# 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.

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

Keywords: