Mathematical Sciences: Radon Transforms and Tomography

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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.