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Stochastic Multigrid Method for Blind Ptychographic Phase Retrieval

Borong Zhang, Junjing Deng, Yi Jiang, Zichao Wendy Di

TL;DR

This work tackles blind ptychographic phase retrieval, where both the object $\bm{z}$ and the probe $\bm{Q}$ must be recovered from phaseless diffraction data. It introduces eMAGPIE, a stochastic multigrid framework that minimizes a majorizing quadratic surrogate $\widetilde{\Phi}(\bm{Q},\bm{z};j)$ of the exit-wave misfit, with a guaranteed descent at each iteration. A joint object–probe update is derived by combining one-variable minimizers through a geometric-mean, phase-aligned rule, and the reconstruction is accelerated by a multilevel surrogate (MAGPIE) that couples fine- and coarse-grid problems. Across simulated and real datasets, the method achieves lower data misfit and phase error, produces smoother, artifact-reduced phase reconstructions, and demonstrates robustness to lower overlaps and higher noise than prior rPIE-based approaches.

Abstract

We present eMAGPIE (extended Multilevel-Adaptive-Guided Ptychographic Iterative Engine), a stochastic multigrid method for blind ptychographic phase retrieval that jointly recovers the object and the probe. We recast the task as the iterative minimization of a quadratic surrogate that majorizes the exit-wave misfit. From this surrogate, we derive closed-form updates, combined in a geometric-mean, phase-aligned joint step, yielding a simultaneous update of the object and probe with guaranteed descent of the sampled surrogate. This formulation naturally admits a multigrid acceleration that speeds up convergence. In experiments, eMAGPIE attains lower data misfit and phase error at comparable compute budgets and produces smoother, artifact-reduced phase reconstructions.

Stochastic Multigrid Method for Blind Ptychographic Phase Retrieval

TL;DR

This work tackles blind ptychographic phase retrieval, where both the object and the probe must be recovered from phaseless diffraction data. It introduces eMAGPIE, a stochastic multigrid framework that minimizes a majorizing quadratic surrogate of the exit-wave misfit, with a guaranteed descent at each iteration. A joint object–probe update is derived by combining one-variable minimizers through a geometric-mean, phase-aligned rule, and the reconstruction is accelerated by a multilevel surrogate (MAGPIE) that couples fine- and coarse-grid problems. Across simulated and real datasets, the method achieves lower data misfit and phase error, produces smoother, artifact-reduced phase reconstructions, and demonstrates robustness to lower overlaps and higher noise than prior rPIE-based approaches.

Abstract

We present eMAGPIE (extended Multilevel-Adaptive-Guided Ptychographic Iterative Engine), a stochastic multigrid method for blind ptychographic phase retrieval that jointly recovers the object and the probe. We recast the task as the iterative minimization of a quadratic surrogate that majorizes the exit-wave misfit. From this surrogate, we derive closed-form updates, combined in a geometric-mean, phase-aligned joint step, yielding a simultaneous update of the object and probe with guaranteed descent of the sampled surrogate. This formulation naturally admits a multigrid acceleration that speeds up convergence. In experiments, eMAGPIE attains lower data misfit and phase error at comparable compute budgets and produces smoother, artifact-reduced phase reconstructions.

Paper Structure

This paper contains 26 sections, 2 theorems, 36 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Organization.
  3. Preliminaries
  4. 2D blind ptychographic phase retrieval
  5. Ptychogrpahical Iterative Engine (PIE)
  6. Surrogate Minimization via a Multigrid Solver
  7. Properties of the Surrogate Minimization
  8. Extended MAGPIE (eMAGPIE)
  9. Numerical Examples
  10. Experiment Setup
  11. Probe
  12. Object
  13. Probe and object initialization
  14. Noise level
  15. Quality metrics
  16. ...and 11 more sections

Key Result

Proposition 3.1

Let $\widetilde{\Phi}(\bm{Q},\bm{z};j)$ be the surrogate in Eq. eq:quadratic_surrogate, and let $\Phi(\bm{Q},\bm{z})$ be the exit-wave misfit in Eq. eq:exit_misfit. Then:

Figures (11)

  • Figure 1: Experimental setup and data acquisition for ptychography.
  • Figure 2: Illustration of the majorization property (adapted from SMM_graph).
  • Figure 3:
  • Figure 4: Synthetic data, $50\%$ overlap. Top: convergence curves for rPIE and eMAGPIE showing the data misfit residual and the reconstruction error versus iteration. Bottom: final reconstructions for each method (magnitude and demeaned phase), together with their corresponding error maps against the ground truth. Both algorithms use the same object regularization constant $\alpha = 0.01$ and ePIE-style probe update, and early stopping based on the moving average of the last five residuals.
  • Figure 5: Synthetic data, $75\%$ overlap. Top: convergence curves for rPIE and eMAGPIE showing the data misfit residual and the reconstruction error versus iteration. Bottom: final reconstructions for each method (magnitude and demeaned phase), together with their corresponding error maps against the ground truth. Both algorithms use the same object regularization constant $\alpha = 0.01$ and ePIE-style probe update, and early stopping based on the moving average of the last five residuals.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Definition 2.1: Revised Exit Wave
  • Proposition 3.1: Majorization
  • proof
  • Proposition 3.2
  • proof