Radon Transforms and Tomography

教授将研究层析成像和Radon变换的数学分析问题。昆图教授将开发有限数据断层扫描的算法。他将继续为外部Radon变换改进自己的算法（ERA）。他对来自工业对象的真实数据进行的测试令人鼓舞，但需要改进以检测某些特定缺陷。他计划利用他的ERA和奇点检测算法开发一种混合算法。他将在火箭上进行测试。昆图教授将继续研究癌症放射治疗的有效剂量计划。他的算法将利用对偶Radon变换的反演和投影到凸集或优化。这项应用研究使用了昆托教授的纯数学专业知识和数值方法。利用微局部分析等技术，他将证明在实解析子流形上具有实解析测度的变换的有限数据定理，包括在欧几里德空间中的曲线和曲面上的变换以及在流形中的测地线球和一般超曲面上的变换。他将使用这些结果来证明近似理论、偏微分方程和复分析中的定理。计算机断层扫描用于检测工业物体的缺陷。昆图教授将开发和改进工业计算机断层扫描的算法，并在火箭的真实CT数据上进行测试。目标是检测火箭本体的裂缝、火箭燃料中的气穴和火箭出口锥的分层。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。需要这些纯数学基础来确保他（或任何其他人）的重构是正确的，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。昆图教授将继续开发数学算法，根据肿瘤的位置和其他医疗限制，告诉医生如何照射肿瘤。在这里，纯数学也将被用来帮助确定好的计划。提供断层扫描基础的纯数学本身就很有趣，首席研究员将追求一些纯数学问题的理论重要性。这种纯粹的研究导致了对应用问题的新见解。

Abstract Quinto Professor Quinto will pursue problems in tomography and the mathematical analysis of Radon transforms. Professor Quinto will develop algorithms for limited data tomography. He will continue refining his own algorithm (ERA) for the exterior Radon transform. His tests on real data from industrial objects have been encouraging, but refinements are needed to detect certain specific defects. He plans to develop a hybrid algorithm using his ERA and a singularity detection algorithm. He will test these on rockets. Professor Quinto will continue his research on finding effective dose plans for cancer radiation therapy. His algorithm will utilize inversion of a dual Radon transform and either projection onto convex sets or optimization. This applied research uses ideas from Professor Quinto's pure mathematical expertise as well as numerical methods. Using techniques including microlocal analysis, he will prove limited data theorems for transforms with real analytic measures on real analytic submanifolds including transforms on curves and surfaces in Euclidean space and on geodesic spheres and general hypersurfaces in manifolds. He will use these results to prove theorems in approximation theory, PDE, and complex analysis. Computed tomography is used to detect defects in industrial objects. Professor Quinto will develop and refine algorithms for industrial computed tomography and test them on real CT data of rockets. The goal is to detect cracks in rocket bodies, air pockets in rocket fuel, and delaminations in rocket exit cones. His current algorithm works quite well 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 could show up. These pure mathematical underpinnings are required to ensure that his (or any other) reconstruct ions are correct and are not lucky guesses. In cancer radiation therapy, doctors irradiate tumors with radiation from different directions. Professor Quinto will continue developing mathematical algorithms that will tell doctors how to irradiate tumors, depending on their location and other medical constraints. Here, too, pure mathematics will be used to help determine good plans. This same pure mathematics that provides the underpinnings of tomography is intriguing in its own right, and the principal investigator will pursue some pure mathematical questions for their theoretical importance. Such pure research has led to new insights about applied problems.