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

光刻是指一种成像技术，通常通过在光源和对象之间放置针孔(可能还有掩膜)并顺序移动针孔来照亮对象的重叠区域。然后对所得的衍射图进行采样，并使用其来计算物体的近似图像。这种成像过程的基本物理原理表明，人们只能直接收集衍射图的强度，而不能收集至关重要的位相信息。这使得恢复准确的图像具有极大的挑战性。然而，通过谨慎地应用启发式算法，实践者已经成功地将这些方法应用于大量重要的应用，如复杂生物分子中的药物释放机制的研究，太阳能电池和电池化学的研究，以及材料科学中的断裂动力学研究。尽管取得了这些令人印象深刻的成果，但仍然存在一些挑战，包括需要以越来越高的分辨率对越来越大的标本进行成像，以及新一代先进成像设备产生的数据集的规模越来越大。该项目致力于开发快速、高效、抗噪声且在数学上严格的计算方法，以支持这种下一代高通量、高分辨率的层析成像。该项目的更广泛影响包括对学生的课程开发和培训，包括来自代表性不足群体的学生，将计算方法应用于实验室中的生物成像应用，以及传播知识以提高公众的科学素养。从数学上讲，最近在理解层析成像和分析从无相位测量中恢复信号的新算法方面取得了很大进展。然而，这些算法及其随之而来的分析通常假设收集的是广义线性测量的模数，其中离散化的测量是高度随机的。结合实际应用，本项目的重点是设计实际用于层析成像的实用测量方案。实际无相位成像应用中的另一个主要困难是成像系统的测量掩模/探头通常只能大致实现和部分已知。因此，该项目的另一个主要目标是为盲层析问题开发新的理论和算法结果。在任何一种情况下，重点都是构建可证明准确的恢复算法，这些算法足够快，可以扩展到多个维度的大问题。这些任务需要开发和使用广泛的数学工具。来自时频分析、框架理论、谱图理论、高维概率和压缩传感的技术将是分析测量方案和为开发的恢复算法提供严格的理论保证所必需的。最后，该项目的一个关键组成部分是将这些计算方法应用到实际的层析无相位成像装置和生物成像应用中。更具体地说，将设计一种新型的广视场、高分辨率无透镜片上显微镜平台，将作为该项目一部分开发的理论技术付诸实践。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.