Computed Tomography and Sampling

该项目涉及计算机层析成像的研究、先进的香农采样理论以及采样理论在层析成像中的应用。它结合了理论研究、数值分析和计算实验与真实世界的数据。计算机断层成像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研究的各种应用中。从数学上讲，它需要从它的线积分重构某个密度函数。普通的层析成像不是局部的：一个点的重建需要在远离该点的直线上进行积分。对于许多应用，如果只需要使用与某些感兴趣区域相交的线上的积分(“感兴趣区域层析成像”或“局部层析成像”)，则是非常可取的。香农采样理论在信号处理中起着基础性的作用。在层析成像中，采样定理被用来识别允许重建图像中的最大分辨率的有效数据收集方案，以及重建算法的误差分析。该项目在层析成像方面的研究涉及跨学科合作，并受到这样一个问题的启发：从给定的线积分集合中可以稳定地恢复密度函数的哪些特征？具体目标包括确定和分析三维层析成像的最佳采样方案；分析具有曲线上的源的三维局部层析成像的重建算法；进一步开发和数值分析二维高分辨率重建算法，以及开发感兴趣区域层析成像软件包。采样理论的研究受到层析成像应用的启发，涉及非等距数据的新采样定理的发展和应用；不同环境下混叠误差的估计；以及在局部紧阿贝尔群的框架下进一步探索采样概念的推广，它提供了许多不同应用的统一观点。

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.