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- F (20)
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.
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 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.