Computed Tomography and Sampling

该项目涉及计算机层析成像的研究、广义香农采样理论以及采样理论在层析成像中的应用。计算机层析成像是一种从大量的线积分重建密度函数的成像技术。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”)，则是非常可取的。一种特殊情况是局部层析成像，其中感兴趣的区域可能任意小。局部层析重建恢复了原始密度函数的一些但不是全部属性。采样定理为带限函数提供了精确的内插公式，在信号处理中起着重要的作用。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。如果像层析成像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，则内插不再精确，并且出现所谓的混叠误差。这类函数的抽样定理的有用性取决于对这一误差的良好估计的可用性。该项目的目标是开发高分辨率重建算法和数值分析，并开发包括局部层析成像在内的感兴趣区域层析成像的软件包；研究三维局部层析成像中的最佳采样；推导混叠误差的新的误差估计；以及在局部紧致阿贝尔群的框架内探索采样概念的推广，其提供了不同应用的统一视图，例如非等距采样、数值积分、多传感器反卷积和滤波器组。

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.