Tomography and Integral Geometry

PI: Eric Todd Quinto， Tufts大学dms - 0200788quinto教授将研究层析成像和radontransform的数学分析。他将开发和完善奇点检测算法（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，他将开始开发这些方法用于合成孔径雷达。理论基础将基于昆托教授的纯数学研究。Radon变换与谐波分析、偏微分方程和复分析有着密切的关系，他将使用积分几何来证明这些领域的定理。首席研究员将证明由多项式扩展定义的Radon变换的唯一性和支持定理，包括欧几里德空间和其他设置中的球变换。他和Agranovsky教授将利用这些结果来表征晶体学域和其他集上波动方程的零集解。解始终为零的零集是重要且难以表征的。他将证明球体和其他表面上Radon变换的唯一性定理，并利用这些定理解决流形上的Morera和均值问题。他将证明x射线变换的支持定理。这项研究包括应用数学和纯数学：层析成像和积分几何。纯研究将用于开发、理解和证明应用的算法，而应用的问题将激励大部分纯研究。计算机断层扫描（CT）算法将用于火箭和其他物体的无损评估。该算法将使科学家能够检测到火箭本体的裂缝、火箭燃料中的气穴以及火箭出口锥的分层。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。目标是产生一种算法，可靠地为SIDEC电子显微镜产生精确的图像。他将发展纯数学，以展示算法的工作原理以及它们的局限性。这些纯粹的数学基础需要确保他（或任何其他人）的重建是正确的，而不是幸运的猜测。声纳数据可以建模为球面上的平均值（声音的球形波前），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，并在海洋中找到物体的边界，并应用于雷达。这种纯数学本身就很有趣。昆图教授将证明关于球面平均的定理，这些定理将用于理解波动方程（描述声光行为的方程）。波动方程描述了鼓头的运动，他将指定鼓上的哪些点根本不动。最后，他将证明定理，他将应用于复数和调和分析，在纯数学领域。

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.