Statistical Inverse Problems

In the time from june 20 th. until july 6 th. 2005 Prof. B. Mair gave an intensive course consisting of nine lectures and exercises about "Statistical Inverse Problems". The lecture took place in the Seminarraum of the Institute of mathematical Stochastics.

Abstract

"Emission tomography is an important method for functional, molecular imaging of living tissue. It can be used to detect abnormalities in cellular activity before there is any observable anatomical change. It is used to identify many forms of cancer, a damaged heart, and many brain disorders. It is based on the metabolism of various naturally occurring chemicals which are tagged with a radio-isotope. As the isotope decays, the emitted photons are detected and numerical algorithms produce an image of the amount of radiotracer at each location. The direct (non-iterative) filtered back projection algorithm has been the mainstay for reconstructing tomographic images since its inception in the seventies. However, due to the variety of applications requiring relatively short imaging times, and the need for accurate quantitative information, modern scanners now incorporate iterative algorithms based on statistical models and methods. The most common iterative algorithms are modifications of the standard expectation maximization algorithm. As a result, we will pay special attention to this method. We will discuss some of the physics motivating the statistical models, convergence characteristics of the algorithms, and indicate interesting open problems in the area."