Blind Ptychographic Phase Retrieval via Convergent Alternating Direction Method of Multipliers

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.

Ptychographic phase retrieval by proximal algorithms

“…For example, adding a regularization term with denoiser has shown suppressed noise in the reconstructed object function [13,38]. With an additional constraint on the probe, it was demonstrated that the periodic aliasing effect seen in grid scans could be mitigated [18]. There is still a lot of room for improvement, and they will be the future work.…”

Advanced denoising for X-ray ptychography

“…During the last decades, researchers have developed several schemes to solve the BP problem. Arguably, the most popular ones are extended Ptychographic Iterative Engine (ePIE) [3], Difference Map [13,16], Maximum Likelihood (ML) method [17], Proximal Splitting algorithm [14], Relaxed Averaged Alternating Reflections (RAAR [18]) based algorithms [19], and generalized Alternating Direction Method of Multipliers (ADMM) [20][21][22]) based BP [15]. Although some implementations of those methods present ad hoc solutions for structural noise removal, there is no formal analysis or in-depth characterization of such experimental problems, even though they can be critical to achieve robust high-resolution images, specially from weakly scattering or low contrast specimens.…”

Iterative Joint Ptychography-Tomography with Total Variation Regularization

“…In order to deal with data contaminated by different noise types, based on the maximum likelihood estimation (MLE), a more general mapping B(·, ·) : R m + × R m + → R + was introduced in [12]. It measures the distance between the recovered intensity g ∈ R m + and the collected intensity f ∈ R m + as…”

“…√ ·, · · denote the element-wise square root and division of a vector, respectively, 1 represents a vector whose elements are all ones, and ·, · denotes the L 2 inner product in Euclidean space. Note that the penalization parameter ε was introduced [12] to guarantee the Lipschitz differentiability of the objective function in order to prove the convergence.…”

“…The smaller the η factor the bigger the noise level. SNR intensity [12] is used to measure the noise level, which is defined as: SNR intensity := . We show the reconstruction results with peak levels 0.1 and 1 in Fig.…”