## Current reconstruction methods for tomographic data

The reconstruction of volumetric data from planar projections is very important for medical applications. Examples are diagnostic methods such as computed tomography (CT), positron emission tomography (PET) or single photon emission computed tomography (SPECT). While the mathematical foundations of reconstruction were laid in the early 20th century, the topic continues to be of great interest because the situations encountered in practice are often far from an ideal mathematical setting, for example when the noise level is very high. This happens routinely in SPECT. Another important practical aspect is the speed of reconstruction algorithms. In this talk I will review some classic mathematical results and some modern extensions which allow for fast analytical reconstruction. In addition, I will present iterative reconstruction schemes which lack the analytical precision but can deal with much more general situations. The fast implementation of such schemes continues to be a major challenge.