Radon Transforms and Geometric Analysis and Tomography

氡气变换、几何分析和断层扫描

• 批准号: 9971828
• 负责人: Eric Grinberg
• 金额: $ 6.36万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971828&HistoricalAwards=false
• 关键词: Radon; Transforms; Geometric; Analysis; Tomography

AbstractAward: DMS-9971828Principal Investigator: Eric L. GrinbergThe P.I. will continue to study problems of analysis and geometryon homogeneous and non-homogeneous spaces. He will use techniquesof group representations when these are available, and microlocalanalysis along with integral transforms, and theirgeneralizations to not-necessarily homogeneous spaces. A majorfocus of his program will include Radon transforms on symmetricspaces, especially their injectivity and restrictionproperties. He will use symmetric spaces as a guide to analysison spaces with less regularity. He will also investigateproblems in geometric inequalities and also several complexvariables using ideas closely related to, but formally distinctfrom Radon transforms. Finally, he will consider orbitfoldanalogs of Radon transforms (or Radon transforms modulo finitegroups) by means of invariant polynomials and enumerationarguments. He will continue an active program of travel andseminar visitors that has had a significant impact on thescientific community in North Philadelphia.Imaging devices are used with increasing frequency in medicine,industry, science, and information technology. While engineering,computational, and ergonomic considerations are critical in thedesign, production and utilization of these devices, thefoundational operating principle is generallymathematical. Typically, a mathematical theorem says thatinformation may be recovered from a geometrically definedintegral transformation or partial differential equation. TheRadon transform played a critical role in A.Cormack's pioneeringwork on the CAT scanner. Today, this transform and itsgeneralizations play a unifying role in an expanding set ofbranches of mathematics. The P.I. will focus on the followingareas: 1) geometric inequalities, which relate properties likevolume, width, cross-sections shadow areas; 2) functions ofseveral complex variables, which may be viewed as data with greatregularity and which play an important role in mathematicalanalysis and physics; 3) homogeneous spaces, where the existenceof a great deal of symmetry is assumed and its exploitation isthe ultimate goal; 4) finite analogs, where the continuousharmonic analysis of smooth spaces is replaced by countingarguments involving finite and discrete sets

Abstractaward：DMS-9971828原理研究员：Eric L. Grinbergthe P.I.将继续研究分析问题和均匀和非均匀空间的遗传问题。当可用的情况下，他将使用组表示技术，微层分析以及整体变换以及它们对不必要的均匀空间的总体化。他的程序的一个重点将包括对对称空间的ra rantress transforts，尤其是它们的注射性和限制性。他将使用对称空间作为较少规律性的分析空间的指南。 他还将使用与rad ra距转换密切相关但正式不同的思想进行几何不平等和几种复杂变量的问题进行研究。最后，他将通过不变的多项式和EnumerationArguments的方式来考虑ra transfss的轨道变换（或ra trancors transice finiteGroups）。他将继续进行一项积极的旅行和网络访客计划，该计划对北费城的特权社区产生了重大影响。成像设备在医学，工业，科学和信息技术中的频率越来越高。尽管工程，计算和人体工程学的考虑在这些设备的特性，生产和利用中至关重要，但基本的工作原理通常是有害的。通常，数学定理说，可以从几何定义的综合转换或部分微分方程中恢复信息。 Theradon Transform在A.Cormack在CAT扫描仪上的开创性工作中发挥了关键作用。如今，这种转变及其总体化在不断扩展的数学组中起着统一的作用。 P.I. 将重点关注以下活动：1）几何不平等现象，这些不平等现象将类似的属性，宽度，横截面阴影区域相关联； 2）肿瘤复杂变量的功能，可以将其视为具有乳清性的数据，并且在数学分析和物理学中起着重要作用； 3）均匀的空间，假定存在很大的对称性，其剥削是最终目标； 4）有限类似物，其中涉及有限和离散集的计数替代了平滑空间的连续谐波分析