Computed Tomography and Sampling

本项目涉及计算机断层扫描、先进的香农采样理论以及采样理论在断层扫描中的应用。它结合了理论调查、数值分析和计算实验与现实世界的数据。计算机断层扫描是一种产生不透明物体内部图像的技术。它被广泛应用于从诊断放射学到量子光学的研究。从数学上讲，它需要从线积分中重建一个密度函数。普通断层扫描不是局部的：在一个点上重建需要在远离该点的直线上积分。对于许多应用来说，如果只需要使用与某个感兴趣区域相交的直线上的积分（“感兴趣区域断层扫描”或“局部断层扫描”），这是非常可取的。香农采样理论在信号处理中起着重要的作用。在断层扫描中，采样定理被用来识别有效的数据收集方案，允许在重建图像中获得最大的分辨率，以及重建算法的误差分析。该项目的断层扫描研究涉及跨学科合作，并受到“密度函数的哪些特征可以从给定的线积分集合中稳定恢复？”这个问题的启发。具体目标包括三维断层扫描中最佳采样方案的识别和分析；具有曲线源的三维局部层析成像重建算法分析进一步发展和二维高分辨率重建算法的数值分析，并开发了一个软件包的兴趣区域断层扫描。采样理论的研究受到层析成像应用的刺激，涉及新的非等距数据采样定理的发展和应用；估计不同设置下的混叠误差；并进一步探讨了在局部紧阿贝尔群的框架下对抽样概念的推广，为许多不同的应用提供了一个统一的观点。

This project involves research in computed tomography, advanced Shannon sampling theory, and applications of sampling theory in tomography. It combines theoretical investigations, numerical analysis, and computational experiments with real world data. Computed tomography is a technique which produces images of the interiors of opaque objects. It is widely used in applications ranging from diagnostic radiology to research in quantum optics. Mathematically it requires the reconstruction of a certain density function from its line integrals. Ordinary tomography is not local: reconstruction at a point requires integrals over lines far from that point. For a number of applications it is highly desirable if only integrals over lines intersecting some region of interest need to be used ("region-of-interest tomography" or "local tomography"). Shannon sampling theory plays a fundamental role in signal processing. In tomography sampling theorems are used to identify efficient data collection schemes allowing for maximum resolution in the reconstructed image, as well as for error analysis of reconstruction algorithms. The project's research in tomography involves interdisciplinary collaboration and is inspired by the question "What features of the density function can be stably recovered from a given collection of its line integrals?". Specific goals include the identification and analysis of optimal sampling schemes in three-dimensional tomography; the analysis of reconstruction algorithms for three-dimensional local tomography with sources on a curve; further development and numerical analysis of high-resolution reconstruction algorithms in two dimensions, and development of a software package for region-of-interest tomography. The research in sampling theory is stimulated by applications in tomography and involves the development and application of new sampling theorems for non-equidistant data;estimates for the aliasing error in various settings; and further exploration of a generalization of the sampling concept in the framework of locally compact abelian groups which provides a unified view of a number of diverse applications.