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- 3D mesh compression (1)
- CAD (1)
- Context-Based Coding (1)
- Discrete Laplace--Beltrami operators (1)
- Discrete Willmore energies (1)
- Frame-Field (1)
- Geometry Synthesis (1)
- Hexagonal Mesh (1)
- Hexagonal Tiling (1)
- Modal analysis (1)

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.

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.

We provide conditions for convergence of polyhedral surfaces and their
discrete geometric properties to smooth surfaces embedded in R^3. The
notion of totally normal convergence is shown to be equivalent to the convergence
of either one of the following: surface area, intrinsic metric, and
Laplace-Beltrami operators. We further show that totally normal convergence
implies convergence results for shortest geodesics, mean curvature,
and solutions to the Dirichlet problem. This work provides the justifi-
cation for a discrete theory of differential geometric operators defined on
polyhedral surfaces based on a variational formulation.

The use of point sets instead ofmeshes becamemore 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.

This work concerns the approximation of the shape operator of smooth surfaces in R^3 from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator. We show experimental results that confirm our estimates.

In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we
derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. Furthermore, we design a quadratic energy, such that the eigenmodes of the Hessian of
this energy are sensitive to the extrinsic curvature of the surface. Based on these spectra and eigenmodes, we derive two shape signatures. One that measures the similarity of points on a surface, and another that
can be used to identify features of the surface. In addition, we discuss a
spectral quadrangulation scheme for surfaces.

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.

We present a novel algorithm for automatic parameterization of tube-like surfaces of arbitrary genus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistent stripe texture. We use the principal curvature frame field of the underlying tube-like surface to guide the creation of a global, topologically consistent stripe parameterization of the surface. Our algorithm extends the QuadCover algorithm and is based, first, on the use of so-called projective vector fields instead of frame fields, and second, on different types of branch points. That does not only simplify the mathematical theory, but also reduces computation time by the decomposition of the underlying stiffness matrices.

We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.

Classical surface parameterization algorithms often place singularities
in order to enhance the quality of the resulting parameter map. Unfortunately, singularities of positive integral index (as the north pole of a sphere) were not handled since they cannot be described with piecewise linear parameter functions on a triangle mesh. Preprocessing is needed to adapt the mesh connectivity. We present an extension to the QuadCover parameterization algorithm [KNP07], which allows to handle those singularities. A singularity of positive integral index can be resolved using bilinear parameter functions on quadrilateral elements. This generalization
of piecewise linear functions for quadrilaterals enriches the space of parameterizations. The resulting parameter map can be visualized by textures using a rendering system which supports quadrilateral elements, or it can be used for remeshing into a pure quad mesh.