Math 582FA-583FA
Inverse Problems

Gunther Uhlmann

Winter/Spring 2003, Monday/Wednesday/Friday 1:30-2:20

This course will be devoted to the study of several inverse problems arising in medical imaging and geophysics. About half of the course will be devoted to the study of the X-ray transform and Radon transform. In this case the inverse problem consists in determining the density of a tissue by knowing its integral along lines or planes. We'll develop the theory of these transforms from scratch. X-ray tomography played a fundamental role in the development of CAT scans and therefore revolutionized the practice of many parts of medicine. We will also study the attenuated X-ray transform and local tomography which are also of importance in medical imaging. The back projection algorithm will be considered in detail.

In the second quarter of the course we will consider several inverse boundary value problems associated to partial differential equations (PDEs). In general terms the physical situation at hand is modeled by a PDE. The problem is to determine the internal parameters of a medium (the coefficients of the PDE) given some information at the boundary of the medium. A prototypical example is Electrical Impedance Tomography which consists in the determination of the electrical conductivity of a body by making current and voltage measurements at the boundary. Another example is Optical Tomography which is based on boundary measurements of near infrared light transmitted through a body. In mathematical terms this consists in studying an inverse boundary value problem for a special (Boltzmann) transport equation.

Prerequisite. The only prerequisite I will assume for the first quarter is knowledge of the Fourier transform and distribution theory at the level of the Linear Analysis course. For the second quarter I will assume that you know the basic theory of PDE as taught in the first quarter of the PDE course. If you have any questions please talk to the instructor.

References. For the first quarter I will use the book by C. Epstein on the Mathematics of Medical Imaging which I hope will be available for winter quarter. I am also recommending the books The Mathematics of Computerized Tomography  by F. Natterer, The Radon Transform and Some of Its Applications  by S. Deans, and The Radon Transform  by S. Helgason.

For the second quarter I will supply notes and I will give further references in those notes. I also highly recommend to read the report Mathematics and Physics of Emerging Biomedical Imaging, National Research Council, Institute of Medicine which was published by the National Academy of Sciences. This available at the URL: http://www.nas.edu/