Multigrid Optimization for Large-Scale Ptychographic Phase Retrieval

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 the cost of additional computational expense. In this work, we propose a multigrid-based optimization framework to reduce the computational burdens of large-scale ptychographic phase retrieval. Our proposed method exploits the inherent hierarchical structures in ptychography through tailored restriction and prolongation operators for the object and data domains. Our numerical results show that our proposed scheme accelerates the convergence of its underlying solver and outperforms the Ptychographic Iterative Engine (PIE), a workhorse in the optics community.

Feasibility-based Fixed Point Networks

“…the zero vector). To identify a faithful mapping N Θ as in (7), we solve a training problem. This is modeled as finding weights that minimize an expected loss, which is typically solved using optimization methods like SGD [69] and Adam [70].…”

“…In practice, the expectation in ( 9) is approximated using a finite subset of data, which is referred to as training data. In addition to minimizing over the training data, we aim for (7) to hold for a set of testing data that was not used during training (which tests the network's ability to generalize).…”

“…Standard results [74][75][76] can be used to guarantee existence [7] of fixed points of nonexpansive T Θ . Iteratively applying T Θ produces a convergent sequence (Theorem 2.1).…”

“…Inverse problems arise in numerous applications such as medical imaging [1][2][3][4], phase retrieval [5][6][7], geophysics [8][9][10][11][12][13], and machine learning [14][15][16][17][18]. The goal of inverse problems is to recover a signal [2] u d from a collection of indirect noisy measurements d. These quantities are typically related by a linear mapping A via…”

Adaptive Uncertainty-Weighted ADMM for Distributed Optimization

“…(1.1) in a distributed manner, where f j : R n → R is smooth and convex, and g : R n → R is proximable. Problems of the form (1.1) arise in many contexts, including machine learning [1,32,30,17], statistics [14], phase retrieval [9,2,6], geophysics [11,7], and image processing [26,13]. These problems often contain many samples, i.e., N is often very large, making the optimization computationally challenging.…”