# Prof. Dr. Matthias Schlottbom

Prof. Egger / Prof. Frank / Prof. Marquardt

### Dissertation

On Forward and Inverse Models in Optical Tomography

## Contact

Dipl. -Comp.Math. Matthias Schlottbom
Aachen Institute for Advanced Study in Computational Engineering Science (AICES)
RWTH Aachen University
Schinkelstraße 2
52056 Aachen
Germany

Phone: +49(0)241 80 99136
Fax: +49(0)241 80 628498
Email: schlottbom (AT) aices (DOT) rwth-aachen (DOT) de

## Education

10/2011 Research assistant at Technical University of Munich here
08/2008 - 09/2011 Postgraduate position at Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen
10/2003 - 03/2008 Diplom in Computermathematics, RWTH Aachen

## Professional Career

05/2008 - 07/2008 Project at 'Verband oeffentlicher Versicherer'

02/2008 - 04/2008 Internship at 'Verband oeffentlicher Versicherer' a service provider for public law insurance companies in Duesseldorf (Germany)
Topic: Comparison of two car insurance tariffs.

Winter 2004 - Student Assistant: Maple Courses, Analysis, Higher Mathematics, Calculus of Variations, Mathematics for CES.

## Research Interests

### Methods for Diffusion Optical Tomography

Diffusion optical tomography (DOT) is a non-invasive imaging technique based on light propagation through a medium and measurement of nearfield data (the light leaving the medium). The corresponding inverse problem consist of determing the spatial distribution of parameters characterizing the scattering and attenuation of light in the unknown medium. The propagation of light through a scattering medium is usually modeled by the radiative transfer equation (RTE). A moment expansion (in terms of spherical harmonics), and truncation of high order terms yields the so-called P_N approximations. In particular for N=1, one obtains the widely used diffusion approximation which is valid when scattering dominates absorption considerably; hence the name DOT. The first order approximation helps to solve the inverse problem efficiently, but the obtained results lack spatial resolution and quantitative information. In order to increase the resolution and obtain quantitative results, one should use better approximations of the underlying RTE model.

### Variational Methods for Radiative Transfer

The radiative transfer equation (RTE) is a basic model for the transport of particles with various applications ranging from astrophysics, over neutron transport, to medical imaging. Since for complex geometries there exist no analytical solutions to the RTE, one has to use numerical approximations. Thus it is not surprising that a huge variety of methods have been proposed and analysed in the last decades, e.g. iteration methods (source iteration, DSA,...), collocation methods (S_N methods) or spherical harmonics expansions (P_N methods). Each of these methods has their own advantages and disadvantages. For example, standard $P_N$ methods can be seen as a Galerkin approximation for the RTE. Unfortunately, if one does not modify the resulting scheme, e.g. transformation in second order form, it becomes unstable due to the hyperbolic nature of transport. In order to circumvent this lack of stability, we have analysed and advised a new mixed variational framework. This formulation allows the construction of Galerkin approximation tailored to the applications under consideration. In particular, we have given sufficient conditions for the scheme to be stable. Comparing this framework to existing methods is a topic of ongoing research.

## Publications

### Recent Work

'A Mixed Variational Framework for the Radiative Transfer Equation'. Herbert Egger and Matthias Schlottbom. to appear in M3AS 03(22)2012, doi:10.1142/S021820251150014X. AICES preprint.

## Reviewed Papers

'Efficient Reliable Image Reconstruction Schemes for Diffuse Optical Tomography'. Herbert Egger and Matthias Schlottbom, Inv. Probl. Sci. Engrg., Vol. 19, Issue 2, pp. 155 - 180, 2011, doi:10.1080/17415977.2010.531469 . AICES preprint.

'Analysis and Regularization of Problems in Diffuse Optical Tomography'. Herbert Egger and Matthias Schlottbom, SIAM J. Math. Anal. Volume 42, Issue 5, pp. 1934-1948 (2010), doi:10.1137/090781590. pdf

'On Forward and Inverse Models in Fluorescence Diffuse Optical Tomography'. Herbert Egger, Manuel Freiberger and Matthias Schlottbom, Inverse Problems and Imaging, Vol. 4, No. 3, 2010, 411-427, doi:10.3934/ipi.2010.4.411. AICES preprint

### Other

Poster 'Numerical Methods for Optical Tomography' at AC.CES. Aachen, July 2011.

Talk 'Analysis of forward and inverse models in fluorescence optical tomography' at Minisymposium 'Non-smooth methods for inverse problems in biomedical imaging', AIP Conference, Texas A&M University. May 2011

Talk 'A Mixed Variational Framework for Radiative Transfer' at the research seminar numerics, Technical University Chemnitz. February 2011.

Talk 'Analysis and Regularization in Diffuse Optical Tomography' at the Chemnitz Symposium on Inverse Problems, Technical University Chemnitz. September 2010.

Talk 'Diffuse Optical Tomography' at the DK-Seminars on Numerical Simulations in Technical Sciences, Technical University Graz. February 2010.

'Numerical Methods in Optical Tomography', Schlottbom, Poster presented at AIP, Vienna, 2009

'C^{2,\alpha}-a-priori Abschaetzungen und ein Existenzsatz fuer harmonische Abbildungen mit Finsler'schem Urbild', Schlottbom, Diploma Thesis, 2008. Advisor: H. von der Mosel

'Evolution von Kurven unter dem Curve Shortening Flow und dem Elastic Flow', Bagh, T. Bedbur, M. Dahmen, Jan Jongen, Hermes, Schlottbom, Project Work, 2006. Advisor: H. von der Mosel