Hilbert transform with incomplete data and applications in Tomography and Optics

在计算机断层扫描（CT）中，探测器通常覆盖患者的整个横截面。即使当患者体内的一个小器官需要可视化时，整个横截面也会受到照射。从截断的CT数据（即，通过仅照射患者体内的感兴趣区域（ROI）获得的数据）将具有许多益处，例如在许多CT扫描中减少对患者的辐射剂量，开辟新的多模态成像平台等。在本项目中，我们将研究截断CT数据重建算法的稳定性。类似的数学方法对于从光学成像中的不完整数据重建图像的研究是有用的，例如在显微镜和光学计量应用中。我们将把为CT开发的方法应用于光学成像，目标是增加视场或减少测量量，同时保持ROI的空间分辨率。这项研究可能为开发廉价的大视场直接相位成像系统铺平道路，这反过来又将有利于细胞生物学研究以及数字病理学和细胞跟踪等应用。在截断数据的CT中，关键的分析工具是Gelfand-Graev公式，该公式将层析重建问题转化为从不完整数据反演衰减系数的有限Hilbert变换（FHT）的问题。当CT数据被截断时，衰减系数的重建常常是非唯一的。另一方面，缺失数据对ROI的贡献是解析的，并且添加关于ROI内的衰减系数的先验知识恢复唯一性。另一个应用，其中反演的FHT与不完整的数据是有用的是光学成像。受光谱学和全息术中压缩光学成像的最新发展的启发，基于FHT的光学系统具有实现高分辨率、高速相衬成像的潜力。在许多显微成像应用中，可以获得关于被研究样品的先验知识。因此，类似的方法可以用于CT和光学成像。研究的目的是发展理论和算法，反演FHT与不完整的数据。我们将通过找到相关算子的奇异值分解来估计不完整数据下FHT反演的稳定性。黎曼-希尔伯特问题的方法、微扰理论和Titchmarsh-Weyl理论是本项目中将使用的一些数学工具。我们还计划开发和测试相应的重建算法，并在模拟和实验数据上进行测试。

In computed tomography (CT), detectors usually cover the entire cross-section of the patient. Even when a small organ inside the patient needs to be visualized, the entire cross-section is irradiated. Development of robust algorithms for image reconstruction from truncated CT data (i.e., the data obtained by irradiating only a region of interest (ROI) inside the patient) will have numerous benefits, such as reducing the radiation dose to patients in many CT scans, opening the way to novel multimodality imaging platforms, etc. In this project, we will investigate stability of algorithms for reconstruction from truncated CT data. Similar mathematical approaches are useful for the study of image reconstruction from incomplete data in optical imaging, such as in microscopy and optical metrology applications. We will apply the methods developed for CT to optical imaging with the goal of increasing the field-of-view or reducing the amount of measurements, while maintaining spatial resolution in the ROI. This research may pave the way to the development of inexpensive, large-field-of-view direct phase imaging systems, which, in turn, would benefit cell biology research, and applications such as digital pathology and cell-tracking. In CT with truncated data, the key analytical tool is the Gelfand-Graev formula, which transforms the tomographic reconstruction problem to the problem of inverting the finite Hilbert transform (FHT) of the attenuation coefficient from incomplete data. When CT data are truncated, reconstruction of the attenuation coefficient is frequently non-unique. On the other hand, the contribution of the missing data to the ROI is analytic, and adding prior knowledge about the attenuation coefficient inside the ROI restores uniqueness. Another application where inversion of the FHT with incomplete data is useful is optical imaging. Inspired by the recent development of compressive optical imaging in spectroscopy and holography, optical systems based on the FHT have the potential to achieve high resolution, high speed phase contrast imaging. In a number of microscopy imaging applications, prior knowledge about the sample being investigated is possible to obtain. Thus, similar approaches can be used both for CT and for optical imaging. The objective of the research is to develop theory and algorithms for inverting the FHT with incomplete data. We will estimate stability of inverting the FHT with incomplete data by finding the singular value decomposition of the relevant operators. The method of the Riemann-Hilbert problem, perturbation theory, and the Titchmarsh-Weyl theory, are some of the mathematical tools that will be used in this project. We also plan to develop and test the corresponding reconstruction algorithms and test them on simulated and experimental data.