SVR: Practical Engineering of a Fast 3D Meshing Algorithm*

Acar, Umut A.; Hudson, Benoît; Phillips, Todd

doi:10.1007/978-3-540-75103-8_3

Cited by 8 publications

(14 citation statements)

References 18 publications

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“…We modified a pre-existing SVR implementation [1] to run in 4D and compute the filtration of Section 3.2. We used the Plex library [27] to compute the persistence diagram.…”

Section: Methodsmentioning

confidence: 99%

Topological inference via meshing

Hudson

Oudot

Sheehy

2010

Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry

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We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud P in Euclidean space R d . Classical approaches rely on theČech, Rips, α-complex, or witness complex filtrations of P , whose complexities scale up very badly with d. For instance, the α-complex filtration incurs the n Ω(d) size of the Delaunay triangulation, where n is the size of P . The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of P , whose sizes are reduced to 2 O(d 2 ) n. A nice property of these filtrations is to be interleaved multiplicatively with the family of offsets of P , so that the persistence diagram of P can be approximated in 23 time in theory, with a near-linear observed running time in practice (ignoring the constant factors depending exponentially on d). Thus, our approach remains tractable in medium dimensions, say 4 to 10.Key-words: Topological persistence, Delaunay triangulation, offsets, sparse Voronoi refinement. * bhudson@tti-c.edu † glmiller@cs.cmu.edu ‡ steve.oudot@inria.fr § dsheehy@cs.cmu.edu Inférence topologique par maillageRésumé : Nous appliquons des idées issues de la litérature sur la génération de maillages afin d'eméliorer la complexité du calcul de l'information topologique persistante associéesà un nuage de poins P dans l'espace euclidien R d . Les méthodes classiques reposent sur l'utilisation de filtrations telles que celle deČech, celle de Rips-Vietoris, celle de l'α-complex ou celle du witness complex, dont les complexités se comportent très mal lorsque la dimension d augmente. Par exemple, la filtration de l'α-complex a la même taille que la triangulation de Delaunay de P , qui est de l'ordre de n Ω(d( dans le pire cas, où n est la taille de P . La solution communément adoptée consisteà tronquer les filtrations avant que leur coût ne devienne prohibitif, mais bien sûr peut-êtreégalement avant que les données topologiques les plus pertinentes n'aientété saisies. Dans cet article nous proposons une nouvelle famille de filtrations, basée sur la triangulation de Delaunay d'un sur-ensemble fini M de P , dont la taille est réduiteà 2 O(d 2 ) n. Une propriété intéressante de ces filtrations est d'être entrelacées multiplicativement avec la famille des offsets de P , si bien que le diagramme de persistance de P peutêtre approché en temps 2en théorie, avec un comportement quasi-linéaire en pratique (en laissant de côté les facteurs dépendant exponentiellement en la dimension d). Ainsi, notre approche demeure praticable en dimensions moyennes, disons entre 4 et 10.

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“…We modified a pre-existing SVR implementation [1] to run in 4D and compute the filtration of Section 3.2. We used the Plex library [27] to compute the persistence diagram.…”

Section: Methodsmentioning

confidence: 99%

Topological inference via meshing

Hudson

Oudot

Sheehy

2010

Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry

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“…Acar et al [1] developed an efficient implementation of SVR in 3-d. It can also be efficiently parallelized [19].…”

Section: Sparse Voronoi Refinementmentioning

confidence: 99%

“…This can be done so that the total complexity of the convex hull of the augmented point set is constant. Second, the Sparse Voronoi Refinement algorithm adds O(n log Δ) Steiner points to produce a τ -well-spaced superset M, for a constant τ > 2 (choosing τ = 3 is reasonable) [1,18]. Third, the facets of the Delaunay triangulation of M are each checked for a potential flip and added to the flip heap accordingly.…”

Section: Preprocessingmentioning

confidence: 99%

“…The quadratic term is from the first phase of the algorithm that solves a d-dimensional linear program for each input point. Matousek and Schwartzkopf [24] showed how to exploit the common structure in these linear programs to improve the running time to O(n 2−2/( d/2 +1) log O (1) n + f log n).…”

mentioning

confidence: 99%

“…1 Seidel [28] gave an algorithm based on polytope shelling that runs in O(n 2 + f log n) time. The quadratic term is from the first phase of the algorithm that solves a d-dimensional linear program for each input point.…”

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confidence: 99%

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A New Approach to Output-Sensitive Construction of Voronoi Diagrams and Delaunay Triangulations

Sheehy

2014

Discrete Comput Geom

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We describe a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is O( f log n log Δ) where f is the output complexity of the Voronoi diagram and Δ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.

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