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We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the one defined by Pinkall and Polthier (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic and holomorphic functions, discrete mean curvature, and discrete minimal surfaces.
Modelling incompressible ideal fluids as a finite collection of
vortex filaments is important in physics (super-fluidity, models for the
onset of turbulence) as well as for numerical algorithms used in computer
graphics for the real time simulation of smoke. Here we introduce
a time-discrete evolution equation for arbitrary closed polygons in 3-
space that is a discretisation of the localised induction approximation of
filament motion. This discretisation shares with its continuum limit the
property that it is a completely integrable system. We apply this polygon
evolution to a significant improvement of the numerical algorithms
used in Computer Graphics.
It is known that for each combinatorial type of convex 3-dimensional
polyhedra, there is a representative with edges tangent to the unit sphere.
This representative is unique up to projective transformations that fix the unit
sphere. We show that there is a unique representative (up to congruence) with
edges tangent to the unit sphere such that the origin is the barycenter of the
points where the edges touch the sphere.
We prove an existence and uniqueness theorem for weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations can also be interpreted as hyperbolic polyhedra with vertices beyond the infinite boundary. The proof is based on a variational principle. This extends similar work by Rivin on Delaunay triangulations and ideal polyhedra to weighted Delaunay triangulations and hyperideal polyhedra.
A new method for noise removal of arbitrary surfaces
meshes is presented which focuses on the preservation
and sharpening of non-linear geometric features such
as curved surface regions and feature lines. Our method
uses a prescribed mean curvature flow (PMC) for simplicial
surfaces which is based on three new contributions:
1. the definition and efficient calculation of a
discrete shape operator and principal curvature properties
on simplicial surfaces that is fully consistent with
the well-known discrete mean curvature formula, 2. an
anisotropic discrete mean curvature vector that combines
the advantages of the mean curvature normal with
the special anisotropic behaviour along feature lines of
a surface, and 3. an anisotropic prescribed mean curvature
flow which converges to surfaces with an estimated
mean curvature distribution and with preserved nonlinear
features. Additionally, the PMC flow prevents
boundary shrinkage at constrained and free boundary
segments.
Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in R3 from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical
projection onto R2 and can therefore be considered as multivalued graphs
over the plane – hence they provide a comprehensible visualization of the
topological structure. Complex functions are elegantly visualized using domain coloring on
a subset of C. By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.
Diffusion weighted imaging is a magnetic resonance based method to investigate
tissue micro-structure especially in the human brain via water diffusion.
Since the standard diffusion tensor model for the acquired data failes in
large portion of the brain voxel more sophisticated models have bee developed.
Here, we report on the package dti and how some of these models
can be used with the package.
We introduce FreeLence, a lossless single-rate connectivity compression algorithm for triangle surface meshes. Based upon a geometry-driven traversal scheme we present two novel and simple concepts: free-valence connectivity encoding and entropy coding based on geometric context. Together these techniques yield signicantly smaller rates for connectivity compression than current state of the art approaches - valence-based algorithms and Angle- Analyzer, with an average of 36% improvement over the former and an average of 18% over the latter on benchmark 3D models, combined with the ability to well adapt to the regularity of meshes. We also prove that our algorithm exhibits a smaller worst case entropy
for a class of ”well-behaved” triangle meshes than valence-driven connectivity encoding approaches.
Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the
mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression
scheme for these adaptive structures that exploits the parent-child relationships inherent to the mesh
hierarchy. We use the rules that correspond to the adaptive refinement scheme and store bits only where some
freedom of choice is left, leading to compact codes that are free of redundancy. Moreover, we extend the coder to
sequences of meshes with varying refinement. The connectivity compression ratio of our method exceeds that of
state-of-the-art coders by a factor of 2 to 7.
For efficient compression of vertex positions we adapt popular wavelet-based coding schemes to the adaptive
triangular and quadrangular cases to demonstrate the compatibility with our method. Akin to state-of-the-art
coders, we use a zerotree to encode the resulting coefficients. Using improved context modeling we enhanced the
zerotree compression, cutting the overall geometry data rate by 7% below those of the successful Progressive
Geometry Compression. More importantly, by exploiting the existing refinement structure we achieve compression
factors that are 4 times greater than those of coders which can handle irregular meshes.
Despite the success of quad-based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. CubeCover is a first approach which provides both, a consistent theoretical framework for volume parameterization plus a full pipeline for generating
a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field.
The input of CubeCover is a tetrahedral volume mesh. First, a frame field is designed with manual input from
the designer. It guides the interior and boundary layout of the parameterization. Then, the parameterization and
the hexahedral mesh are computed so as to align with the given frame field.
CubeCover has similarities to the QuadCover algorithm and extends it from 2D surfaces to 3D volumes. The
paper also provides theoretical results for 3D hexahedral parameterizations and analyses topological properties
of the appropriate function space.