The Expected Number of 3D Visibility Events Is Linear

In this paper, we show that, amongst n uniformly distributed unit balls in R , the expected number of maximal non-occluded line segments tangent to four balls is linear, considerably improving the previously known upper bound. 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. 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. Key-words: computational geometry, 3D visibility, visibility complex, visual events, probabilistic analysis, expected complexity INRIA Sophia-Antipolis. [email protected]. http://www-sop.inria.fr/prisme/. Partially supported by the IST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-2000-26473 (ECG E ective Computational Geometry for Curves and Surfaces). y School of Computer Science, McGill University; [email protected]. Research supported by FCAR. z LORIA INRIA Lorraine, CNRS, Univ. Nancy 2. feverett, goaoc, lazard, [email protected]. http://www.loria.fr/ feverett, goaoc, lazard, petitjeag/. Research supported by the McGill-ISA collaborative INRIA project. x Hong Kong University of Science and Technology. [email protected]. This author's research was done during a post-doctoral tenure at LORIA INRIA Lorraine. Le nombre moyen d' ev enements visuels 3D est lin eaire R esum e : Nous montrons que, pour une sc ene de R compos ee de n boules de rayon 1 uniform ement distribu ees, le nombre moyen de segments de droites non occult es, maximaux et tangents a quatre boules est lin eaire. Nous montrons que la borne lin eaire s'applique egalement a la taille moyenne du complexe de visibilit e, une structure de donn ees encodant l'information de visibilit e de la sc ene, donnant a penser que l'espace m emoire n ecessaire au stockage de cette structure de donn ees n'est pas n ecessairement prohibitif. Nous g en eralisons ces r esultats dans di erentes directions. Nous montrons que la borne lin eaire sur le nombre moyen de segments de droites non occult es, maximaux et tangents a quatre boules unit es suÆsamment eloign ees du bord de la sc ene s'applique egalement a des sc enes compos ees de boules de rayon di erent mais born e, de poly edres de proportion born ee et aussi a des objets non epais comme des polygones de proportion born ee. Nous prouvons egalement que nos r esultats se g en eralisent a d'autres distributions, notamment la distribution de Poisson. En n, nous montrons comment nos r esultats ouvrent de nouvelles perspectives concernant l' etude de la taille moyenne d'autres structures de visibilit e globale comme le graphe d'aspects. Mots-cl es : g eom etrie algorithmique, visibilit e 3D, complexe de visibilit e, ev enements visuels, analyse probabiliste, complexit e en moyenne The Expected Number of 3D Visibility Events is Linear 3