Tomography and Integral Geometry

该项目继续得到美国国家科学基金资助的研究项目(DMS 9622947)。昆托教授将继续研究断层摄影术和积分几何方面的问题。他将开发和改进用于工业有限数据层析成像的算法，包括他目前的算法和Lambda CT，并在火箭数据上测试它们。昆托教授将开发一种浅水声纳边界检测方法，并对地表附近的结构进行地球物理检查。该算法将在使用不同物理模型的模拟上进行测试。昆托教授将与布尔杰斯教授合作制定医疗辐射剂量计划，其目标是使用X射线杀死体内的肿瘤，并尽可能减少对周围组织的损害。研究人员将刻画半离散和完全离散问题的剂量算符的零空间。他们将研究剂量算符的奇异值与衰减的关系。昆托教授将发现广义Radon变换的性质，并将其应用于数学分析的其他领域。他将证明由多项式的展开定义的Radon变换的唯一性和支持定理，包括欧氏空间中的球面变换和欧氏空间中由其他多项式定义的双曲平面和展开。他将使用这些结果来回答逼近理论和偏微分方程式中的问题。他将证明欧几里德空间中X射线变换的性质。他将证明球面和流形上其他曲面上的Radon变换的支持定理，并用这些定理来解决复杂流形上的Morera问题。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。纯粹的研究将用于开发、理解和证明应用算法，而应用问题将激励一些纯粹的研究。计算机断层成像算法将被开发和改进，以检测工业对象中的缺陷。它们将在真实的火箭数据上进行测试。这些算法将允许科学家检测火箭身体的裂缝，火箭燃料的气囊，以及火箭出口锥体的脱层。总体而言，他目前的算法运行得很好，但它没有足够好地检测某些类型的火箭缺陷。因此，他将开发其他算法，这些算法将与他的算法一起工作，以获得最佳重建。他将开发纯粹的数学，显示算法的工作效果如何，以及它们的局限性可能会在哪里显现。这些纯粹的数学基础是需要的，以确保他(或任何其他)重建是正确的，而不是幸运的猜测。在癌症放射治疗中，医生从不同的方向对肿瘤进行照射。昆托教授将开发和分析数学，帮助医生选择如何有效地照射肿瘤。声纳数据可以被建模为球面上的平均(声音的球面波阵面)，昆托教授将证明关于这些平均的定理。他将利用这些定理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。这些算法也可能对地球物理检测有用。这些算法将在使用不同物理模型的模拟上进行测试。这种纯粹的数学本身就很耐人寻味。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)和声纳。他还将证明有关X射线变换的定理，这是最新的X射线断层扫描仪背后的数学原理。最后，他将证明他将应用于复数分析的定理，这是纯数学中的一个领域。

ABSTRACTQuintoThis project continues research supported by National Science Foundation grant DMS 9622947. Professor Quinto will pursue problems in tomography and in integral geometry. He will develop and refine algorithms for industrial limited data tomography, including his current algorithm and Lambda CT, and test them on rocket data. Professor Quinto will develop a boundary detection method for SONAR in shallow water and geophysical examination of structures near the surface. The algorithm will be tested on simulations using different physical models. Professor Quinto will work jointly with Professor Boergers on medical radiation dose planning, the goal of which is to use X-rays to kill tumors in the body and to damage as little surrounding tissue as possible. The researchers will characterize the null space of the dose operator for both the semi-discrete and completely discrete problems. They will investigate the dependence of the singular values of the dose operator on the attenuation. Professor Quinto will discover properties of generalized Radon transforms and apply them to other areas of mathematical analysis. He will prove uniqueness and support theorems for Radon transforms defined by spreads of polynomials, including spherical transforms in Euclidean space and the hyperbolic plane and spreads defined by other polynomials in Euclidean space. He will use these results to answer questions in approximation theory and partial differential equations. He will prove properties of the X-ray transform in Euclidean space. He will prove support theorems for Radon transforms on spheres and other surfaces on manifolds and use these to solve Morera problems on complex manifolds.This research encompasses both applied and pure mathematics: tomography and integral geometry. The pure research will be used to develop, understand, and justify the applied algorithms, and the applied problems will motivate some of the pure research. Computed tomography algorithms will be developed and refined to to detect defects in industrial objects. They will be tested on real rocket data. The algorithms will allow scientists to detect cracks in rocket bodies, air pockets in rocket fuel, and delaminations in rocket exit cones. His current algorithm works well, in general, but it does not detect some types of rocket defects well enough. So, he will develop other algorithms that will work in conjunction with his algorithm to get optimal reconstructions. He will develop pure mathematics that will show how well the algorithms work and where their limitations might show up. These pure mathematical underpinnings are required to ensure that his (or any other) reconstructions are correct and are not lucky guesses. In cancer radiation therapy, doctors irradiate tumors with radiation from different directions. Professor Quinto will develop and analyze mathematics that will help doctors choose how to irradiate tumors effectively. SONAR data can be modeled as averages over spheres (the spherical wave fronts of the sound) and Professor Quinto will prove theorems 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. These algorithms could also be useful for geophysical examination. The algorithms will be tested on simulations using different physical models. 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) and for SONAR. He will also prove theorems about the X-ray transform, the mathematics behind the newest X-ray tomography scanners. Finally, he will prove theorems he will apply to complex analysis, a field in pure mathematics.