Radon Transforms and Tomography

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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.