RUI: Geometric Tomography

几何层析成像利用几何物体的平面切面和平面上的投影数据来获取这些物体的信息。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逆问题更有可能有唯一解。通常，未知物体具有均匀密度的先验知识可以被利用来检索比其他方法更多的信息。这可能导致算法在可用的测量值较少时更有效，并且对测量误差或噪声不太敏感。几何断层扫描与功能分析、凸几何、闵可夫斯基几何和组合学有关。该项目将继续发展几何层析成像。新的研究方向包括涉及（对偶）仿射quermassintegral的猜想强仿射不等式的研究；几何层析成像，用高斯测度或其他测度代替勒贝格测度；关于离散层析成像和离散布伦-闵可夫斯基理论的几个项目；用固定的有限x射线集以任意精度重建凸体的Hammer x射线问题的完全解；利用噪声支持函数重构任意维凸体的新算法以及几何层析成像中其他新的重建算法，包括经验过程理论应用的收敛性证明。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。他们从几个不同的方向拍摄x射线，并综合这些信息来创建身体部分的二维图像。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；重建后的图像只是近似值，为了用同样的程序获得更好的图像，人们必须拍摄更多的x光片，这就造成了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。人们可以利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宽得多。任何通过线或面来测量均匀物体的部分或其在线或面上的阴影的测量都可以考虑。正因为如此，它与其他领域有许多联系，无论是在数学中（与凸几何，无孔或凹痕的形状的几何有很大的重叠）还是在外部。例如，一种称为局部立体学的新技术依赖于对生物组织平面切片的测量；每一部分都经过一个固定点，通常是细胞核，测量可以用光学方法而不是物理方法。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.