Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P , the aperture angle of x with respect to Q is defined as the angle subtended by the cone that contains Q, has apex at x, and has its two rays emanating from x tangent to Q. We present algorithms with complexities O(n log m) and O(n+m) for computing the maximum aperture angle with respect to Q when x is allowed to vary in P . To compute the minimum aperture angle we modify the latter algorithm obtaining an O(n + m) algorithm. In fact, this is optimal as we show that \Omega\Gammaat/ fm; ng) is a lower bound for the minimization problem. Finally, we establish an \Omega\Gamma n) time lower bound for the maximization problem.

Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point

Abstract. We present a stochastic state-space quality metric for use in controlling active camera networks aimed at 3D vision tasks such as surveillance, motion tracking, and 3D shape/appearance reconstruction. Specifically, the metric provides an estimate of the aggregate steady-state uncertainty of the 3D resolution of the objects of interest, as a function of camera parameters such as pan, tilt, and zoom. The use of stochastic state-space models for the quality metric results in the ability to model and accommodate virtually all traditional quality factors, such as visibility, field of view, occlusion, resolution, surface normals, image contrast, focus, and depth of field. In addition, the stochastic state-space approach naturally addresses camera networks that are aided by other sensing modalities. We begin by surveying the traditional quality factors. We then present our new quality metric, aided by some background in the relevant stochastic state-space models, and an evaluation strategy that scales the computation of our metric to allow its use in a real-time active camera network system. Finally we present some simulation results that illustrate the incorporation of some traditional quality factors, and the use of our metric and evaluation strategy for some simulated scenes containing multiple objects of interest. 1

Let [a; b] be a line segment with end points a, b and a point at which a viewer is located, all in R 3 . The aperture angle of [a; b] from point , denoted by `(), is the interior angle at of the triangle \Delta(a; b; ). Given a convex polyhedron P not intersecting a given segment [a; b] we consider the problem of computing ` max () and ` min (), the maximum and minimum values of `() as varies over all points in P . We obtain two characterizations of ` max (). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation. 1 Introduction Visibility plays an important role in the manufacturing industry in such problems as accessibility analysis in machining [28], [23], [6] and visual inspection [22] as well as computer graphics, robotics, computer vision, operations research and several other disciplines of computing science and computer engineering [17], [21]. The t...