The Expected Number of 3D Visibility Events Is Linear

Abstract

In this paper, we show that, amongst n uniformly distributed unit balls in $\mathbb{R}^3$, the expected number of maximal nonoccluded 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 $O(n^{8/3})$ [F. Durand, G. Drettakis, and C. Puech, {ACM Transactions on Graphics}, 21 (2002), pp. 176-206]. Our results generalize in various directions. We show that the linear bound on the expected number of maximal nonoccluded 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 nonfat three-dimensional 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.