Tomography and Integral Geometry

项目：断层扫描和积分几何PI： 埃里克托德昆托，塔夫茨大学DMS-0200788摘要：昆托教授将继续研究层析成像和Radon变换的数学分析问题。 他将开发和改进奇异性检测算法（包括有限数据Lambda CT和小波方法），并将其应用于电子显微镜和有限数据工业和医疗X射线断层扫描，目标是在真实数据上测试算法并将其用于扫描仪。 Quinto教授将继续开发一种适用于浅水声纳的边界检测方法，他将开始为合成孔径雷达开发这些方法。 理论基础将基于昆托教授的纯数学研究。 拉东变换有密切的关系，调和分析，偏微分方程，复杂的分析，他将使用积分几何证明定理在这些领域。 主要研究者将证明由多项式扩展定义的Radon变换的唯一性和支持定理，包括欧氏空间和其他设置中的球面变换。 他和教授Agranovsky将使用这些结果来表征零套解的波动方程的结晶域和其他集。 零集，即解始终为零的集合，是重要的，也是难以描述的。 他将证明唯一性定理的拉东变换领域和其他表面，并使用这些解决莫雷拉和平均值问题的流形。 他将证明X射线变换的支持定理。这项研究涵盖应用数学和纯数学：层析成像和积分几何。 纯理论研究将用于开发、理解和证明应用算法，而应用问题将激发大部分纯理论研究。 将为火箭和其他物体的非破坏性评价开发计算机断层摄影术（CT）算法。 这些算法将使科学家能够检测火箭主体的裂缝，火箭燃料中的气穴以及火箭出口锥体的分层。作为联合工业项目的一部分，将为电子显微镜开发有限的数据算法。 我们的目标是产生一个算法，可靠地产生精确的图片SIDEC电子显微镜。 他将开发纯数学，以展示算法的工作效果以及它们的局限性。 这些纯粹的数学基础是必要的，以确保他（或任何其他）的重建是正确的，而不是侥幸的猜测。 声纳数据可以被建模为球体（声音的球面波阵面）的平均值，昆托教授将证明这些平均值的定理。 他将使用这些定理来开发简单的算法来绘制海底地图，并找到海洋中物体的边界，并将其应用于雷达。 这种纯数学本身就很有趣。 昆托教授将证明有关球面平均值的定理，这些定理将用于理解波动方程（描述声音和光线行为的方程）。波动方程描述了鼓头的运动，他将指定鼓上哪些点根本不运动。 最后，他将证明他将应用于纯数学领域的复分析和调和分析的定理。

PROJECT: Tomography and Integral Geometry PI: Eric Todd Quinto, Tufts UniversityDMS-0200788ABSTRACT: Professor Quinto will pursue problems in tomography and the mathematical analysis of Radontransforms. He will develop and refine singularity detection algorithms (including limited data Lambda CT and wavelet methods) and apply them to electron microscopy and limited data industrial and medical X-ray tomography, with the goal of testing the algorithms onreal data and using them in scanners. Professor Quinto will continue to develop a boundary detection method that is applicable to Sonar in shallow water, and he will begin to develop these methods for synthetic aperture Radar. The theoretical underpinnings will be based on Professor Quinto's pure mathematical research. Radon transforms have an intimate relation to harmonic analysis, partial differential equations, and complex analysis, and he will use integral geometry to prove theorems in these fields. The principal investigator will prove uniqueness and support theorems for Radon transforms defined by spreads of polynomials, including sphericaltransforms in Euclidean space and other settings. He and Professor Agranovsky will use these results to characterize zero sets of solutions of the wave equation on crystallographic domains and other sets. Zero sets, where the solution is zero for all time, are important and difficult to characterize. He will prove uniqueness theorems for Radon transforms on spheres and other surfaces and use these to solve Morera and mean value problems on manifolds. He willprove support theorems for X-ray transforms.This research encompasses both applied and pure mathematics: tomography and integral geometry. The pure research will be used to develop, understand, and justify the appliedalgorithms, and the applied problems will motivate much of the pure research. Computed tomography (CT) algorithms will be developed for the nondestructive evaluation of rockets and other objects. The algorithms will allow scientists to detect cracks in rocket bodies, air pockets in rocket fuel, and delaminations in rocket exit cones. Limited data algorithms will be developed for electron microscopy as a part of a joint industrial project. The goal is to produce an algorithm that reliably produces accurate pictures for a SIDEC electron microscope. He will develop pure mathematics that will show how well the algorithms work and where their limitations might showup. These pure mathematical underpinnings are required to ensure that his (or any other) reconstructions are correct and are not lucky guesses. SONAR data can be modeled as averages over spheres (the spherical wave fronts of the sound) and Professor Quinto will provetheorems about these averages. He will use these theorems to develop simple algorithms to map the ocean floor and to find boundaries of objects in the ocean and to apply to Radar. This pure mathematics is intriguing in its own right. Professor Quinto will prove theorems about spherical averages which will be used to understand the wave equation (the equation that describes how sound and light behave). The wave equation describes the motion of a drum head, and he willspecify which points on the drum do not move at all. Finally, he will prove theorems he will apply to complex and harmonic analysis, fields in pure mathematics.