Tomography and Integral Geometry

断层扫描和积分几何

• 批准号: 9877155
• 负责人: Eric Todd Quinto
• 金额: $ 7.87万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2002-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9877155&HistoricalAwards=false
• 关键词: Tomography; Integral; Geometry

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.

昆托：本项目由美国国家科学基金资助项目DMS 9622947继续进行研究。昆图教授将研究断层摄影和积分几何方面的问题。他将开发和完善工业有限数据断层扫描的算法，包括他目前的算法和Lambda CT，并在火箭数据上进行测试。Quinto教授将开发一种浅水声纳边界检测方法和近地表结构的地球物理检测方法。该算法将在使用不同物理模型的模拟中进行测试。昆图教授将与博格斯教授共同研究医疗辐射剂量规划，其目标是利用x射线杀死体内的肿瘤，并尽可能减少对周围组织的损害。研究人员将描述半离散和完全离散问题的剂量算子的零空间。他们将研究剂量算子的奇异值对衰减的依赖性。Quinto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。他将证明由多项式扩展定义的Radon变换的唯一性和支持定理，包括欧几里得空间和双曲平面上的球面变换以及欧几里得空间中其他多项式定义的扩展。他将用这些结果来回答近似理论和偏微分方程中的问题。他将证明欧几里得空间中x射线变换的性质。他将证明球体和流形表面上Radon变换的支持定理，并利用这些定理解决复杂流形上的Morera问题。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、理解和证明应用算法，而应用问题将激励一些纯研究。计算机断层扫描算法将得到发展和改进，以检测工业物体的缺陷。它们将在真实的火箭数据上进行测试。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。总的来说，他目前的算法工作得很好，但它不能很好地检测某些类型的火箭缺陷。因此，他将开发其他算法，与他的算法一起工作，以获得最佳的重建。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。在癌症放射治疗中，医生用不同方向的辐射照射肿瘤。昆图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。声纳数据可以建模为球面上的平均值（声音的球形波阵面），昆托教授将证明这些平均值的定理。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。这些算法也可用于地球物理检测。这些算法将在使用不同物理模型的模拟中进行测试。这种纯数学本身就很有趣。昆托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）和声纳。他还将证明关于x射线变换的定理，这是最新的x射线断层扫描扫描仪背后的数学原理。最后，他将证明一些定理，这些定理将应用于复数分析，这是纯数学中的一个领域。