Geometric Tomography

摘要- DMS 0203527几何层析成像使用有关几何对象的平面和平面投影的截面数据来获得有关这些对象的信息。 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，它使逆问题更有可能有唯一的解。通常，可以利用未知物体具有均匀密度的先验知识来检索比其他情况下更多的信息。 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Lutwak的双Brunn-Minkowski理论的延伸，通过引入集合的矩;工作朝着一个连贯的discrete Brunn-Minkowski理论，适用于有限集，用基数取代体积;和调查的可能性的Brunn-Minkowski理论的能力。在相关的应用中，该计划是巩固几何体层摄影术和一种称为局部体视学的实用方法之间的联系，并解决一个关于有限集的逆问题，并将其应用于医学成像。期望能有新的发现，并可应用于从截面到原点的体积重建星星体。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光片，并将信息合成为身体部分的二维截面图像。 这个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，这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。 人们可以利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宽得多。 任何涉及到由线或平面或其在线或平面上的阴影的均匀物体的截面的测量都可以被考虑。正因为如此，它与其他领域有许多联系，无论是在数学（与凸几何有很大的重叠，没有孔或凹痕的形状的几何）还是外部。 例如，一种被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;每一个切片都通过一个固定点，通常是细胞核，并且可以用光学方法而不是物理方法进行测量。这个项目继续发展几何层析成像数学的几个方面。还包括一个旨在促进本科生研究的项目。

ABSTRACT - DMS 0203527Geometric tomography uses data concerning sections by planes andprojections on planes of geometric objects to obtain informationabout 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 becomesmore probable that inverse problems have a unique solution.Generally, the a priori knowledge that the unknown object is ofuniform density can be exploited to retrieve more informationthan would otherwise be possible. This can lead to algorithmsthat are more effective when few measurements are available, andless sensitive to measurement errors or noise. Geometrictomography has links to functional analysis, convex geometry,Minkowski geometry, and combinatorics. The project will continuethe development of geometric tomography. New directions includean extension of Lutwak's dual Brunn-Minkowski theory by theintroduction of moments of sets; work towards a coherent discreteBrunn-Minkowski theory that applies to finite sets, with volumereplaced by cardinality; and an investigation into thepossibility of a Brunn-Minkowski theory for capacity. In relatedapplications, the plan is to consolidate links between geometrictomography and a practical method called local stereology, and tosolve an inverse problem concerning finite sets with anapplication to medical imaging. New discoveries aboutintersection bodies are anticipated, with an application to thereconstruction of star bodies from the volumes of their sectionsthrough the origin. Also included is a program designed tostimulate undergraduate research.CAT scanners are machines that save lives daily. They takeX-rays in a number of different directions, and synthesize theinformation to create an image of a two-dimensional section ofpart of the body. The mathematics behind this process is calledcomputerized tomography. It is very successful, but not perfect;the reconstructed image is only approximate, and to get a betterpicture with the same procedure one has to take more X-rays,causing greater expense and likelihood of side effects. Ingeometric tomography, only homogeneous objects are allowed - thedensity of the object is the same everywhere inside it. Anexample from medicine would be a bone or a kidney. One can usethis information to find better reconstruction procedures. Thescope of geometric tomography is actually much wider. Anymeasurement involving sections of a homogeneous object by linesor planes or its shadows on lines or planes can be considered.Because of this, it has many links to other areas, both inmathematics (there is a large overlap with convex geometry, thegeometry of shapes without holes or dents) and outside. Forexample, a new technique called local stereology depends onmeasurements of planar sections of biological tissue; each sectionpasses through a fixed point, usually the nucleus of a cell, andthe measurements can be made optically rather than physically.This project continues the development of several aspects of themathematics of geometric tomography. Also included is a programdesigned to stimulate undergraduate research.