Address:
Faculty of Mathematics
Vienna
University
Nordbergstr. 15, C, 7th floor (room 717)
A 1090
Vienna, Austria
Telephone: (0043-1) 4277 50717
Fax: (0043-1)
4277 50665
Email: carola.schoenlieb@univie.ac.at
homepage:
http://homepage.univie.ac.at/carola.schoenlieb
Advisor:
P. A. Markowich
Research: Partial differential
equations, image processing
My main research interest are higher-order PDE´s, especially their application in image processing. The study of higher order PDE´s is still very young and therefore both their analytical analysis and suitable numerical solutions are challenging problems.
A famous higher-order PDE in material sciences is the so called Cahn-Hilliard equation. This equation models phase separation and subsequent coarsening in binary alloys. In a joint work with Martin Burger from the University of Münster, Shun-Yin Chu from RICAM in Linz and my advisor Peter Markowich we studied instabilities of solutions of the Cahn-Hilliard equation and their connection to the Willmore functional. An article on the results of our considerations on Cahn-Hilliard is in preparation.
Additionally I consider higher-order flows in image processing. Motivated by a recent work of Andrea Bertozzi, Selim Esedoglu and Alan Gillette on the application of the Cahn-Hilliard equation for inpainting, I turned my mind on this topic. The behavior of solutions of the Cahn-Hilliard equation is mainly influenced by the double-well potential in the equation. Solutions of Cahn-Hilliard therefore asymptotically tend to two constant states corresponding to the wells of the potential. Choosing a potential with wells at the values 0 and 1, a modified Cahn-Hilliard equation provides a simple model for inpainting of binary images. Together with Martin Burger and Peter Markowich I am now working on the analysis of the Cahn-Hilliard inpainting approach and its generalization for grayvalue images.
Publications:
J. Fernández Bonder, J. D. Rossi, C. Schönlieb. An optimization problem related to the best Sobolev trace constant in thin domains, submitted.
J. Fernández Bonder, J. D. Rossi, C. Schönlieb. The best constant and extremals of the Sobolev embeddings in domains with holes: the $L^\infty$ case, submitted.
M.Burger, S.-Y.Chu, P.Markowich, C.Schönlieb. The Willmore Functional and Instabilities in the Cahn-Hilliard equation, submitted.
C.Schönlieb. Emmy Noethers Idealtheorie in Ringbereichen. Mathematische Berichte, Fachbereich Mathematik, Universität Salzburg. Bericht 30, 2005.
C.Schönlieb and K.C.Wang. Line Segmentation and Analysis With Special Interest to the Duct of a Line. Conference Proceedings Fourth Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP 2004), Kolkata 2004, pp. 550-555.
P.Gerl, C.Schönlieb and K.C.Wang: The Use of Fractal Dimension in Arts Analysis. HarFA - Harmonic and Fractal Image Analysis Journal 2004, pp. 70-73.
P.Gerl, R.Huber, C.Schönlieb and K.C.Wang. Automatisierte Analyse und Klassifikation von Zeichnungen und Gemälden. Konferenzband Electronic Imaging and the Visual Arts (EVA 2003), Berlin 2003, pp 123-126.
C.Schönlieb and K.C.Wang.
Feature Selection and Clustering in Arts Analysis. Conference
Proceedings Irish Machine Vision and Image Processing Conference
(IMVIP 2003), Coleraine 2003, pp. 25-31.
Masters thesis: C.Schönlieb. Invariante Momente. Salzburg, 2004.
Activities:
Participation in Conferences & Workshops:
Dezember 11-15, 2006. Escuela de Matemática sobre Métodos Geométricos y Analíticos para Ecuaciones en Derivadas Parciales. Buenos Aires, Argentina.
November 13-17, 2006. Encuentro Nacional de Ecuaciones Diferenciales (ENED 2006). La Falda (Córdoba), Argentina.
August 22-30, 2006. International Congress of Mathematicians (ICM 2006). Madrid, Spain.
May 15-17, 2006. SIAM Conference on Imaging Science 2006. Minneapolis, Minnesota.
September 26-30, 2005. Herbstschule: Modellierung und Optimierung mit Partiellen Differentialgleichungen. University of Hamburg, Germany.
December 16-18, 2004. Fourth Indian Conference on Computer Vision, Graphics & Image Processing (ICVGIP 2004). Indian Statistical Institute & Indian Unit for Pattern Recognition and Artificial Intelligence, Calcutta, India.
May 14-15, 2004. Tagung: Frauen - Mathematik -Männer, eine interdisziplinäre Initiative. University of Bielefeld, Germany.
Sept. 3-5, 2003. Irish Machine Vision and Image Processing Conference (IMVIP 2003). University of Ulster, Northern Ireland.
Research Stays:
March-June 2007. Institute for Pure and Applied Mathematics (IPAM), University of California, Los Angeles (UCLA). Los Angeles, USA.
September-Dezember, 2006. University of Buenos Aires (UBA). Buenos Aires, Argentina.
Poster Presentations:
Line Segmentation and Analysis, With Special Interest to the Duct of a Line. Fourth Indian Conference on Computer Vision, Graphics & Image Processing (ICVGIP 2004). Indian Statistical Institute & Indian Unit for Pattern Recognition and Artificial Intelligence, Calcutta, India, December 2004.
Lectures at International Conferences:
The Behavior of the Optimal Hole in the Steklov Eigenvalue Problem in Thin Domains. Encuentro Nacional de Ecuaciones Diferenciales (ENED 2006). La Falda (Córdoba), Argentina, November 2006.
Numerical Study of Stability Properties of the Cahn-Hilliard equation. International Congress of Mathematicians (ICM 2006). Madrid, Spain, August 2006
Cahn-Hilliard Inpainting and the Willmore Functional. SIAM Conference on Imaging Science 2006. Minneapolis, Minnesota, May 2006.
Feature Selection and Clustering in Arts Analysis. Irish Machine Vision and Image Processing Conference (IMVIP 2003). University of Ulster, Northern Ireland, September 2003.