The expected number of 3D visibility events is linear

1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
3 ISA - Models, algorithms and geometry for computer graphics and vision
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : In this paper, we show that, amongst $n$ uniformly distributed unit balls in $\mathbb{R}^3$, the expected number of maximal non-occluded line segments tangent to four balls is linear. Using our techniques we show a linear bound on the expected size of the visibility complex, a data structure encoding the visibility information of a scene, providing evidence that the storage requirement for this data structure is not necessarily prohibitive. These results significantly improve the best previously known bounds of $O(n^{8/3})$. Our results generalize in various directions. We show that the linear bound on the expected number of maximal non-occluded line segments that are not too close to the boundary of the scene and tangent to four unit balls extends to balls of various but bounded radii, to polyhedra of bounded aspect ratio, and even to non-fat 3D objects such as polygons of bounded aspect ratio. We also prove that our results extend to other distributions such as the Poisson distribution. Finally, we indicate how our probabilistic analysis provides new insight on the expected size of other global visibility data structures, notably the aspect graph.
Mots-clés :
Type de document :
Article dans une revue
SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2003, 32 (6), pp.1586-1620. 〈10.1137/S0097539702419662〉
Domaine :

https://hal.inria.fr/inria-00099810
Contributeur : Sylvain Lazard <>
Soumis le : mardi 15 décembre 2009 - 14:43:55
Dernière modification le : samedi 27 janvier 2018 - 01:31:49
Document(s) archivé(s) le : lundi 5 avril 2010 - 23:20:46

Fichier

SIAM-revised.pdf
Fichiers produits par l'(les) auteur(s)

Citation

Olivier Devillers, Vida Dujmovic, Hazel Everett, Xavier Goaoc, Sylvain Lazard, et al.. The expected number of 3D visibility events is linear. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2003, 32 (6), pp.1586-1620. 〈10.1137/S0097539702419662〉. 〈inria-00099810〉

Métriques

Consultations de la notice

470

Téléchargements de fichiers