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 half 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. An example of this 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.  We will also consider inverse boundary problems associated to the wave equation arising in several practical applications like reflection seismology and ultrasound.

Prerequisites: The only prerequisite I will assume is knowledge of the Fourier transform and distribution theory at the level of the Linear Analysis course. If you have any questions please talk to the instructor.

References: For the first part of the course I will use the book by  F. Natterer, The Mathematics of Computerized Tomography.  I am also recommending the books The Radon Transform and Some of Its Applications by S. Deans and The Radon Transform by S. Helgason. I also highly recommend that you  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 is available at the URL http:///www.nas.edu/.