Tomography and Integral Geometry

摘要：昆托教授将致力于层析成像和Radon变换的数学分析。他将开发和改进断层扫描奇点检测算法，并将其应用于电子显微镜。昆托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算法，他将利用这些结果作为进一步改进的基础，目标是考虑散射和其他影响。他将继续研究三维SONAR中的近似逆。昆托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。在纯数学中，《原理调查者》计划证明唯一性，并支持球面Radon变换的定理。他和阿格拉诺夫斯基教授将利用这些结果来刻画波动方程的定常解集。其中解始终为零的平稳集是非常重要且很难刻画的。他计划证明实解析流形上球面上Radon变换的唯一性定理，并利用这些结果开始刻画秩一对称空间上的平稳集。他正在证明关于多圆盘区别边界的Morera定理，并计划将其推广到复流形上。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。这项研究既包括应用数学，也包括纯数学：层析成像和积分几何。纯研究将用于开发、理解和证明应用算法，而应用问题将激励大部分纯研究。一种用于电子显微镜的计算机断层扫描算法将被开发出来，并与瑞典卡罗林斯卡研究所的同事一起进行测试。我们的目标是用电子显微镜制作准确的病毒和小分子图片。他将与一名同事和一名本科生共同开发一种放射断层成像算法，这是一种定位新陈代谢过程的断层扫描。他将开发纯数学，这些数学将决定算法的工作情况以及它们的局限性可能出现在哪里。这些纯粹的数学基础是确保他的(或任何其他)算法尽可能有效的所必需的。声纳数据可以被建模为球面上的平均(声波的球面波阵面)，昆托教授将共同开发声纳的新算法。这种纯粹的数学本身很耐人寻味。昆托教授将证明关于球面平均的定理，这些定理将被用来理解波动方程(描述声音和光的行为的方程)。波动方程描述了鼓头的运动，他将指明鼓上的哪些点根本不动。最后，他将证明他将应用于复数和调和分析的定理，这两个领域都是纯数学领域。

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.