Collaborative Research: Computational Ptychography: Fast Algorithms, Recovery Guarantees, and Applications to Bio-Imaging

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Ptychography refers to an imaging technique where overlapping regions of an object are illuminated, usually by placing a pinhole (and possibly a mask) between a light source and the object, and sequentially moving the pinhole. The resulting diffraction patterns are then sampled and used to calculate an approximate image of the object. The underlying physics of this imaging process dictates that one can only directly collect the intensity of the diffraction patterns, and not the critically important phase information. This makes the recovery of an accurate image extremely challenging. Nevertheless, through careful application of heuristic algorithms, practitioners have successfully employed these methods in a vast array of important applications such as the study of drug delivery mechanisms in complex bio-molecules, study of solar cells and battery chemistry, and the study of fracture dynamics in materials science. Despite these impressive results, several challenges remain, including the need to image larger and larger specimens at increasingly higher resolutions, and the growing size of datasets generated by a new generation of advanced imaging apparatus. This project seeks to develop fast, highly efficient, noise-robust, and mathematically rigorous computational methods in support of this next generation of high-throughput, high-resolution ptychographic imaging. The broader impacts of this project include curriculum development and training of students, including those from underrepresented groups, application of the computational methods to bio-imaging applications in the lab, and knowledge dissemination to raise the scientific literacy of the public.Mathematically, much progress has been recently made in understanding ptychographic imaging and in analyzing novel algorithms for signal recovery from phase-less measurements. However, these algorithms and their attendant analysis often assume one collects the modulus of generalized linear measurements, where the discretized measurements are highly random. In line with applications, a focus of this project is on designing practical measurement schemes of the type actually used in ptychographic imaging. Another major difficulty in realistic phase-less imaging applications is that the imaging system's measurement masks/probes can often only be approximately implemented and partially known. Hence, another major objective of this project is the development of novel theoretical and algorithmic results for the blind ptychography problem. In either case, the emphasis is on constructing provably accurate recovery algorithms that are fast enough to scale to large problems in multiple dimensions. These tasks require developing and using a broad range of mathematical tools. Techniques from time-frequency analysis, frame theory, spectral graph theory, high-dimensional probability, and compressive sensing will be necessary for analyzing the measurement schemes and for providing rigorous theoretical guarantees for the developed recovery algorithms. Finally, a key component of this project is the application of these computational methods to real ptychographic phase-less imaging setups and bio-imaging applications. More specifically, a novel wide-field, high-resolution lense-less on-chip microscopy platform will be designed, which puts the theoretical techniques developed as part of this project into practice.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.