Computed Tomography and Sampling

本项目涉及计算机断层扫描、广义香农采样理论以及采样理论在断层扫描中的应用。计算机断层扫描是一种利用大量线积分重建密度函数的成像技术。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”），则是非常理想的。一个特殊的例子是局部断层扫描，其中感兴趣的区域可能是任意小的。局部层析重建恢复了原始密度函数的部分性质，但不是全部性质。采样定理为带限函数提供了精确的插值公式，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。如果像断层扫描和其他应用中的情况一样，将采样定理应用于非带宽限制函数，则插值不再精确，并发生所谓的混叠误差。这些函数的抽样定理的有用性取决于对该误差的良好估计的可用性。该项目的目标是高分辨率重建算法的开发和数值分析，以及用于包括局部断层扫描在内的感兴趣区域断层扫描的软件包的开发；三维局部层析成像中最优采样方法的研究新的混叠误差估计的推导；并探索了在局部紧阿贝尔群框架下采样概念的推广，为非等距采样、数值积分、多传感器反褶积和滤波器组等不同应用提供了统一的观点。

This project involves research in computed tomography, generalized Shannon sampling theory, and applications of sampling theory in tomography. Computed tomography is an imaging technique which reconstructs a density function from a large number of 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"). A special case is local tomography where the region of interest may be arbitrarily small. Local tomographic reconstructions recover some but not all properties of the original density function. Sampling theorems provide exact interpolation formulas for bandlimited functions and play 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. If, as is the case in tomography and other applications, sampling theorems are applied to non-bandlimited functions, interpolation is no longer exact and a so-called aliasing error occurs. The usefulness of sampling theorems for such functions depends on the availability of good estimates for this error. The goals of this project are development and numerical analysis of high-resolution reconstruction algorithms and development of a software package for region-of-interest tomography including local tomography; investigation of optimal sampling in three-dimensional local tomography; derivation of new error estimates for the aliasing error; and exploration of a generalization of the sampling concept in the framework of locally compact abelian groups which provides a unified view of diverse applications such as non-equidistant sampling, numerical integration, multisensor deconvolution, and filter banks.