Mathematical Sciences: Radon Transforms and Tomography

数学层析成像，或氡的应用 变换和积分几何，是在最前沿的 科学调查 例如，可以引用Cormack的 医学影像学方面的诺贝尔奖。 图像重建是一种 X射线断层扫描的主要科学和工程前沿， 发射断层扫描，电子显微镜，射电天文学， 核磁共振 它的数学基础是 Radon变换及其算法实现 这是本提案的主题。 昆托教授是一位理论分析专家， 他坚持科学的观点， 理论的进步必须与计算的进步齐头并进。 执行。 正如昆托教授所指出的， 他在这两个领域的贡献，他命令的角度来看， 整个领域。 因此，该项目具有双重推动力。 上 理论方面，昆托教授将继续他的 广义Radon支撑定理研究 变形 他将考虑这样的转换问题， 真实的解析子流形上的真实的解析测度，包括 X射线和k平面变换的有限数据定理，以及 来自双纤维化和子群不变量的变换 变形 在计算工作中，昆托教授将 完成他的算法的实现， 外Radon变换 解决这个问题可以让 周围器官的良好计算机断层成像 跳动的心脏，骨内金属，以及非破坏性 环形或锥形的评估，如火箭喷管。

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.