On the Size of the 3D Visibility Skeleton: Experimental Results

Linqiao Zhang (1) , Hazel Everett (1) , Sylvain Lazard (1) , Christophe Weibel (2) , Sue Whitesides

1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
2 Department of Mathematics and Statistics [Montréal]

The 3D visibility skeleton is a data structure used to encode global visibility information about a set of objects. Previous theoretical results have shown that for $k$ convex polytopes with $n$ edges in total, the worst case size complexity of this data structure is $\Theta(n^2 k^2) $ [Brˆnnimann et al. 07]; whereas for $k$ uniformly distributed unit spheres, the expected size is $\Theta(k)$ [Devillers et al. 03]. In this paper, we study the size of the visibility skeleton experimentally. Our results indicate that the size of the 3D visibility skeleton, in our setting, is $ C\,k\sqrt{n\,k}$, where $C$ varies with the scene density but remains small. % This is the first experimentally determined asymptotic estimate of the size of the 3D visibility skeleton for reasonably large $n$ and expressed in terms of both $n$ and $k$. We suggest theoretical explanations for the experimental results we obtained. Our experiments also indicate that the running time of our implementation is $O(n^{3/2} k\log k)$, while its worst-case running time complexity is $O(n^2k^2 \log k)$.

Domaines: Géométrie algorithmique [cs.CG]

Linqiao Zhang, Hazel Everett, Sylvain Lazard, Christophe Weibel, Sue Whitesides. On the Size of the 3D Visibility Skeleton: Experimental Results. 16th Annual European Symposium on Algorithms - ESA 2008, Sep 2008, Karlsruhe, Germany. pp.805--816, ⟨10.1007/978-3-540-87744-8_67⟩. ⟨inria-00336502⟩