Mathematical Sciences: Radon Transforms and Tomography

昆托教授将继续研究数学领域的问题 Radon变换和层析成像的分析 目标.第一个目标是证明注入性， 广义Radon变换的支撑定理 在真实的解析子流形上的真实的解析测度，可能 在集体诉讼中。这些结果对于经典 Radon变换对于群 表示，偏微分方程和断层扫描。 第二个目的是解决断层摄影中的两个问题， 即癌症剂量计划算法的开发 放射治疗，以及现有算法的调整 对于外部Radon变换到非破坏性 环和火箭出口锥的评价。 这个项目的数学要点如下。 取一个流形（表面、固体或高维模拟） 和一个定义在其上的函数。假设流形被切开 在某种程度上，每个切片上的函数的平均值 是已知的。问题是重建原始函数。一 一个很好的例子是给出每一点密度的函数 指固体内部不能直接进入。x射线 图像给出了在所述方向上沿着沿着的平均密度。 梁断层摄影是一门推断物体内部密度的科学， 从这些图像的经济数量的对象。

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.