Applications of Fourier analysis to convex geometry and geometric tomography

傅里叶分析在凸几何和几何断层扫描中的应用

基本信息

批准号：
RGPIN-2014-03874
负责人：
Yaskin, Vladyslav
金额：
$ 1.31万
依托单位：
University of Alberta
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2017
资助国家：
加拿大
起止时间：
2017-01-01 至 2018-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635231
关键词：
Applications Fourier analysis convex geometry

项目摘要

The main theme of the proposed research project is centered around the study of sections, projections, volumes of convex and star bodies using methods of Fourier analysis. The area dealing with the retrieval of information about convex or star bodies based on data obtained from various measurements is known as geometric tomography. The book "Geometric Tomography" by R. J. Gardner gives an excellent overview of known results, techniques, and open problems in this area. The area lies at the intersection of several branches of mathematics: functional analysis, harmonic analysis, convex geometry, differential geometry, integral geometry etc. This explains the diversity of methods employed in geometric tomography. Our approach is based on techniques of harmonic analysis. Fundamental discoveries in this direction were made by A. Koldobsky at the end of the last century, who applied the Fourier transform to the study of sections of convex bodies (see the book "Fourier Analysis in Convex Geometry"). Since then such methods have been successfully used to solve a variety of problems in geometric tomography, convex geometry, geometric functional analysis etc. The goal of the proposed research project is to find new applications of such techniques. Even though this project concerns purely theoretical aspects of geometric tomography, results and methods from this area find applications in many scientific fields. One of the most well-known applications is computer assisted tomography, which allows to generate images from X-rays of human patients.

拟议的研究项目的主题围绕着使用傅立叶分析方法的部分，预测，凸形和星体的研究的研究。根据从各种测量结果获得的数据来检索有关凸或恒星物体信息的区域称为几何层析成像。 R. J. Gardner的《几何层析成像》一书概述了该领域的已知结果，技术和开放问题。该区域位于数学几个分支的交集：功能分析，谐波分析，凸几何，差异几何形状，积分几何等。这解释了几何层析成像中采用的方法的多样性。我们的方法基于谐波分析技术。上个世纪末，A。Koldobsky在这个方向上的基本发现将傅立叶变换应用于凸体部分的研究（请参阅《凸几何形状中的傅立叶分析》一书）。从那时起，这种方法已成功地用于解决几何层析成像，凸几何，几何功能分析等各种问题。拟议的研究项目的目的是找到此类技术的新应用。即使该项目涉及几何层析成像的理论方面，但该领域的结果和方法在许多科学领域中找到了应用。最著名的应用之一是计算机辅助断层扫描，它允许从人类患者的X射线射线产生图像。