Computed Tomography and Sampling

本课题的研究内容涉及计算机层析成像、先进的香农采样理论、采样理论在层析成像中的应用以及图像重建与图像处理的结合。计算机x射线断层摄影术产生不透明物体的图像。基本的数学问题包括从其线积分的有限多个测量值重构密度函数f(X)。目前该领域的大部分焦点都集中在全三维成像上，这是受到近年来重大算法进步的刺激。香农采样理论为带限函数提供了精确的内插公式，在信号处理中起着基础性的作用。它在层析成像中被用来识别允许重建图像中的最大分辨率的有效采样方案，以及重建算法的误差分析。本项目中提出的研究处于这些领域的交叉点，并受到应用的推动。其相互关联的主题包括新采样定理的推导；3D断层扫描的最佳数据获取；对最近开发的3D断层扫描算法进行严格的数学和实用的分析；二维和三维的局部断层扫描；高分辨率重建；以及断层扫描重建与图像处理的密切交互，其动机是高分辨率重建通常带有更多噪声，并可通过最先进的图像处理进一步改进。在追求这些主题时，该项目将把重点放在跨学科合作上。计算机X射线断层扫描(CT)是一种日益流行的技术，用于不透明物体的二维和三维(3D)成像。物体被照射，并且根据所传输的辐射的测量来计算图像。除了在医学成像中的著名用途外，层析成像还在科学、工程和技术中发现了越来越多的应用。在这个项目中进行的研究将有助于使这项技术更加有效。它将发展和使用香农型采样定理来确定3D层析成像的最佳数据采集方案，并分析和改进图像重建和图像处理的算法。这将有助于开发全三维图像重建方法，这种方法需要最少的测量次数，并具有高精度、最佳分辨率和最小噪声。该项目采取的一般方法旨在将对基础的研究与由断层摄影术用户提出的实际挑战直接推动的跨学科努力相结合。该项目还将努力为研究生提供以多维成像为中心的跨学科环境。

AbstractThe research proposed in this project involves Computed Tomography, Advanced Shannon Sampling Theory, applications of sampling theory in tomography, and combination of image reconstruction with image processing. Computed x-ray tomography produces images of opaque objects. The underlying mathematical problem consists of reconstructing a density function f(x) from finitely many measurements of its line integrals. Much of the present focus of the field is on fully three-dimensional imaging, stimulated by major algorithmic advances in recent years. Shannon sampling theory provides exact interpolation formulas for band-limited functions and plays a fundamental role in signal processing. It is used in tomography to identify efficient sampling schemes that allow for maximum resolution in the reconstructed images, as well as for error analysis of reconstruction algorithms. The research proposed in this project lies in the intersection of these areas and is motivated by applications. Its interconnected themes include the derivation of new sampling theorems; optimal data acquisition in 3D tomography; mathematically rigorous and practically useful analysis of recently developed 3D tomographic algorithms; local tomography in two and three dimensions; high-resolution reconstructions; and the close interaction of tomographic reconstruction with image processing, motivated by the experience that high-resolution reconstructions often carry more noise and can be further improved by state of the art image processing. In pursuing these themes the project will place an emphasis on interdisciplinary collaboration.Computed x-ray tomography (CT) is an increasingly popular technique for two- and three-dimensional (3D) imaging of opaque objects. An object is irradiated and the image is computed from measurements of the transmitted radiation. Beyond its famous use in medical imaging, tomography has found an increasing number of applications in science, engineering, and technology. The research undertaken in this project will contribute to make this technique even more efficient. It will develop and use Shannon type sampling theorems to identify optimal data acquisition schemes for 3D tomography and to analyze and improve algorithms for image reconstruction and image processing. This will contribute to the development of fully three-dimensional image reconstruction methods that require a minimal number of measurements and have high accuracy, optimal resolution, and minimal noise. The general approach taken in this project aims to unite research on fundamentals with interdisciplinary efforts driven directly by practical challenges presented by users of tomography. The project will also involve efforts to provide graduate students with an interdisciplinary environment centered around multi-dimensional imaging.