maximize

Dr. Nadine Olischläger

 

Address: Rheinische Friedrich-Wilhelms-Universität Bonn
Institut für Numerische Simulation
Endenicher Allee 60
D-53115 Bonn
Room:2.047
Phone: +49 (0)228 73-3301
Fax:+49 (0)228 73-9015
eMail:


Dr. Nadine Olischläger is currently a visitor in the Computer Science Group of Prof. M. Desbrun, California Institute of Technology (Caltech).

 

 

Research interests

Two step time discretization of the anisotropic Willmore flow

Using the Willmore functional does not lead to satisfying results if an edge or a corner of the surface is destroyed. The anisotropic Willmore energy is a natural generalization of the Willmore energy which has crystal-shaped surfaces like cubes or octahedra as minimizers. We extend the two step time discretization for discrete isotropic Willmore flow to the anisotropic case. To evaluate the anisotropic Willmore energy on the unknown surface of the next time step, we first solve an inner, secondary variational problem describing a time step of anisotropic mean curvature motion, which only involves the anisotropy but not its derivatives. In the anisotropic case we end up with a scheme for a single fully implicit time step of anisotropic mean curvature motion to be solved with a Newton approach, instead solving a linear system of equations in the isotropic model. The difference quotient in time between the given surface and the next time step surface of the anisotropic mean curvature motion can again be regarded as a time discrete, fully implicit approximation of the anisotropic mean curvature vector. Based on this anisotropic mean curvature vector, the generalized Willmore functional can be approximated. The approach is applied to polygonal curves, where the anisotropy could be chosen almost crystalline. Various numerical examples underline again the stability of the new scheme, which enables time steps of the order of the spatial grid size.


Surface restoration based on the two step time discrete isotropic Willmore flow

Extending the two step time discretization of the isotropic Willmore flow to boundary conditions, we are able to restore surfaces with smooth boundary conditions. E.g. we apply the new scheme to a real world restoration problem, where we reconstruct damaged regions of an Egea sculpture. Since the corresponding flow leads to a system of fourth order partial differential equations, we can prescribe Dirichlet and Neumann boundary conditions to achieve C1-continuity at the patch boundary. We incorporate boundary conditions in our nested variational minimization in each time step as follows. In the inner problem, on the new time step we solve a fully implicit time discrete problem for the mean curvature motion of the unknown surface at the next time step with prescribed Neumann boundary condition. Then, in the outer problem, the actual implicit variational formulation of Willmore flow, we prescribe Dirichlet boundary conditions for the new time step. If the inner time step size converges to zero, the co-normal for the new time step converges to the prescribed co-normal of the inner problem. For the discretization of the Neumann boundary condition, we introduce two different numerical methods. The first one considers the boundary conditions to be explicitly calculated whereas in the second one the co-normal is implicitly incorporated to avoid its calculation.





Two step time discretization of Willmore Flow, (poster download), (curve video download), (surface video download)

Based on a natural approach for the time discretization of gradient flows we develop a new time discretization for discrete Willmore flow of polygonal curves and triangulated surfaces. The approach is variational and takes into account an approximation of the L2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the Willmore functional at the new time step. To evaluate the Willmore energy on the unknown surface of the next time step, we first ask for the solution of a inner, secondary variational problem describing a time step of mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the mean curvature vector and enters the approximation of the actual Willmore functional. To solve the resulting nested variational problem in each time step numerically relaxation theory from PDE constraint optimization are taken into account. The approach is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. The new scheme is stable and enables time steps of the order of the spatial grid size.

Flow Visualization via Segmentation in E-Dur

The visualization of flow is an important and challenging topic in scientific visualization. In the third project Weiterentwicklung der Rechenprogramme d3f und r3t (E-DuR) funded by the German Federal Ministry of Education and Research, we develop a Mumford-Shah model for visualizing flow fields via segmentation.

Visualisation in GRAPE

The software package GRAPE has been developed at the Collaborative Research Center 256 at the University of Bonn and at the Institute for Applied Mathematics at the University of Freiburg. My main interests in scientific visualisation using GRAPE are adaptively hierachical postprocessing and visualization methods for large data sets accessed via a procedural interface.

A number of these methods are included in the postprocessing and visualization tool that has been developed in cooperation with the Gesellschaft für Anlagen- und Reaktorsicherheit in three projects funded by the German Federal Ministry of Education and Research, Entwicklung eines schnellen Programms zur Modellierung von Grundwasserströmungen mit variabler Dichte (d3f), Entwicklung eines Programmes zur dreidimensionalen Modellierung des Schadstofftransportes (r3t) and Weiterentwicklung der Rechenprogramme d3f und r3t (E-DuR).

