Mathematical Sciences: Radon Transforms and Tomography

点击翻译按钮获取中文摘要

Mathematical tomography, or the application of Radon transforms and integral geometry, is in the forefront of scientific investigation. For example, one may cite Cormack's Nobel prize work in medical imaging. Image reconstruction is a major scientific and engineering frontier for X-ray tomography, emission tomography, electron microscopy, radio astronomy, and nuclear magnetic resonance. Its mathematical foundation in the Radon transform and its computational execution via algorithms form the subject matter of this proposal. Professor Quinto is an expert in theoretical analysis of the Radon transform, who holds to the scientific perspectve that theoretical progress must go hand in hand with computational implemention. As Professor Quinto has made noteworthy contributions in both areas he commands a perspective of the entire field. Thus, the project has a dual thrust. On the theoretical side, Professor Quinto will continue his investigation of support theorems for generalized Radon transforms. He will consider such problems for transforms with real analytic measures on real analytic submanifolds, including limited data theorems for X-ray and k-plane transforms as well as transforms coming from a double fibration and subgroup invariant transforms. In computational work, Professor Quinto will complete the implementation of his algorithm to invert the exterior Radon transform. Solution to this problem could allow good computer tomographic imaging of the organs around the beating heart, of metal in bone, as well as nondestructive evaluation of rings or cones such as rocket nozzles.