Campuses:

Travel Time Tomography, Boundary Rigidity and Electrical Impedance Tomography<br/><br/>

Wednesday, October 19, 2005 - 9:00am - 9:50am
EE/CS 3-180
Gunther Uhlmann (University of Washington)
In inverse boundary problems one attempts to determine the properties of a
medium by making measurements at the boundary of the medium. In the
lecture we will concentrate on two inverse boundary problems, Electrical
Impedance Tomography and Travel Tomography in anisotropic media. These
problems arise in medical imaging, geophysics and other fields. We will
also discuss a surprising connection between these two inverse problems.

Travel Time Tomography, consists in determining the index of refraction or
sound speed of a medium by measuring the travel times of waves going
through the medium. In differential geometry this is known as the the
boundary rigidity problem. In this case the information is encoded in the
boundary distance function which measures the lengths of geodesics joining
points of the boundary of a compact Riemannian manifold with boundary. The
inverse boundary problem consists in determining the Riemannian metric
from the boundary distance function.

Calderön's inverse boundary problem consists in determining the
electrical conductivity inside a body by making voltage and current
measurements at the boundary. This inverse problem is also called
Electrical Impedance Tomography (EIT). The boundary information is

Calderön's inverse boundary problem consists in determining the
electrical conductivity inside a body by making voltage and current
measurements at the boundary. This inverse problem is also called
Electrical Impedance Tomography (EIT). The boundary information is
encoded in the Dirichlet-to-Neumann (DN) map and the inverse problem is to
determine the coefficients of the conductivity equation (an elliptic
partial differential equation) knowing the DN map.

A connection between these two inverse problems has led to a solution of
the boundary rigidity problem in two dimensions for simple Riemannian
metrics. We will also discuss a reconstruction method in two dimensions
for the sound speed from first arrival times of waves.
MSC Code: 
31A25