Fast Partial Fourier Transforms for Large-Scale Ptychography

TL;DR

The paper tackles the computational bottleneck of large-scale ptychographic phase retrieval by introducing a fast partial Fourier Transform (PFT) to warm up the Ptychographic Iterative Engine (ePIE). By first capturing low-frequency, large-scale features with the PFT-based PIE and then refining with standard FFT-based PIE, the method accelerates convergence while preserving reconstruction quality. A differentiable PyTorch implementation of the PFT enables seamless differentiation through the operator, facilitating integration with learning-based and gradient-based optimization. Numerical experiments in both non-blind and blind settings demonstrate reduced time-to-solution on large-scale problems (e.g., up to 16384×16384) with comparable or improved reconstruction metrics, highlighting practical impact for high-resolution, large-field ptychography.

Abstract

Ptychography is a popular imaging technique that combines diffractive imaging with scanning microscopy. The technique consists of a coherent beam that is scanned across an object in a series of overlapping positions, leading to reliable and improved reconstructions. Ptychographic microscopes allow for large fields to be imaged at high resolution at additional computational expense. In this work, we explore the use of the fast Partial Fourier Transforms (PFTs), which efficiently compute Fourier coefficients corresponding to low frequencies. The core idea is to use the PFT in a plug-and-play manner to warm-start existing ptychography algorithms such as the ptychographic iterative engine (PIE). This approach reduces the computational budget required to solve the ptychography problem. Our numerical results show that our scheme accelerates the convergence of traditional solvers without sacrificing quality of reconstruction.

Figures (16)

• Figure 1: Illustration of the ptychography experiment with three overlapping scans.
• Figure 2: Illustration of coefficients computed by the PFT. On the left, we show the full FFT applied to an image of size $512 \times 512$. The red square shows the frequencies that the PFT computes on the right without requiring one to take the FFT of the original image and then cropping as one might naively attempt.
• Figure 3: The ground truth used to simulate data in numerical experiments. The baboon image is used as the magnitude and the cameraman image is used as the phase of the object of interest.
• Figure 4: Scanning positions generated by the illumination windows described in Section \ref{['subsubsec: nonblind_distribution_relerrs']}. For the small problem where $n_1 = n_2 = 512$ (Section \ref{['subsubsec: nonblind_experimental_setup']}), each illumination window $Q_i$ has size $256 \times 256$ and shifts $128$ pixels at a time. Similarly, for the large-scale problem where $n_1 = n_2 = 16384$ (Section \ref{['subsubsec: nonblind_large_scale']}, each illumination window $Q_i$ has size $8192 \times 8192$ and shifts $4096$ pixels at a time. In both setups, there is a $50\%$ overlap between consecutive probes.
• Figure 5: Histogram of the final reconstruction relative errors. The blue histogram shows the relative error frequency for PIE and the orange histogram shows the relative error frequency for hybrid PIE. a) shows the relative error of the reconstructed object, b) shows relative errors of only the magnitude of the object, and c) shows the relative errors of the phase of the object.