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。在数学中，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。该项目将集中于该领域的几个主题，并继续发展在前两个NSF提案中开始的几何层析成像。新的方向包括从平面上投影的体积测量或平面上的并行剖面重建物体，与剖面相关的稳定性问题，以及离散层析成像的系统发展（涉及有关有限集的逆问题）。所提出的技术的例子是使用最近发现的傅立叶变换公式和由度量量的第p均值定义的特殊物体。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。在几何断层扫描中，只允许使用均匀物体——物体内部各处的密度都是一样的。医学上的一个例子是骨头或肾脏。人们可以利用这些信息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，电子显微镜中的一项新技术可以测量晶体在某些线上的原子数量。材料科学家想根据这些信息重建晶体。这个例子中的物体——晶体——在数学上是由一组有限的点来表示的。这个项目继续发展几何层析成像数学的几个方面。

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. Within mathematics, geometric tomography has links to functional analysis, convex geometry, Minkowski geometry, and combinatorics. The project will focus on several topics in this area and continue the development of geometric tomography begun in two previous NSF proposals. New directions include reconstruction of objects from measurements of volumes of projections onto planes or concurrent sections by planes, stability questions related to sections, and the systematic development of discrete tomography (involving inverse problems concerning finite sets). Examples of proposed techniques are the use of recently discovered Fourier transform formulas and of special bodies defined by $p$th means of metric quantities. 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 ar e 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 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 in electron microscopy allows measurement of the number of atoms in a crystal lying on certain lines. The material scientist wants to reconstruct the crystal from this information. The object in this case-the crystal-is mathematically represented by a finite set of points. This project continues the development of several aspects of the mathematics of geometric tomography.