Mathematical Tools for Imaging Reconstruction

图像重建的数学工具

• 批准号: 0603901
• 负责人: F. Alberto Grunbaum
• 金额: $ 19.8万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603901&HistoricalAwards=false
• 关键词: Mathematical; Tools; Imaging; Reconstruction

This research project consists of using ideas initially developed in the context of medical imaging in two much more difficult areas: network tomography and the imaging of tensor valued quantities. The first area contemplates solving a complex system of nonlinear equations, and the second one goes well beyond the more standard problem of reconstructing scalar valued quantities. In the first case the major step consists in taking into account loops or cycles in the network. Mathematically this is the analog of taking multiple scattering events into account, a difficulty that is ignored in standard tomographic approaches. brbrThe first part of this project deals with the issue of monitoring a complicated network -- such as the Internet or more restricted networks -- by making only measurements at a few monitoring stations in the network. A typical problem to be addressed by these methods is to detect if some inaccessible part of the network has been compromised either accidentally or maliciously. The second part of the project deals with the study of stresses and strains in different parts of the human anatomy, such as the heart or the brain, by using tools such as Magnetic Resonance Imaging. These same mathematical developments should be useful in the study of stress and elasticity tensors on the surface of the earth.

该研究项目包括使用最初在两个更困难的领域的医学成像背景下开发的想法：网络断层扫描和张量值的成像。 第一个领域设想解决一个复杂的非线性方程组，第二个远远超出了重建标量值的数量更标准的问题。 在第一种情况下，主要步骤包括考虑网络中的环路或循环。 在数学上，这是考虑多个散射事件的模拟，这是一个在标准层析成像方法中被忽略的困难。brbr本项目的第一部分涉及通过仅在网络中的几个监测站进行测量来监测复杂网络（如Internet或更受限的网络）的问题。 这些方法要解决的一个典型问题是检测网络的某些不可访问部分是否意外或恶意地受到损害。 该项目的第二部分涉及使用磁共振成像等工具研究人体解剖学不同部位（如心脏或大脑）的应力和应变。 这些数学上的发展在地球表面的应力和弹性张量的研究中应该是有用的。