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。新的方向包括研究涉及(对偶)仿射查询积分的猜想的强仿射不等式；几何层析成像中的勒贝格度量被高斯或其他测量取代；离散层析成像和离散Brunn-Minkowski理论的几个项目；Hammer的X射线问题的完整解决方案，从固定的有限X射线集重建凸体到任意精度；从噪声支持函数重建任意维度的凸体的新算法；以及几何层析成像中的其他新的重建算法，包括应用经验过程理论证明收敛。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 the study of conjectured strong affine inequalities involving (dual) affine quermassintegrals; geometric tomography where Lebesgue measure is replaced by Gaussian or other measures; several projects on discrete tomography and a discrete Brunn-Minkowski theory; a complete solution to Hammer's X-ray problem to reconstruct convex bodies to arbitrary accuracy from a fixed finite set of X-rays; a new algorithm for reconstructing convex bodies in any dimension from noisy support functions; and other new reconstruction algorithms in geometric tomography, including proofs of convergence by an application of the theory of empirical processes. 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.