Tomography and Integral Geometry

摘要：昆托教授将继续研究断层摄影和Radon变换的数学分析问题。 他将开发和完善断层奇异性检测算法，并将其应用于电子显微镜。 Quinto教授已经成功地在电子显微镜数据上测试了他的有限数据层析成像算法，他将使用这些结果作为进一步改进的基础，目标是将散射和其他效应结合起来。 他将继续研究三维声纳中的近似逆。Quinto教授计划证明合成孔径雷达模型的唯一性定理，这项研究基于他的纯数学工作。 在纯数学，principalinvestigator计划证明唯一性和支持定理球面Radon变换。 他和Agranovsky教授将利用这些结果来描述波动方程的定态解集。 平稳集，其中的解决方案是零的所有时间，是非常重要的和难以表征。 他计划证明唯一性定理的Radon变换领域的真正analyticmanifold和使用这些结果开始的特点stationarysets秩一对称空间。 他正在证明莫雷拉定理的区别边界的polydisks，他计划将其推广到复杂的流形。 他希望证明一类重要的局部有限数据问题的逆连续性，或者澄清它们何时不成立。这项研究包括应用和纯数学：层析成像和积分几何。 纯理论研究将被用来开发、理解和证明应用算法，应用问题将激发大量的纯理论研究。 一种计算机断层扫描算法将被开发用于电子显微镜，并与瑞典卡罗林斯卡研究所的同事共同测试。 我们的目标是用电子显微镜制作病毒和小分子的精确图像。 他将与一位同事和一名本科生共同开发一种发射断层扫描算法，这是一种定位代谢过程的断层扫描。 他将发展纯数学，以展示算法如何工作以及它们的局限性可能出现在哪里。这些纯数学基础是确保他的（或任何其他）算法尽可能有效所必需的。 SONAR可以被建模为球体（声波的球面波阵面）的平均值，昆托教授将共同开发SONAR的新租赁。 这种纯数学本身就很有趣。 昆托教授将证明有关球面平均数的定理，这些定理将用于理解波动方程（描述声音和光的行为的方程）。 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运动。 最后，他将证明定理，他将适用于复杂和调和分析，领域在纯数学。

ABSTRACT: Professor Quinto will pursue problems in tomographyand the mathematical analysis of Radon transforms. He will developand refine tomographic singularity detection algorithms and apply themto electron microscopy. Professor Quinto has successfully tested hislimited data tomography algorithm on electron microscope data, and hewill use these results as a basis for further refinement with the goalof incorporating scattering and other effects. He will continueresearch on the approximate inverse in three-dimensional SONAR.Professor Quinto plans to prove uniqueness theorems for a model ofsynthetic aperture RADAR, research which is based on his puremathematical work. Within pure mathematics, the principalinvestigator plans to prove uniqueness and support theorems forspherical Radon transforms. He and Professor Agranovsky will usethese results to characterize stationary sets of solutions of the waveequation. Stationary sets, where the solution is zero for all time,are very important and difficult to characterize. He plans to proveuniqueness theorems for Radon transforms on spheres on real-analyticmanifolds and use these results to begin to characterize stationarysets on rank-one symmetric spaces. He is proving Morera theorems forthe distinguished boundary of polydisks, and he plans to generalizethem to complex manifolds. He hopes to prove inverse continuity foran important class of local limited data problems or clarify when theydo not hold.This research encompasses both applied and puremathematics: tomography and integral geometry. The pure research willbe used to develop, understand, and justify the applied algorithms,and the applied problems will motivate much of the pure research. Acomputed tomography algorithm will be developed for electronmicroscopy and tested jointly with colleagues at the KarolinskaInstitute in Sweden. Our goal is to produce accurate pictures ofviruses and small molecules using an electron microscope. Jointlywith a colleague and an undergraduate student, he will develop analgorithm for emission tomography, a type of tomography that locatesmetabolic processes. He will develop pure mathematics that will showhow well the algorithms work and where their limitations might occur.These pure mathematical underpinnings are required to ensure that his(or any other) algorithms are as effective as they can be. SONAR datacan be modeled as averages over spheres (the spherical wave fronts ofthe sound wave), and Professor Quinto will jointly develop newalgorithms for SONAR. This pure mathematics is intriguing in its ownright. Professor Quinto will prove theorems about spherical averageswhich will be used to understand the wave equation (the equation thatdescribes how sound and light behave). The wave equation describesthe motion of a drum head, and he will specify which points on thedrum do not move at all. Finally, he will prove theorems he willapply to complex and harmonic analysis, fields in pure mathematics.