Mathematical Tools for Imaging Reconstruction

这个研究项目包括使用最初在医学成像背景下发展的两个更困难的领域的想法：网络断层扫描和张量成像。第一个领域考虑解决一个复杂的非线性方程组，第二个领域远远超出了重建标量值的更标准的问题。在第一种情况下，主要的步骤是考虑网络中的环路或循环。从数学上讲，这是考虑多重散射事件的类比，这是标准层析方法中忽略的一个困难。这个项目的第一部分处理的问题是监测一个复杂的网络，如互联网或更有限的网络，通过只在网络中的几个监测站进行测量。这些方法要解决的一个典型问题是检测网络中某些不可访问的部分是否被意外或恶意破坏。该项目的第二部分是通过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。这些相同的数学发展在研究地球表面的应力和弹性张量时应该是有用的。

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.