RUI: Geometric Tomography

RUI：几何断层扫描

• 批准号: 9802388
• 负责人: Richard Gardner
• 金额: $ 11.55万
• 依托单位: Western Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802388&HistoricalAwards=false
• 关键词: RUI; Geometric; Tomography

Geometric tomography uses data concerning sections by planes and projections on planes of geometric objects to obtain information about these objects. The latter include general compact sets, but often they are convex bodies, polytopes, star-shaped bodies, or finite sets. One advantage of this setting is that it becomes more probable that inverse problems have a unique solution. Generally, the a priori knowledge that the unknown object is of uniform density can be exploited to retrieve more information than would otherwise be possible. This can lead to algorithms that are more effective when few measurements are available, and less sensitive to measurement errors or noise. Within mathematics, geometric tomography has links to functional analysis, convex geometry, Minkowski geometry, and combinatorics. The project will focus on several topics in this area and continue the development of geometric tomography begun in two previous NSF proposals. New directions include reconstruction of objects from measurements of volumes of projections onto planes or concurrent sections by planes, stability questions related to sections, and the systematic development of discrete tomography (involving inverse problems concerning finite sets). Examples of proposed techniques are the use of recently discovered Fourier transform formulas and of special bodies defined by $p$th means of metric quantities. Also included is a program designed to stimulate undergraduate research. CAT scanners are machines that save lives daily. They take X-rays in a number of different directions, and synthesize the information to create an image of a two-dimensional section of part of the body. The mathematics behind this process is called computerized tomography. It is very successful, but not perfect; the reconstructed image is only approximate, and to get a better picture with the same procedure one has to take more X-rays, causing greater expense and likelihood of side effects. In geometric tomography, only homogeneous objects ar e allowed-the density of the object is the same everywhere inside it. An example from medicine would be a bone or a kidney. One can use this information to find better reconstruction procedures. The scope of geometric tomography is actually much wider. Any measurement involving sections of a homogeneous object by lines or planes or its shadows on lines or planes can be considered. Because of this, it has many links to other areas, both in mathematics (there is a large overlap with convex geometry, the geometry of shapes without holes or dents) and outside. For example, a new technique in electron microscopy allows measurement of the number of atoms in a crystal lying on certain lines. The material scientist wants to reconstruct the crystal from this information. The object in this case-the crystal-is mathematically represented by a finite set of points. This project continues the development of several aspects of the mathematics of geometric tomography.

几何层析成像使用关于几何对象的平面上的截面和投影的数据来获得关于这些对象的信息。后者包括一般的紧集，但它们通常是凸体、多面体、星形体或有限集。这种设置的一个优点是，它变得更有可能，逆问题有一个唯一的解决方案。 通常，可以利用未知对象具有均匀密度的先验知识来检索比否则可能的更多的信息。这可以导致算法在很少测量可用时更有效，并且对测量误差或噪声不太敏感。在数学中，几何层析成像与泛函分析、凸几何、闵可夫斯基几何和组合学有联系。该项目将集中在这一领域的几个主题，并继续在两个以前的NSF提案开始的几何层析成像的发展。新的方向包括重建的物体从测量体积的投影到平面或并发部分的平面，稳定性问题有关的部分，和系统的发展离散断层扫描（涉及逆问题有关有限集）。所提出的技术的例子是使用最近发现的傅立叶变换公式和特殊机构所定义的$p$th手段的度量数量。还包括一个旨在刺激本科生研究的项目。 CAT扫描仪是每天拯救生命的机器。他们从多个不同的方向拍摄X射线，并将信息合成为身体部分的二维截面图像。这一过程背后的数学原理被称为计算机断层扫描。它非常成功，但并不完美;重建的图像只是近似的，并且为了用相同的程序获得更好的图像，必须拍摄更多的X射线，导致更大的费用和副作用的可能性。在几何层析成像中，只允许使用均匀的物体--物体内部各处的密度都是相同的。医学上的一个例子是骨头或肾脏。人们可以使用这些信息来找到更好的重建程序。几何层析成像的范围实际上要广得多。可以考虑任何涉及由线或平面或其在线或平面上的阴影的均质物体的截面的测量。正因为如此，它与其他领域有许多联系，无论是在数学（与凸几何，没有孔或凹痕的形状的几何有很大的重叠）还是外部。例如，电子显微镜中的一种新技术可以测量晶体中位于某些线上的原子数。材料科学家想根据这些信息重建晶体。在这种情况下的对象-晶体-是数学上表示的一组有限的点。 该项目继续发展几何层析成像数学的几个方面。