Summary:
By computerized tomography (CT) we mean the reconstruction of a function from
its line or plane integrals. This mathematical problem is encountered in a
growing number of diverse settings in medicine, science and technology, ranging
from the famous application in diagnostic radiology. In the first chapter we
give an idea of the scope and limitations of CT and discuss some physical
principles which lead to CT.In the second chapter we introduce various integral
transforms , with most important the Radon transform. In more detail, we give
relations and theorems which connect the Radon with the rest of transforms, and
the definitions of the dual transforms, which are useful for the inversion
formulas. Also, we explore questions of uniqueness and stability, and we define
the ranges of the transforms.In the third chapter we want to find out how to
sample Pf and Rf for some function f, which is basically undetermined for
finitely many directions even in the semi-discrete case. It turns out that
positive and practically useful results are obtained for band limited functions
f. These functions and their sampling properties are summarized in the first
section. In the second section we study the possible resolution if the Radon
transform is available for finitely many directions.In the fourth chapter we
give a detailed description of some well known reconstruction algorithms. We
start with the widely used filtered backprojection algorithm and study the
possible resolution. In the second section we give an error analysis of the
Fourier algorithm which leads to an improved algorithm comparable in accuracy
with filtered backprojection.
Keywords:
Radon transform, Fourier transform, Computerized Tomography, Band-limited functions, Algorithms
