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VC-dimension of exterior visibility

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4 Author(s)
Isler, V. ; Dept. of Comput. & Inf. Sci., Pennsylvania Univ., Philadelphia, PA, USA ; Kannan, S. ; Daniilidis, K. ; Valtr, P.

In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VC-dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes. Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary. We present results for the cases of cameras lying on a circle containing a polygon (VC-dimension=2) or lying outside the convex hull of a polygon (VC-dimension=5). The main result of this paper concerns the 3D case: We prove that the VC-dimension is unbounded if the cameras lie on a sphere containing the polyhedron, hence the term exterior visibility.

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Pattern Analysis and Machine Intelligence, IEEE Transactions on  (Volume:26 ,  Issue: 5 )