Tomography and Integral Geometry

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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.