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Hilbert transform with incomplete data and applications in Tomography and Optics

不完整数据的希尔伯特变换及其在层析成像和光学中的应用

基本信息

批准号：
1615124
负责人：
Alexander Katsevich
金额：
$ 33.5万
依托单位：
The University of Central Florida Board of Trustees
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615124&HistoricalAwards=false
关键词：
Hilbert transform incomplete data applications

项目摘要

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.

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。即使需要对患者体内的一个小器官进行可视化，整个横截面也会受到照射。开发用于从截断的 CT 数据（即通过仅照射患者体内感兴趣区域 (ROI) 获得的数据）进行图像重建的稳健算法将具有许多好处，例如减少许多 CT 扫描中对患者的辐射剂量，为新型多模态成像平台开辟道路等。在这个项目中，我们将研究从截断的 CT 数据重建算法的稳定性。类似的数学方法可用于研究光学成像中不完整数据的图像重建，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，目的是增加视野或减少测量量，同时保持 ROI 的空间分辨率。这项研究可能为开发廉价、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CT中，关键的分析工具是Gelfand-Graev公式，它将层析成像重建问题转化为从不完整数据中求出衰减系数的有限希尔伯特变换（FHT）的反演问题。当 CT 数据被截断时，衰减系数的重建通常是不唯一的。另一方面，缺失数据对 ROI 的贡献是分析性的，并且添加有关 ROI 内部衰减系数的先验知识可以恢复唯一性。光学成像是利用不完整数据进行 FHT 反演的另一个有用应用。受光谱学和全息术中压缩光学成像最新发展的启发，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，可以获得有关所研究样品的先验知识。因此，类似的方法可用于 CT 和光学成像。该研究的目的是开发用不完整数据反演 FHT 的理论和算法。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反演 FHT 的稳定性。 Riemann-Hilbert 问题的方法、微扰理论和 Titchmarsh-Weyl 理论是该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。