Invariant Theory and Imaging

该项目旨在研究成像和光学中四种重要方法的数学基础：X射线晶体学，低温电子显微镜（Cryo-EM），重叠关联学和超短脉冲检测。虽然X射线晶体学是确定分子三维原子结构的普遍方法，但该理论的数学基础尚未完全发展。Cryo-EM是生物成像中的最新技术，其中样品（通常是蛋白质）在液体溶液中快速冷冻，然后用低强度电子束引导。重叠关联成像是一种通过用移动掩模扫描样品来获得高分辨率图像的方法。它可以应用于许多环境中，包括活细胞的成像。超短脉冲探测是时间分辨超快现象（如化学反应）的关键组成部分。这项研究将创建一套统一的数学技术来理解这些方法。这些技术的发展将在生物化学、医学和工程等不同领域具有无数应用的潜力。这项研究也将为研究生提供一个培训机会。这项研究将利用不变量理论和代数几何的技术来建立成像和光学中四种方法的数学框架。X射线晶体学对应于从其功率谱中恢复信号。这可以说是最具挑战性的相位检索问题。第一个项目旨在制定严格的数学标准，以确定何时可以从其功率谱中恢复离散周期信号。由于冷冻EM测量具有非常高的噪声水平，因此构建高分辨率图像需要大量数据。第二个项目将使用来自不变量理论的技术来估计冷冻EM和相关实验的样本复杂性。重叠关联测量可以使用短时傅里叶变换（STFT）来建模。第三个项目将获得信号恢复所需的STFT测量数量的信息理论界。最后一个项目将侧重于使用多模光纤进行脉冲表征的新技术的数学基础。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to investigate the mathematical foundations of four important methods in imaging and optics: X-ray crystallography, Cryogenic electron microscopy (Cryo-EM), ptychography, and ultra-short pulse detection. Although X-ray crystallography is the prevalent method for determining the 3-dimensional atomic structure of molecules, the mathematical foundations of this theory are not fully developed. Cryo-EM is a recent technique in biological imaging where a sample (typically a protein) is flash-frozen in a liquid solution and then directed with a low-intensity electron beam. Ptychography is a method of obtaining a high-resolution image by scanning across a sample with a moving mask. It can be applied in a number of contexts, including the imaging of live cells. Ultra-short pulse detection is a key ingredient in time-resolved ultrafast phenomena such as chemical reactions. This research will create a unified set of mathematical techniques to understand these methods. The development of these techniques will have the potential for a myriad of applications in areas as diverse as biochemistry, medicine, and engineering. This research will also provide a training opportunity for graduate students. This research will exploit techniques from invariant theory and algebraic geometry to build a mathematical framework for four methods in imaging and optics. X-ray crystallography corresponds to recovering a signal from its power spectrum. This is arguably the most challenging phase-retrieval problem. The first project aims to develop rigorous mathematical criteria to determine when a discrete periodic signal can be recovered from its power spectrum. Because cryo-EM measurements have a very high noise level, constructing a high-resolution image requires massive data. The second project will use techniques from invariant theory to estimate the sample complexity of cryo-EM and related experiments. Ptychographic measurements can be modeled using the short-time Fourier transform (STFT). The third project will obtain information-theoretic bounds on the number of STFT measurements needed for signal recovery. The last project will focus on the mathematical foundations of a novel technique for pulse characterization using multi-mode fibers.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.