AIR Tools - A MATLAB Package for Algebraic Iterative ...

Chapter 1
Introduction
In the beginning of the 20th century the Polish mathematician Stefan Kaczmarz
[27] and the Italian mathematician Gianfranco Cimmino [8] independently developed
iterative algorithms for solving linear systems. In 1970 Gordon, Bender
and Herman rediscovered Kaczmarz’s method applied in medical imaging [17].
They called the method ART (Algebraic Reconstruction Technique) and when
Houndsfield patented the first CT-scanner in 1972, which awarded him together
with Cormark the Nobel Prize in 1979, the classical methods found their practical
purpose in tomograpgy [24]. The word tomography means reconstruction
from slices. After the invention of the CT-scanner several new methods familiar
with the old classical methods were developed.
This master thesis deals with the classical Kaczmarz’s and Cimmino’s methods
but also with the methods familiar with these methods. We divide the gathered
methods into two main categories, the SIRT and the ART methods, and present
strategies for choosing the relaxation parameter and different stopping rules.
We will compare the performance of different methods and different strategies
on a test problem derived from medical tomography.