Background Denoising for Ptychography via Wigner Distribution Deconvolution

TL;DR

This work tackles background noise in ptychographic phase retrieval by extending the Wigner Distribution Deconvolution (WDD) framework to denoise measurements with shift-invariant background. The authors propose two reconstruction procedures, one for general objects and one tailored to phase objects, and prove a uniqueness guarantee for almost every phase object under realistic window conditions. The approach exploits the shift-invariance of the background to separate corrupted frequencies, and uses the redundancy of the high-dimensional WDD representation to recover the lost information, followed by standard WDD-based object recovery. Numerical experiments on synthetic 2D data demonstrate robust noise suppression and competitive reconstruction accuracy, highlighting practical impact for improving ptychographic imaging in noisy environments.

Abstract

Ptychography is a computational imaging technique that aims to reconstruct the object of interest from a set of diffraction patterns. Each of these is obtained by a localized illumination of the object, which is shifted after each illumination to cover its whole domain. As in the resulting measurements the phase information is lost, ptychography gives rise to solving a phase retrieval problem. In this work, we consider ptychographic measurements corrupted with background noise, a type of additive noise that is independent of the shift, i.e., it is the same for all diffraction patterns. Two algorithms are provided, for arbitrary objects and for so-called phase objects that do not absorb the light but only scatter it. For the second type, a uniqueness of reconstruction is established for almost every object. Our approach is based on the Wigner Distribution Deconvolution, which lifts the object to a higher-dimensional matrix space where the recovery can be reformulated as a linear problem. Background noise only affects a few equations of the linear system that are therefore discarded. The lost information is then restored using redundancy in the higher-dimensional space. Keywords: phase retrieval, ptychography, background noise, Wigner Distribution Deconvolution, uniqueness of reconstruction.

Figures (5)

• Figure 1: (a) Ground-truth object, (b) ground-truth phase object and (c) window.
• Figure 2: (a) Background and (b) exemplary diffraction pattern perturbed by background noise in logarithmic scale.
• Figure 3: Comparison of the performance of the vanilla WDD algorithm (\ref{['alg: algorithm_WDD']}) without and with preprocessing, and \ref{['alg: algorithm general object']} for different noise levels and different amounts of diagonals incorporated into the experiment (all or $\gamma = 3$). Below the reconstruction results, find the respective (relative reconstruction error/measurement error).
• Figure 4: Comparison of the performance of the vanilla WDD algorithm (\ref{['alg: algorithm_WDD']}) without and with preprocessing, and \ref{['alg: algorithm 2']} for different noise levels and different amounts of diagonals incorporated into the experiment (all or $\gamma = 3$). Below the reconstruction results, find the respective (relative reconstruction error / measurement error).
• Figure 5: Geometric visualization of \ref{['l: shift_1_rank_1_no_unique_recovery']}. By \ref{['prop: abs_zero_freq_phase_object']}, $\vert f_0^q\vert$ is known. This guarantees that the possible Fourier coefficients $f^q_0$ lie on the blue circle. By definition, $f_0^{q} + a_\ell = d (F^{-1} f^q)_\ell$ and $\vert (F^{-1} f^q)_\ell \vert = 1$. Hence, $f_0^q + a_\ell$ has to lie on the green circle. Consequently, the possible solutions $f^q_0$ lie on the orange circle, or, more precisely, on its intersections with the blue circle. It can be shown that neither the blue and the green circle coincide nor the orange circle is tangent to the blue circle, so that there are precisely two intersection points, corresponding to two solutions. The lightgray dotted lines are parallels in the direction of $a_0$ containing the solutions $f^q_0$. Their two intersection with the green circle show that there are two distinct values the entries of the diagonal can take.

Theorems & Definitions (28)

• Lemma 1
• Corollary 2
• Theorem 3: perlmutter2021inverting
• Theorem 4: bojarovska2016phase
• Theorem 5: bojarovska2016phase
• Corollary 6
• Theorem 7: preskitt2018phase
• Proposition 8
• proof : Proof.
• Proposition 9