3: Compute uk+1 by (3.17) where M 2k satisfies (3.16). 4: Compute z1k+1, z2k+1 by (3.18)

5: Update multipliers as

Extensive tests

Conclusions

M2 with