Tandem
Workshop

on Geometry, Numerics
and Visualization

26-28
May 2003

Organized by TU-Berlin,
ZIB and WIAS as part of the FZT86 research center

Overview of
Talks

Detection of Shape Features by Skeletonization

*Shape matching
between triangulated surfaces requires detection and identification of mutually
corresponding features on shapes. In order to localize such features, a
set of discrete
curvature operators can be applied. First we translate various types/notions of
curvatures
known from differential geometry to discrete, piecewise linear surfaces,
and discuss some of the problems arising from generalizing curvature notions to
triangulated
surfaces.
Thresholding the curvature field defines which nodes
of the surface
potentially are shape features. The final step comprises the skeletonization of the feature
field, resulting in a set of feature lines.*

Interior Point Methods: An Introduction

*The modern era of interior-point
methods dates to 1984, when Karmarkar
proposed his algorithm for linear programming. In the years since then,
algorithms and software for linear programming have become
quite sophisticated, while extensions to more general classes of problems,
such as quadratic programming, nonconvex and
nonlinear problems
have reached varying levels of maturity. In this talk we will introduce
interior-point methods for linear programming.
This talk is in preparation for the following talk from Martin Weiser:
Adaptive Multilevel Methods for Optimal Control Problems.*

Biomechanical Modeling of Deformable Soft Tissues

*The realistic simulation of
interactions with virtual bodies
is essential to many computer assisted medical applications.
Since soft tissues are non-rigid, any interaction with
biological structures causes their deformation. Thus, modeling
tissue deformations under the impact of external forces is
of general importance.
In this talk, a brief overview over biomechanical modeling of
deformable biological tissues with emphasis on the quasi-static
FE-approach for the long term soft tissue prediction in
the craniofacial surgery planning is presented.*

Creating 3D Geometrical Models from 3D
Image Data

*3D imaging
methods are widely used in various areas like medicine,
biology, material sciences and earth sciences. In this talk we focus
on biology and medicine, where 3D image data form the basis of
morphological and functional analyses, as well as of image-guided
diagnosis and computer assisted therapy.
A central task in such applications is the creation of 3D models
that faithfully represent the objects depicted in the image data.
Some problems of this task will be sketched. A set of practically
useful methods will be presented which enables researchers to segment
biomedical images and to reconstruct smooth 3D geometrical models.
This core technology provides the basis of a wealth of new quantitative
methods -- in the long term turning many biomedical fields into more
quantitative areas. Examples from various areas like medical treatment
and surgery planning, neuroanatomy and gene
expression analysis will
be presented.*

Denoising and enhencement
of surface features

*Noise
is an omnipresent artifact in 2d and 3d meshes due to resolution problems in
mesh acquisition process. For example, meshes extracted from image data or
supplied by laser scanning devices often carry high-frequency noise in the
position of the vertices. Many filtering techniques have been suggested in
recent years, among them Laplace smoothing is the
most prominent example. We present a method for anisotropic denoising
that concentrates on the preservation and enhencement
of linear and curved surface features. The method havily
relies on explicit measures for curvature of discrete surfaces.*

Reorientation of a liquid surface -
solving the Navier Stokes equations in a
time-dependent domain

*The prediction of the dynamic
behavior of liquids with free capillary
surfaces under various acceleration conditions is of high interest for
the construction of space vehicles using liquid propulsion. This
behavior may be numerically simulated by solving the incompressible
Navier-Stokes equations.
Since the free surface is moving, the computional
domain has to change
as well with the time during the simulation. This talk shall present
and discuss one technique to solve PDEs in
time-dependent domains.*

Mathematische
Modellierung von Zweikörperkontakt (I)

*This talk consists of two
parts. Part I deals with the*

*one-sided** contact problem, which
is also known as the*

*Signorini**
problem.** Its aim is to
simulate the mechanical*

*behaviour** of an elastic body in
the presence of a rigid*

*obstacle**. We explain the standard discretizations*

*used** to tackle this problem
and, if time permits, introduce*

*the** audience to fast multigrid solvers for the*

*Signorini**
problem.** For Part II see Oliver Sander.*

3d Statistical Shape Models for Image
Segmentation

*Image segmentation is a prerequisite
for computer assisted
medical diagnosis and therapy planning. Since manual image
segmentation is rather time consuming, automatic and robust
segmentation techniques are of great practical importance.
For this, a-priori knowledge about the anatomy has to be
considered. Statistical shape models have proven
to be effective yet difficult to construct in 3D,
because of the problem of identifying corresponding
points that are densely distributed on two shapes.
In this talk the mathematical problems involved with
shape matching are sketched. A solution for matching
geometrical objects of arbitrary topology based on their
representation as triangular meshes is presented. It will
then be shown how to construct statistical shape models
and how to use them in medical image segmentation.*

Anisotropic fairing of point sets

*The use of point sets instead of
meshes became more popular during the*

*last** years. We present a
new method for anisotropic fairing of a point*

*sampled** surface using an
anisotropic geometric mean curvature flow.*

*The main advantage of our approach
is that the evolution removes noise*

*from** a point set while it
detects and enhances geometric features of*

*the** surface such as edges
and corners. We derive a shape operator,*

*principal** curvature properties
of a point set, and an anisotropic*

*Laplacian**
of the surface.** This anisotropic Laplacian
reflects curvature*

*properties** which can be
understood as the point set analogue of*

*Taubin's**
curvature-tensor for polyhedral surfaces.** We combine these*

*discrete** tools with techniques
from geometric diffusion and image*

*processing**. Several applications
demonstrate the efficiency and*

*accuracy** of our method.*

On the approximation of geometric
invariants of a smooth surface

*If S is a smooth surface
approximated by a triangulation T, we
shall study the relationship between the geometry of S and the
geometry of T. In particular, we shall compare the area and the
curvatures of S and T. Moreover, we shall explain why these kind
of
problems appear in different fields of science, and we
shall deal with an example in structural geology.*

