RUI: Geometric Tomography

RUI：几何断层扫描

• 批准号: 1103612
• 负责人: Richard Gardner
• 金额: $ 16.02万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103612&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. Geometric tomography has links to functional analysis, convex geometry, Minkowski geometry, and combinatorics. The project will continue the development of geometric tomography. New directions include fundamental new classification theorems for binary operations between, and symmetrization operations on, compact convex sets or star bodies; development of new tools for dealing with intersections of convex bodies, including an attack on a conjectured stronger form of the Brunn-Minkowski inequality; formulating and proving restricted Brunn-Minkowski inequalities, where the sets involved are contained in some fixed region of Euclidean space or the unit sphere; new algorithms for reconstruction from noisy data, in particular, from section functions, with proof of convergence for origin-symmetric convex bodies; progress towards a solution of Hammer's X-ray problem in higher dimensions and a conjecture on spherical X-rays of convex bodies; extension of previous results on convex and star bodies to log concave and radial measures; various other projects, for example renewed attacks on the discrete Aleksandrov projection problem and new inequalities of the Loomis-Whitney type or their reverses. 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 are 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 extra 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 called local stereology depends on measurements of planar sections of biological tissue; each section passes through a fixed point, usually the nucleus of a cell, and the measurements can be made optically rather than physically. This project continues the development of several aspects of the mathematics of geometric tomography. Also included is a program designed to stimulate undergraduate research.

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。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。该项目将继续发展几何层析成像。新的方向包括基本的新的分类定理之间的二元运算，对称化操作，紧凑的凸集或星星机构的发展，新的工具，用于处理凸体的交叉，包括攻击的一个更强形式的Brunn-Minkowski不等式;建立并证明了约束Brunn-Minkowski不等式，其中所涉及的集合包含在欧氏空间的某个固定区域或单位球面内;新的算法重建从嘈杂的数据，特别是，从部分功能，与收敛的证明为原点对称凸体;进展走向解决锤的X射线问题在更高的维度和一个猜想球面X射线的凸体;扩展以前的结果凸和星星机构，以记录凹和径向措施;各种其他项目，例如重新攻击离散亚历山德罗夫投影问题和新的不等式的卢米斯-惠特尼型或其逆转。还包括一个旨在刺激大学生研究的项目。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，只允许均匀的物体-物体的密度在它内部的任何地方都是相同的。医学上的一个例子是骨头或肾脏。人们可以使用这些额外的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，一种称为局部体视学的新技术依赖于对生物组织平面切片的测量;每个切片都通过一个固定点，通常是细胞核，并且可以通过光学而不是物理方式进行测量。该项目继续发展几何层析成像数学的几个方面。还包括一个旨在刺激本科生研究的项目。