Mathematical Sciences: Mathematical Tools for Imaging Reconstruction

数学科学：图像重建的数学工具

• 批准号: 9101224
• 负责人: F. Alberto Grunbaum
• 金额: $ 12万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-15 至 1994-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101224&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Tools; Imaging; Reconstruction

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.

这个项目将处理图像重建中出现的逆问题。本文将讨论未知介质对低能探针的衰减和散射、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，将付出相当大的努力，进一步发展PI引入的一种技术，以获得某些偏微分方程的二阶解。这种技术避免了使用复杂的代数几何方法。这个项目的动机是在成像重建过程中出现的问题，如X射线和磁共振。要发展的数学对构建用于医学成像的算法和设备有直接的影响。