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。在数学中，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。该项目将集中在这一领域的几个主题，并继续发展几何层析成像开始于前两个国家科学基金会的建议。新的方向包括由平面到平面或平行截面上的投影体积的测量重建物体，与截面相关的稳定性问题，以及离散层析成像的系统发展(涉及有限集的反问题)。提出的技术的例子是使用最近发现的傅里叶变换公式和由度量值的$p$平均定义的特殊物体。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，只允许均匀的物体--物体内部的每个地方的密度都是相同的。医学上的一个例子是骨头或肾脏。人们可以利用这些信息来寻找更好的重建程序。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，电子显微镜中的一项新技术可以测量晶体中位于特定谱线上的原子数量。这位材料科学家想要根据这些信息重建晶体。在这种情况下，物体--水晶--在数学上由一组有限的点表示。该项目继续几何层析成像数学的几个方面的发展。

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.