Computerized Tomography - Week 1
Frank Natterer
WWU Muenster
math.uni-muenster.de/u/natterer/
We start with a short description and a brief account of the history of CT. Then the basic integral transforms, in particular the Radon transform, are introduced. Their basic properties, in particular their relation to the Fourier transform, are studied.
In preparation of the treatment of reconstruction algorithms, and as a complement to Cheney's lab, an introduction to sampling theory and discrete Fourier transforms is given.
The various inversion formulas are derived. The stability properties of the inverse transforms are studied in a Sobolev space framework. Based on these results, reconstruction algorithms for various tomographic problems are derived.
The spatial resolution of these algorithms is analyzed in the light of Shannon's sampling theorem. As basis for the treatment of iterative reconstruction algorithms we give a convergence analysis of the Kaczmarz method for linear systems.
We finish with an outlook on related imaging techniques, such as MRI, PET and SPECT.