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FlowTIE: Flow-based Transport of Intensity Equation for Phase Gradient Estimation from 4D-STEM Data

Arya Bangun, Maximilian Töllner, Xuan Zhao, Christian Kübel, Hanno Scharr

TL;DR

FlowTIE addresses phase retrieval in 4D-STEM by integrating the Transport of Intensity Equation with a flow-based neural model to learn phase gradients under a continuity constraint. It uses defocused triplets to approximate $\partial I/\partial z$ and optimizes a composite loss that enforces physically plausible flow, continuity, and phase reconstruction. On simulated 4D-STEM data for GaAs and SrTiO$_3$, FlowTIE achieves lower phase MSE than classical TIE and a gradient-descent baseline, especially for thicker specimens, while maintaining sub-second runtimes on GPUs. The method offers a practical, scalable, physics-informed framework that can be extended to experimental data and other microscopy modalities.

Abstract

We introduce FlowTIE, a neural-network-based framework for phase reconstruction from 4D-Scanning Transmission Electron Microscopy (STEM) data, which integrates the Transport of Intensity Equation (TIE) with a flow-based representation of the phase gradient. This formulation allows the model to bridge data-driven learning with physics-based priors, improving robustness under dynamical scattering conditions for thick specimen. The validation on simulated datasets of crystalline materials, benchmarking to classical TIE and gradient-based optimization methods are presented. The results demonstrate that FlowTIE improves phase reconstruction accuracy, fast, and can be integrated with a thick specimen model, namely multislice method.

FlowTIE: Flow-based Transport of Intensity Equation for Phase Gradient Estimation from 4D-STEM Data

TL;DR

FlowTIE addresses phase retrieval in 4D-STEM by integrating the Transport of Intensity Equation with a flow-based neural model to learn phase gradients under a continuity constraint. It uses defocused triplets to approximate and optimizes a composite loss that enforces physically plausible flow, continuity, and phase reconstruction. On simulated 4D-STEM data for GaAs and SrTiO, FlowTIE achieves lower phase MSE than classical TIE and a gradient-descent baseline, especially for thicker specimens, while maintaining sub-second runtimes on GPUs. The method offers a practical, scalable, physics-informed framework that can be extended to experimental data and other microscopy modalities.

Abstract

We introduce FlowTIE, a neural-network-based framework for phase reconstruction from 4D-Scanning Transmission Electron Microscopy (STEM) data, which integrates the Transport of Intensity Equation (TIE) with a flow-based representation of the phase gradient. This formulation allows the model to bridge data-driven learning with physics-based priors, improving robustness under dynamical scattering conditions for thick specimen. The validation on simulated datasets of crystalline materials, benchmarking to classical TIE and gradient-based optimization methods are presented. The results demonstrate that FlowTIE improves phase reconstruction accuracy, fast, and can be integrated with a thick specimen model, namely multislice method.

Paper Structure

This paper contains 12 sections, 10 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Notation
  4. Multislice Model
  5. Transport of Intensity Equation (TIE)
  6. FlowTIE
  7. Numerical Results
  8. Dataset and Model Architecture
  9. Description of the comparative methods
  10. Estimation of Projected Phase and Gradient Phase
  11. Computational Time
  12. Conclusion and Future Work

Figures (3)

  • Figure 1: 4D-STEM diffraction data and vector field of the phase are simulated from multislice models, which serve as training data for the neural network model shown in (b). After training, the model can generate vector fields based on the intensity distribution of diffraction patterns (c).
  • Figure 2: Material used for test data, (a) Gallium Arsenide (GaAs) projection over $z-$ axis, (b) 3D view of GaAs, (c) Strontium Titanate (SrTiO$_3$) projection over $z-$ axis, (d) 3D view of SrTiO$_3$. The unit cell dimension for both SrTiO$_3$ and GaAs are $(a,b,c)= ( 5.6533,5.6533,5.6533)$ and $(a,b,c)= (3.905, 3.905, 3.905)$ all units are in Å, respectively
  • Figure 3: Reconstruction result (a) Ground truth of phase gradient, (b) FlowTIE, (c) Ground truth of projected phase, (d) TIE, (e) FlowTIE, (f) Gradient Descent.