Mathematical Sciences: Radon Transforms and Tomography

数学断层扫描，或氡变换和积分几何的应用，是科学研究的前沿。例如，人们可能会引用科马克在医学成像方面获得诺贝尔奖的工作。图像重建是x射线断层扫描、发射断层扫描、电子显微镜、射电天文学和核磁共振的主要科学和工程前沿。它在Radon变换中的数学基础及其通过算法的计算执行构成了本提案的主题。Quinto教授是Radon变换理论分析方面的专家，他坚持理论进步必须与计算实现齐头并进的科学观点。由于昆图教授在这两个领域都做出了显著的贡献，他掌握了整个领域的视角。因此，该项目具有双重推力。在理论方面，Quinto教授将继续研究广义Radon变换的支持定理。他将考虑实解析子流形上具有实解析测度的变换的这类问题，包括x -射线和k-平面变换的有限数据定理，以及由双颤振和子群不变变换引起的变换。在计算工作中，Quinto教授将完成他的反外部Radon变换算法的实现。这个问题的解决方案可以让计算机对跳动的心脏周围的器官、骨骼中的金属进行良好的断层成像，以及对火箭喷嘴等环或锥体进行无损评估。

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.