Topic
Delaunay triangulation
About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.
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23 Oct 2000
TL;DR: A novel and efficient invertible transform for shape segmentation is defined that serves to localize and extract shape characteristics and causes the migration of a shape from its affine description to an invariant, and semantically salient feature-based representation in the form of attributed planar graphs.
Abstract: A novel and efficient invertible transform for shape segmentation is defined that serves to localize and extract shape characteristics. This transform -- the chordal axis transform (CAT) -- remedies the deficiencies of the well-known medial axis transform (MAT). The CAT is applicable to shapes with discretized boundaries without restriction on the sparsity or regularity of the discretization. Using Delaunay triangulations of shape interiors, the CAT induces structural segmentation of shapes into limb and torso chain complexes of triangles. This enables the localization, extraction, and characterization of the morphological features of shapes. It also yields a pruning scheme for excising morphologically insignificant features and simplifying shape boundaries and descriptions. Furthermore, it enables the explicit characterization and exhaustive enumeration of primary, semantically salient, shape features. Finally, a process to characterize and represent a shape in terms of its morphological features is presented. This results in the migration of a shape from its affine description to an invariant, and semantically salient feature-based representation in the form of attributed planar graphs. The research described here is part of a larger effort aimed at automating image understanding and computer vision tasks.
29 citations
TL;DR: This paper forms a very robust geometry and topology optimization of Bézier meshes based on a new simple functional promoting isotropic and uniform Jacobians throughout the domain and demonstrates that the resulting curved meshes can adapt to complex domains with high precision even for a small count of elements.
Abstract: Meshes with curvilinear elements hold the appealing promise of enhanced geometric flexibility and higher-order numerical accuracy compared to their commonly-used straight-edge counterparts. However, the generation of curved meshes remains a computationally expensive endeavor with current meshing approaches: high-order parametric elements are notoriously difficult to conform to a given boundary geometry, and enforcing a smooth and non-degenerate Jacobian everywhere brings additional numerical difficulties to the meshing of complex domains. In this paper, we propose an extension of Optimal Delaunay Triangulations (ODT) to curved and graded isotropic meshes. By exploiting a continuum mechanics interpretation of ODT instead of the usual approximation theoretical foundations, we formulate a very robust geometry and topology optimization of Bezier meshes based on a new simple functional promoting isotropic and uniform Jacobians throughout the domain. We demonstrate that our resulting curved meshes can adapt to complex domains with high precision even for a small count of elements thanks to the added flexibility afforded by more control points and higher order basis functions.
29 citations
TL;DR: This work presents an algorithm that approximates a skin surface with a topologically correct mesh and adapt two existing refinement algorithms to improve the quality of the mesh and shows that the same algorithm can be used for meshing a union of balls.
Abstract: Skin surfaces are used for the visualization of molecules. They form a class of tangent continuous surfaces defined in terms of a set of balls (the atoms of the molecule) and a shrink factor. More recently, skin surfaces have been used for approximation purposes. We present an algorithm that approximates a skin surface with a topologically correct mesh. The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal. We also adapt two existing refinement algorithms to improve the quality of the mesh and show that the same algorithm can be used for meshing a union of balls.
29 citations
TL;DR: It turned out that a new variant of mesh relaxation was most effective in improving an existing triangulation, which was a constrained Delaunay triangulations of two-dimensional domains.
Abstract: The paper describes a method for generating triangulations of two-dimensional domains. Firstly, a description is given of a reliable algorithm that creates a constrained Delaunay triangulation for a multiply-connected planer domain, without performing any explicit visibility computations. Then, several techniques to improve an existing triangulation are discussed. It turned out that a new variant of mesh relaxation was most effective in improving an existing triangulation. Several examples are included to illustrate the method's overall behaviour.
29 citations
TL;DR: It is proved that local transformations can be used to construct a Delaunay triangulation of a set of nk-dimensional points for any $k \geq 2$ and algorithms using this approach are presented.
Abstract: In [SIAM J. Sci. Statist. Comput.,10 (1989), pp. 718–741] and [Comput. Aided Geom. Des., 8 (1991), pp. 123–142] the author presented algorithms that use local transformations to construct a Delaunay triangulation of a set of n three-dimensional points. This paper proves that local transformations can be used to construct a Delaunay triangulation of a set of nk-dimensional points for any $k \geq 2$, and presents algorithms using this approach. The empirical time complexities of these algorithms are discussed for sets of random points from the uniform distribution as well as worst-case time complexities. These time complexities are about the same or better than those of other algorithms for constructing k-dimensional Delaunay triangulations (when $k \geq 3$).
29 citations