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26
A package for exact kinetic data structures and sweepline algorithms
, 2007
"... In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to ex ..."
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In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the operations involved, the software can also be used to compute arrangements of polynomial objects using a sweepline approach. The package consists of three main parts, the kinetic data structure framework support code, an algebraic kernel which implements the set of algebraic operations required for kinetic data structure processing, and kinetic data structures for Delaunay triangulations in one and two dimensions, and Delaunay and regular triangulations in three dimensions. The models provided for the algebraic kernel support both exact operations and inexact approximations with heuristics to improve numerical stability. 1
Kinetic convex hulls and delaunay triangulations in the black-box model
- In Proc. 27th Annu. Sympos. Comput. Geom
, 2011
"... Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the ob-jects are known in advance. This assumption severely limits the applicability ..."
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Cited by 6 (1 self)
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Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the ob-jects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the tradi-tional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum dis-placement of any point in one time step. We study the maintenance of the convex hull and the De-launay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ∈ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∆k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(k∆k log 2 n) amortized time. For the Delaunay triangu-lation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2∆2k) at each time step.
Filtering Relocations on a Delaunay Triangulation
- COMPUTER GRAPHICS FORUM (2009)
, 2009
"... Updating a Delaunay triangulation when its vertices move is a bottleneck in several domains of application. Rebuilding the whole triangulation from scratch is surprisingly a very viable option compared to relocating the vertices. This can be explained by several recent advances in efficient construc ..."
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Cited by 6 (2 self)
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Updating a Delaunay triangulation when its vertices move is a bottleneck in several domains of application. Rebuilding the whole triangulation from scratch is surprisingly a very viable option compared to relocating the vertices. This can be explained by several recent advances in efficient construction of Delaunay triangulations. However, when all points move with a small magnitude, or when only a fraction of the vertices move, rebuilding is no longer the best option. This paper considers the problem of efficiently updating a Delaunay triangulation when its vertices are moving under small perturbations. The main contribution is a set of filters based upon the concept of vertex tolerances. Experiments show that filtering relocations is faster than rebuilding the whole triangulation from scratch under certain conditions.
Delaunay edge flips in dense surface triangulations. arXiv:cs.CG/0712.1959, 2007. Available from authors’ web-pages
"... Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface tr ..."
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Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface triangulations in three dimensions. In this paper we address the question of converting a surface triangulation to a subcomplex of the Delaunay triangulation with edge flips. We show that the surface triangulations which closely approximate a smooth surface with uniform density can be transformed to a Delaunay triangulation with a simple edge flip algorithm. The condition on uniformity becomes less stringent with increasing density of the triangulation. If the condition is dropped completely, the flip algorithm still terminates although the output surface triangulation becomes “almost Delaunay ” instead of exactly Delaunay.
GPU-assisted surface reconstruction on locally-uniform samples
, 2008
"... In point-based graphics, surfaces are represented by point clouds without explicit connectivity. If the distribution of the points can be carefully controlled, surface reconstruction becomes a much easier problem. We present a simple, completely local surface reconstruction algorithm for input poi ..."
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Cited by 5 (0 self)
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In point-based graphics, surfaces are represented by point clouds without explicit connectivity. If the distribution of the points can be carefully controlled, surface reconstruction becomes a much easier problem. We present a simple, completely local surface reconstruction algorithm for input point distributions that are locally uniform. The locality of the computation lets us handle large point sets using parallel and out-of-core methods. The algorithm can be implemented robustly with floating-point arithmetic. We demonstrate the simplicity, efficiency, and numerical stability of our algorithm with an out-of-core and parallel implementation using graphics hardware.
From Segmented Images to Good Quality Meshes using Delaunay Refinement
"... Abstract. This paper surveys Delaunay-based meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the so-called Delaunay refinement technique allows to mesh surfaces and volumes ..."
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Abstract. This paper surveys Delaunay-based meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images. We show that the so-called Delaunay refinement technique allows to mesh surfaces and volumes bounded by surfaces, with theoretical guarantees on the quality of the approximation, from a geometrical and a topological point of view. Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly non-uniform) sizing field. We show how this general paradigm can be adapted to produce anisotropic meshes, i.e. meshes elongated along prescribed directions. Lastly, we discuss extensions to higher dimensions, and especially to space-time for producing time-varying 3D models. This is also of interest when input images are transformed into data points in some higher dimensional space as is common practice in machine learning. 1
DYNAMIC FIELD PROCESS SIMULATION WITHIN GIS: THE VORONOI APPROCHE
"... Simulation of a dynamic and continuous phenomenon (field) is a difficult task for GISs as their data structures are 2D and static and are not well-adapted to manage neither the dynamic behavior of the phenomenon nor its geometrical and topological information. In this paper we present the Voronoi di ..."
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Simulation of a dynamic and continuous phenomenon (field) is a difficult task for GISs as their data structures are 2D and static and are not well-adapted to manage neither the dynamic behavior of the phenomenon nor its geometrical and topological information. In this paper we present the Voronoi diagram as an alternative data structure that, through its useful geometrical and topological properties, provides an adequate discretization of a field and can represent its temporal changes by providing numerical integration methods on either dynamic or kinetic mesh. 1.
