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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.
Discrete Laplace--Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace--Beltrami operator are well-studied, less is known about the strong form. We present a principle for constructing strongly consistent discrete Laplace--Beltrami operators based on the cotan weights. The consistency order we obtain, improves previous results reported for the mesh Laplacian. Furthermore, we prove consistency of the discrete Willmore energies corresponding to the discrete Laplace--Beltrami operators.
This work concerns the approximation of the shape operator of smooth surfaces in $\mathbb{R}^{3}$ 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, including mean and Gaussian curvature, principal curvatures, and principal curvature directions.
The estimates are confirmed by numerical experiments.
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.
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.
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.
Structuring of surface meshes is a labor intensive task in reverse engineering. For example in CAD, scanned triangle meshes must be divided into characteristic/uniform patches to enable conversion into high-level spline surfaces. Typical industrial techniques, like rolling ball blends, are very labor intensive.
We provide a novel, robust and quick algorithm for the automatic generation of a patch layout based on a topology consistent feature graph. The graph separates the surface along feature lines into functional and geometric building blocks. Our algorithm then thickens thickens the edges of the feature graph and forms new regions with low varying curvature. Further these new regions - so called fillets and node patches - will have highly smooth boundary curves making it an ideal preprocessor for a subsequent spline fitting algorithm.
Lifted Domain Colorings
(2009)
Complex-valued functions are fundamental objects in complex analysis, algebra, differential geometry and in many other areas such as numerical mathematics and physics. Visualizing complex functions is a non-trivial task since maps between two-dimensional spaces are involved whose graph would be an unhandy submanifold
in four-dimensional space. The present paper improves the technique of “domain coloring” in several aspects: First, we lift domain coloring from the complex plane to branched Riemann surfaces, which are essentially the
correct domain for most complex functions. Second, we extend domain coloring to the visualization of general 2-valued maps on surfaces. As an application of such general maps we visualize the Gauss map of surfaces as domain colored plots and establish a link to current surface parametrization techniques and texture maps. Third, we adjust the color pattern in domain and in image space to produce higher quality domain colorings. The new color schemes specifically enhance the display of singularities, symmetries and path integrals, and give better qualitative measures of the complex map.
Abstract—We present a novel coder for lossless compression
of adaptive multiresolution meshes that exploits their special
hierarchical structure. The heart of our method is a new
progressive connectivity coder that can be combined with
leading geometry encoding techniques. The compressor uses
the parent/child relationships inherent to the hierarchical mesh.
We use the rules that accord to the refinement scheme and
store bits only where it leaves freedom of choice, leading to
compact codes that are free of redundancy. To illustrate our
scheme we chose the widespread red-green refinement, but
the underlying concepts can be directly transferred to other
adaptive refinement schemes as well. The compression ratio of
our method exceeds that of state-of-the-art coders by a factor
of 2 to 3 on most of our benchmark models.
In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to discrete Laplacians for applications in shape analysis. We construct such operators as the Hessians of surface energies or deformation energies. In particular, we design a quadratic energy such that, on the one hand, its Hessian equals the Laplace operator if the surface is a part of the Euclidean plane, and, on the other hand, the Hessian eigenfunctions are sensitive to the extrinsic curvature (e.g. sharp bends) on curved
surfaces. Furthermore, we consider eigenvibrations induced by deformation energies, and we derive a closed form representation for the Hessian (at the rest state of the energy) for a general class of deformation energies. 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 surfaces.