RUI: Geometric Tomography

几何层析成像使用关于几何对象的按平面的截面和平面上的投影的数据来获得关于这些对象的信息。后者包括一般紧集，但通常是凸体、多面体、星形体或有限集。这种设置的一个优点是，反问题更有可能有唯一的解。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。这可以导致算法在可用测量较少时更有效，并且对测量误差或噪声不那么敏感。几何层析与泛函分析、凸几何、明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技术。新的方向包括：紧凸集或星体之间的二元运算和对称化运算的基本新分类定理；处理凸体相交的新工具的开发，包括对猜想的更强形式的Brunn-Minkowski不等式的攻击；提出和证明受限的Brunn-Minkowski不等式，其中涉及的集合包含在欧几里德空间或单位球面的某个固定区域；由噪声数据重建的新算法，特别是从截面函数重建的新算法，以及原点对称凸体的收敛证明；解决Hammer的高维X射线问题和关于凸体的球面X射线的猜想的进展；将以前关于凸体和星体的结果推广到对数凹和径向测量；各种其他项目，例如，对离散的亚历山大罗夫投影问题的新攻击和新的Loomis-Whitney类型的不等式或它们的逆转。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。他们在许多不同的方向上拍摄X射线，并合成信息来创建身体部分的二维部分的图像。这一过程背后的数学原理被称为计算机化断层扫描。这是非常成功的，但并不完美；重建的图像只是近似值，要用同样的程序获得更好的图像，必须拍摄更多的X光，这会导致更大的费用和可能的副作用。在几何层析成像中，只允许均匀的对象-对象的密度在其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。任何涉及由直线或平面或其在直线或平面上的阴影的均匀对象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，无论是在数学上(与凸几何学有很大重叠，即没有洞或凹痕的形状的几何学)，还是在外部。例如，一种名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；每一切片都通过一个固定点，通常是细胞的细胞核，测量可以通过光学而不是物理方式进行。该项目继续几何层析成像数学的几个方面的发展。还包括一个旨在刺激本科生研究的项目。

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.