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VC-Dimension of Exterior Visibility
May 2004 (vol. 26 no. 5)
pp. 667-671

Abstract—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 [1]. 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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Index Terms:
VC-dimension, sensor placement, sampling, visibility.
Volkan Isler, Sampath Kannan, Kostas Daniilidis, Pavel Valtr, "VC-Dimension of Exterior Visibility," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 5, pp. 667-671, May 2004, doi:10.1109/TPAMI.2004.1273987
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