Mathematical Tools for Imaging Reconstruction

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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.