Mathematical Sciences: Radon Transforms and Tomography

昆托教授将在Radon变换和断层扫描的数学分析中研究两个主要目标。第一个目标是证明实解析子流形上的实解析测度集合中广义Radon变换的注入性和支持性定理，可能存在群作用。这些经典Radon变换的结果对于群表示、偏微分方程和层析成像非常有用。第二个目标是解决断层扫描中的两个问题，即开发用于癌症放射治疗的剂量计划算法，以及将现有的外部Radon变换算法应用于环和火箭出口锥的无损评估。这个项目的数学要点如下。取一个流形（曲面、实体或高维模拟物）和在其上定义的函数。假设流形以某种方式被分割，并且函数在每个分割上的平均值是已知的。问题是重建原始函数。一个很好的例子是一个函数，它给出了一个内部不能直接进入的固体的每个点的密度。x射线图像给出沿光束方向的平均密度。断层扫描是一门从大量图像中推断出物体内部密度的科学。

Professor Quinto will pursue problems in the mathematical analysis of Radon transforms and in tomography with two major objectives. The first objective is to prove injectivity and support theorems for generalized Radon transforms in the setting of real analytic measures on real analytic submanifolds, possibly in the presence of a group action. Such results for classical Radon transforms have been extremely useful for group representations, partial differential equations, and tomography. The second objective is to solve two problems in tomography, namely the development of a dose plan algorithm for cancer radiation therapy, and the adaptation of an existing algorithm for the exterior Radon transform to the non-destructive evaluation of rings and rocket exit cones. The gist of the mathematics in this project is as follows. Take a manifold (surface, solid, or higher-dimensional analogue) and a function defined on it. Suppose the manifold is sliced up in some fashion and the average of the function over each slice is known. The problem is to reconstruct the original function. A good example is the function that gives the density at each point of a solid body whose inside is not directly accessible. X-ray images give average densities along lines in the direction of the beam. Tomography is the science of deducing densities inside the object from an economical number of these images.