Mathematical Sciences: Mathematical Tools for Imaging Reconstruction

这个项目将处理在图像重建中出现的逆问题。将讨论低能探头在未知介质中的衰减和散射，孤子理论在磁共振成像切片选择中的应用，以及群表示在信号处理中的应用。此外，还将付出相当大的努力来进一步发展PI引入的技术，以获得某些偏微分方程组的二阶解。该技术避免了使用复杂的代数几何方法。这个项目的动机是成像重建过程中出现的问题，如X射线和磁共振。即将开发的数学对构建用于医学成像的算法和设备具有直接影响。

This project will deal with inverse problems arising in image reconstruction. Attenuation and scattering of low energy probes by unknown media, soliton theory applied to slice selection in magnetic resonance imaging, and application of group representation to signal processing will be considered. In addition, considerable effort will be placed in further developing a technique introduced by the PI to obtain rank two solutions to certain partial differential equations. This technique avoids using complicated algebraic geometry methods. This project has been motivated by problems arising in imaging reconstruction processes such as X rays and magnetic resonance. The mathematics to be developed has direct implications in the construction of algorithms and devices used in medical imaging.