Dr. Nadine Olischläger
Address: 
Rheinische FriedrichWilhelmsUniversität Bonn
Institut für Numerische Simulation Endenicher Allee
60 D53115 Bonn 
Room:  2.047 
Phone: 
+49 (0)228
733301 
Fax:  +49 (0)228
739015 
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 crystalshaped 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 C1continuity
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 conormal for the new time step
converges to the prescribed conormal 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
conormal is implicitly incorporated to avoid its calculation.
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
L2distance 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 codimension. The new
scheme is stable and enables time steps of the order of the spatial
grid size.
Flow Visualization via Segmentation
in EDur
The visualization of
flow is an important and challenging topic in scientific
visualization. In the third project Weiterentwicklung der Rechenprogramme
d3f und r3t (EDuR) funded by the German Federal Ministry of
Education and Research, we develop a MumfordShah 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 (EDuR).
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 nonrigid 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.
For a twodimensional surface in IR3 that has gender zero,
lowdistortion 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 nonlinear 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.
The
visualization of timedependent flow is an important and challenging
topic in scientific visualization. Its aim is to represent transport
phenomena governed by timedependent 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 longtime,
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 278292. 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 111151.
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