557 research outputs found
Two triangulations methods based on edge refinement
In this paper two curvature adaptive methods of surface triangulation
are presented. Both methods are based on edge refinement to obtain a
triangulation compatible with the curvature requirements. The first
method applies an incremental and constrained Delaunay triangulation
and uses curvature bounds to determine if an edge of the triangulation
is admissible. The second method uses this function also in the edge
refinement process, i.e. in the computation of the location of a
refining point, and in the re-triangulation needed after the insertion
of this refining point. Results are presented, comparing both
approachesPostprint (published version
A curvature-adapted anisotropic surface remeshing method
We present a new method for remeshing surfaces that respect the intrinsic anisotropy of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a higher dimensional space (here we use 6d). This allow us to form an isotropic mesh optimization problem in this embedded space. Starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by combining common local modifications operations, i.e., edge flip, edge contraction, vertex smoothing, and vertex insertion. All operations are applied directly on the 3d surface mesh. This method results a curvature-adapted mesh of the surface. This method can be easily adapted to mesh multi-patches surfaces, i.e., containing corner singularities and sharp features. We present examples of remeshed surfaces from implicit functions and CAD models
Vorosweep: a fast generalized crystal growing Voronoi diagram generation algorithm
We propose a new algorithm for generating quickly approximate generalized Voronoi diagrams of point sites associated to arbitrary convex distance metric in the Euclidian plane. This algorithm produces connected cells by emulating the growth of crystals starting at the point sites, in order to reduce the complexity of the diagram. The main practical contribution is the Vorosweep package which is the reference implementation of the algorithm. Experimental results and benchmarks are given to demonstrate the versatility of this approach.WIST 3 grant 1017074 DOMHEX (Dominant Hexahedral Mesh Generation
Optimal Voronoi Tessellations with Hessian-based Anisotropy
International audienceThis paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
Generalized balanced power diagrams for 3D representations of polycrystals
Characterizing the grain structure of polycrystalline material is an
important task in material science. The present paper introduces the concept of
generalized balanced power diagrams as a concise alternative to voxelated
mappings. Here, each grain is represented by (measured approximations of) its
center-of-mass position, its volume and, if available, by its second-order
moments (in the non-equiaxed case). Such parameters may be obtained from 3D
x-ray diffraction. As the exact global optimum of our model results from the
solution of a suitable linear program it can be computed quite efficiently.
Based on verified real-world measurements we show that from the few parameters
per grain (3, respectively 6 in 2D and 4, respectively 10 in 3D) we obtain
excellent representations of both equiaxed and non-equiaxed structures. Hence
our approach seems to capture the physical principles governing the forming of
such polycrystals in the underlying process quite well
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