Efficient Ptychography Reconstruction using the Hessian operator

TL;DR

This work introduces a Hessian-based, vectorization-free framework for efficient ptychography reconstruction, enabling simultaneous estimation of the object $\psi$, probe $p$, and, when desired, scan positions. By deriving the gradient, bilinear Hessian, and Hessian operator through a chain-rule for bilinear Hessians, the authors implement gradient-based, conjugate-gradient, and quasi-Newton schemes without forming the full Hessian. Synthetic and experimental near-field data demonstrate that second-order methods (notably BH-CG) converge an order of magnitude faster and with higher fidelity than first-order approaches like LSQML or Adam. The framework is extendable to far-field, multimode probes, batch processing, and 3D ptychography, offering practical avenues to accelerate high-resolution coherent imaging at synchrotrons and in-situ experiments.

Abstract

X-ray ptychography is a powerful and robust coherent imaging method providing access to the complex object and probe (illumination). Ptychography reconstruction is typically performed using first-order methods due to their computational efficiency. Higher-order methods, while potentially more accurate, are often prohibitively expensive in terms of computation. In this study, we present a mathematical framework for reconstruction using second-order information, derived from an efficient computation of the bilinear Hessian and Hessian operator. The formulation is provided for Gaussian based models, enabling the simultaneous reconstruction of the object, probe, and object positions. Synthetic data tests, along with experimental near-field ptychography data processing, demonstrate a ten-fold reduction in computation time compared to first-order methods. The derived formulas for computing the Hessians, along with the strategies for incorporating them into optimization schemes, are well-structured and easily adaptable to various ptychography problem formulations.

Figures (10)

• Figure 1: Experimental setups for (a) near-field ptychography, where diffraction patterns are recorded in the Fresnel regime and modeled using the Fresnel transform of the scattered/exit wave after propagation through the object, and (b) far-field ptychography, where diffraction patterns are recorded in the Fraunhofer regime and modeled using the Fourier transform of the scattered/exit wave after propagation through the object. In both cases, beam focusing can be achieved using Kirkpatrick-Baez (KB) mirrors or a zone plate. The schematic setups also illustrate examples of the recovered probe amplitude and phase distributions.
• Figure 2: Settings and the probe for near-field ptychography simulations.
• Figure 3: Objective functional value vs. iteration number (left) and vs. computation time (right) when reconstructing the object and probe from synthetic data.
• Figure 4: Reconstruction results (object phase) for a synthetic siemens star after 6 s execution of different methods. Corresponding states of the objective functional are marked with line 'visualization' in Figure \ref{['fig:fig_syn_plots']}.
• Figure 5: Probes recovered during reconstruction with BH-CG of a) synthetic data from Figure \ref{['fig:settings_syn']}, b) experimental Siemens star data from Figure \ref{['fig:settings_real']}, and c) experimental coded aperture data from Figure \ref{['fig:settings_real_ca']}. The red dashed frame in c) outlines the detector size.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2