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射线结晶学、低温电子显微镜(Cryo-EM)、薄层摄影术和超短脉冲检测。虽然X射线结晶学是确定分子三维原子结构的流行方法，但这一理论的数学基础还没有完全发展起来。冷冻-EM是一种最新的生物成像技术，即样品(通常是蛋白质)在液体溶液中闪速冷冻，然后用低强度电子束引导。平版印刷术是一种通过移动掩模扫描样本来获得高分辨率图像的方法。它可以应用在许多环境中，包括活细胞的成像。超短脉冲探测是化学反应等时间分辨超快现象的关键组成部分。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、医学和工程等领域具有广泛应用的潜力。本研究也将为研究生提供培训机会。这项研究将利用不变量理论和代数几何的技术，为成像和光学的四种方法建立一个数学框架。X射线结晶学相当于从其功率谱中恢复信号。这可以说是最具挑战性的相位恢复问题。第一个项目旨在制定严格的数学标准，以确定离散周期信号何时可以从其功率谱中恢复。由于低温电磁测量的噪声水平非常高，构建高分辨率图像需要大量数据。第二个项目将使用不变量理论中的技术来估计低温EM和相关实验的样本复杂性。可以使用短时傅立叶变换(STFT)对层析测量进行建模。第三个项目将获得信号恢复所需的短时傅立叶变换测量次数的信息论界限。最后一个项目将专注于一种使用多模光纤的脉冲表征新技术的数学基础。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。

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.