Surface Matching

Establishing a correspondence between two surfaces is a basic ingredient in many geometry processing applications. Existing approaches, which attempt to match two embedded meshes directly, can be cumbersome to implement and it is often hard to produce accurate results in reasonable time. A new variational method for matching surfaces that addresses these issues is presented. Instead of matching two surfaces via a non-rigid deformation directly in R3, we apply well established matching methods from image processing in the parameter domains of the surfaces. A matching energy is introduced which may depend on curvature, feature demarcations or surface textures, and a regularization energy controls length and area changes in the induced deformation between the two surfaces. The metric on both surfaces is properly incorporated into the formulation of the energy. This approach reduces all computations to the 2D setting while accounting for the original geometries. Consequently a fast multiresolution numerical algorithm for regular image grids can be applied to solve the global optimization problem. The final algorithm is robust, generically much simpler than direct matching methods, and computationally very fast for highly resolved triangle meshes.

    

 

Optimal Conformal Parameterization of Topological Spheres, Diploma thesis, February 2005, (poster download)  

For a two-dimensional surface in IR3 that has gender zero, low-distortion conformal parameterizations are described in terms of minimizers of suitable energy functionals. Appropriate distortion measures are derived from principles of rational mechanics, closely related to the theory of non-linear elasticity. The parameterization can be optimized with respect to the varying importance of length preservation and area preservation. A finite element discretization is introduced and a constrained Newton method is used to minimize a corresponding discrete energy. The obtained parameterization can be used to improve the triangulation of a given parametrization. A good parametrization is a basic ingredient in many geometry processing applications.




Transport and Anisotropic Diffusion in Time Dependent Flow Visualization

The visualization of time-dependent flow is an important and challenging topic in scientific visualization. Its aim is to represent transport phenomena governed by time-dependent vector fields in an intuitively understandable way, using images and animations. Here we pick up the recently presented anisotropic diffusion method, expand and generalize it to allow a multiscale visualization of long-time, complex transport problems. Instead of streamline type patterns generated by the original method now streakline patterns are generated and advected. This process obeys a nonlinear transport diffusion equation with typically dominant transport. Starting from some noisy initial image, the diffusion actually generates and enhances patterns which are then transported in the direction of the flow field. Simultaneously the image is again sharpened in the direction orthogonal to the flow field. A careful adjustment of the models parameters is derived to balance diffusion and transport effects in a reasonable way. Properties of the method can be discussed for the continuous model, which is solved by an efficient upwind finite element discretization. As characteristic for the class of multiscale image processing methods, we can in advance select a suitable scale for representing the flow field.

                           

 

Publications

[1] N. Olischläger. Processing Elastic Surfaces and Related Gradient Flows. Dissertation, University Bonn, 2010. [ bib | http | .pdf | Abstract ]
[2] N. Olischläger and M. Rumpf. Two step time discretization of Willmore flow. In E. R. Hancock, R. R. Martin, and M. A. Sabin, editors, IMA Conference on the Mathematics of Surfaces, volume 5654 of Lecture Notes in Computer Science, pages 278-292. Springer, 2009. [ bib | DOI | .pdf ]
[3] J. F. Acker, B. Berkels, K. Bredies, M. S. Diallo, M. Droske, C. S. Garbe, M. Holschneider, J. Hron, C. Kondermann, M. Kulesh, P. Mass, N. Olischläger, H.-O. Peitgen, T. Preusser, M. Rumpf, K. Schaller, F. Scherbaum, and S. Turek. Mathematical Methods in Time Series Analysis and Digital Image Processing, chapter Inverse Problems and Parameter Identification in Image Processing, pages 111-151. Understanding Complex Systems. Springer, 2008. [ bib | DOI | .pdf ]
[4] N. Olischläger. Optimale konforme Parametrisierungen von topologischen Sphären. Diploma thesis, University Duisburg, 2005. [ bib | .pdf ]

 

Teaching

SS 2010Übungen zu Ingenieurmathematik II von Martin Rumpf zusammen mit Ole Schwen
WS 2009Übungen zu Ingenieurmathematik I von Sven Beuchler zusammen mit Benedict Geihe
SS 2009Hauptseminar Numerik von Martin Rumpf zusammen mit Benjamin Berkels
SS 2008Übungen zu Numerische Mathematik von Martin Rumpf zusammen mit Martin Lenz
SS 2007Softwarepraktikum Mathematische Methoden in der Bildverarbeitung in der Geodäsie
WS 2006 Organisation des Doktorandenkolloquium der Bigs