Simulation of particle systems

*In** this talk I will present an algorithm to simulate a system
of particles or rigid objects interacting on each other.
Given such a set of rigid objects. Next define some
forces applied on them. They could for example attract themselves. The
simulation should now calculate what happens to the objects in real time.
With this algorithm it is possible to visualize physical processes, like a
spinning gyro, as mathematical ones, like the moving of a flexible Steffen
Sphere.*

Konrad Polthier (TU)

Conformal Maps and
Non-conforming Meshes

*In
non-conforming simplicial meshes the continuity
of adjacent triangles is required at edge midpoints only rather
than along whole edges. This relaxation provides additional freedom
which allows us to successfully study some optimization problems for
regular triangle meshes including minimization of the conformal energy.*

Recognition of 2D Vector Field
Singularities Using a Discrete Hodge Decomposition

*Singularities** of vector fields are
among the most important features of
flows. Vortices can influence the flying abilities of aircrafts, higher
order singularities often appear in magnetic fields. One approach for
the detection and analyzation of 2d singularities is
to decompose the
vector field into a rotation free, a divergence free and a harmonic
component. The potential respectively co-potential of the
first two
components offer an easy way to get some insight into the vector field
structures.
In this talk a discrete Hodge method is presented and an overview of
vector field singularities and their decomposition and detection is given.*

Smoothing subdivision of discrete
surfaces using the Rivara algorithm

*As the Rivara algorithm itself does not smooth surfaces,
it is combined
with the Butterfly subdivision scheme to a smoothing local subdivision
algorithm. The Butterfly subdivison scheme works
usually with a 4-to-1
split of triangles. A great disadvantage of the 4-to-1 split is the need
of temporary refinements to hold the conformity of the triangulaiton;
these must be removed again, if the subdivision process shall be
continued later. In contrast a Rivara refined
triangulation is
conformized in a way, that does not require to distinguish between
triangles refined to improve the geometrical properties of the surface
and triangles refined to gain conformity. Using Rivara
refinement and
Butterfly scheme together combines the good smoothing properties of the
Butterfly scheme with the very pleasant triangulation handling of the
Rivara algorithm. Thus the Rivara
refinement becomes an interesting
algorithm for a local subdivision not only for two-dimensional
triangulations, for what it is mainly used up to now, but as well for
triangulaitons of surfaces in higher-dimensional
space.*

Mathematische
Modellierung von Zweikörperkontakt (II)

*This talk consists of two
parts. For Part I see Rolf Krause.*

*Part II treats the behaviour of two elastic bodies*

*in** contact with each
other. This two-body contact problem*

*is** considerably less well
understood. We discuss several ways*

*of
formalizing** and discretizing
the contact conditions and*

*explain** some of the problems
encountered on the way.*

*We will further show where two-body
contact fits into the framework*

*of** human-gait simulation.*

Adaptive finite element
methods for phase transition problems

*During** solidification of an undercooled melt, anisotropic interface
effects lead to a geometrical law of motion for the rapidly moving
front between liquid and solid material, the Gibbs-Thomson equation
which couples velocity, anisotropic curvature and temperature.
The interface motion is coupled to heat conduction in the solid and
liquid material.
Different models can be used for the moving interface:
- a parametric surface representing a sharp interface,
- a level set approach,
- a diffuse interface modeled by a phase variable.
All those models lead to degenerate parabolic partial differential
equations for the interface motion.
After introducing the models, we present adaptive finite element
methods for sharp and diffuse interfaces.
The talk presents joint work with E. Bänsch (Berlin),
Z. Chen (Beijing),
G. Dziuk (Freiburg), D. Kessler, R.H. Nochetto (UMD College Park), and
K.G. Siebert (Augsburg).*

TetGen, a 3D tetrahedral mesh generator
based on Delaunay method

*TetGen** is a C++ program for three-dimensional tetrahedral
mesh
generation. It generates boundary constrained meshes and quality meshes
for 3D piecewise linear domains. It uses Delaunay
based mesh generation
algorithms. It's a public domain program and runs almost all platforms
with a C++ compiler.
This talk discusses the mesh problems TetGen solves
and the algorithms
TetGen used. Some main features of TetGen are introduced by simple
examples.*

Symmetric Differential Operators on
Polyhedral Manifolds

*Whereas the
traces of differential geometric operators in the discrete
setting (e.g. Laplacian and mean curvature) are well
understood, the
underlying operators themselves are not as thoroughly studied.
We present a novel simple approach towards the discrete Hessian and
Weingarten operators by considering a weak (integral) version of
vector-valued normal curvatures on piecewise linear models.
In contrast to the smooth setting where these operators are a priori
symmetric this is not the case in the discrete world. Hence we briefly
discuss their symmetrization using a certain
1-parameter family of
convolution operators and show that the traces as well as the principal
directions remain unchanged throughout this symmetrizing
family.
In particular, the traces (Laplacian and mean
curvature) agree
with what has been well established for polyhedral surfaces. We will
shortly discuss how to recover the symmetrization at vertices from the
symmetrization at incident edges.*

Adaptive Multilevel Methods for Optimal
Control Problems

*A
function space oriented approach to solving optimal control problems
with interior point methods is presented. The algorithm combines
advantages from both direct and indirect methods by obtaining accurate
solutions without analytical preparation.
Convergence theory and the algorithmic realization via inexact Newton
methods and adaptive refinement are sketched, and ODE examples are given.
Finally, the prospective extension to PDEs with
application to
hyperthermia treatment planning is discussed.*

*Last updated on **23/05/2003 09:21*