[Mathkoll] Defensio Mag. Boukhgueim / 26.11., 15:00, ESI

Institut für Mathematik sekr.mathematik@univie.ac.at
Wed, 12 Nov 2003 07:24:36 +0100

Institut fuer Mathematik der Universitaet Wien

Vorstand: Univ.-Prof. Dr. Harald Rindler

Einladung zum öffentlichen Rigorosum (Defensio) von

Alexander A. Boukhgueim

Thema der Dissertation: "Numerical  Algorithms for Attenuated Tomography in
Medicine and Industry"

The given dissertation is devoted to the numerical solution of a number of
planar tomography prob-lems and its application to the solution of inverse
kinematic problems of seismology. The aim of this work was the derivation of
new inversion formulae and its effective numerical implementation.
Tomography problems arise in different areas of medicine and industry that
require the layer-by-layer reconstruction of the images of 3D heterogeneous
In the first chapter the author writes down the inversion formula for the
fan-beam Radon transform, obtained by the method of A-analytic functions,
and estimates the error of the projection method that uses only the finite
number of Fourier coefficients of the initial data. In the second chapter
new inver-sion formulae for the emission tomography problem are derived (for
the scalar and vector cases) that don't formally use the theory of
A-analytic functions. In the vector case it becomes possible to reconstruct
the full vector field (and not merely its solenoidal part like in the
unattenuated case), provided that the attenuation function doesn't vanish.
In the third chapter a singular value decomposition of the Radon transform
of tensor fields was obtained in the framework of the fan-beam scanning
geometry, it allows to characterize the range of the tensorial Radon
transform, invert it and estimate the level of incorrectness. In the fourth
chapter the author considers several statements of the inverse kinematic
problem of seismology, chooses the stable ones and derives inversion
formulae for the case of reflected rays (in a 2D layer) and for the case of
refracted rays (in a 3D volume). All the listed above algorithms were
numerically implemented.

Vorsitz: Univ.-Prof. Dr. Harald Rindler (Institut für Mathematik,
Universitaet Wien)

1.	Pruefer: ao. Prof.  Dr. Hans Georg Feichtinger
     (Institut fuer Mathematik, Universitaet Wien)
2.	Pruefer: o. Prof. Dr. Arnold Neumaier
(Institut fuer Mathematik, Universitaet Wien)

Zeit:	Mittwoch, den  26. November 2003, 15:00 Uhr
Ort:	Institut fuer Mathematik, ESI - Hoersaal, Boltzmanngasse 9, A-1090 Wien