Computed Tomography and Sampling

计算机断层扫描和采样

• 批准号: 0206752
• 负责人: Adel Faridani
• 金额: $ 20.81万
• 依托单位: Oregon State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0206752&HistoricalAwards=false
• 关键词: Computed; Tomography; Sampling

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.

本计画内容包括电脑断层摄影、先进香农取样理论及取样理论于断层摄影之应用。它结合了理论研究，数值分析和计算实验与真实的世界的数据。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研究的应用。在数学上，它需要从其线积分重建一定的密度函数。普通的层析成像不是局部的：在一个点的重建需要在远离该点的线上积分。对于许多应用，如果仅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"），则是非常期望的。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，采样定理用于识别有效的数据收集方案，允许在重建图像中的最大分辨率，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，并受到以下问题的启发："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断层摄影的最佳取样方案;分析曲线上有源的三维局部断层摄影的重建算法;进一步发展和数值分析二维高分辨率重建算法，以及开发感兴趣区域断层摄影的软件包。在采样理论的研究是刺激应用在层析成像，并涉及新的采样定理的发展和应用的非等距数据;估计在各种设置的混叠误差;和进一步探索的采样概念的泛化的框架内的局部紧凑阿贝尔群，提供了一个统一的观点，一些不同的应用。