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Hence we can write n x qu c kkx k2 bx u 19 i i u1 the

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Unformatted text preview: ^ (19) i i u=1 The geometric interpretation of (17) is the cosine of the angle p between thepm-dimensional, unit vectors ; ;p p^ p1 : : : pm > and q1 : : : qm > . ^ ^ ^ Using now (17) the distance between two distributions can be de ned asp ^^ d(y) = 1 ; p(y) q] : (18) The statistical measure (18) is well suited for the task of target localization since: 1. It is nearly optimal, due to its link to the Bayes error. Note that the widely used histogram intersection technique 26] has no such theoretical foundation. 2. It imposes a metric structure (see Appendix). The Bhattacharyya distance 15, p.99] or Kullback divergence 8, p.18] are not metrics since they violate at least one of the distance axioms. 3. Using discrete densities, (18) is invariant to the scale of the target (up to quantization e ects). Histogram intersection is scale variant 26]. 4. Being valid for arbitrary distributions, the distance (18) is superior to the Fisher linear discriminant, which yields useful results only for distributions that are separated by the mean-di erence 15, p.132]. Similar measures were already used in computer vision. The Cherno and Bhattacharyya bounds have been employed in 20] to determine the e ectiveness of edge detectors. The Kullback divergence has been used in 27] for nding the pose of an object in an image. The next section shows how to minimize (18) as a function of y in the neighborhood of a given location, by exploiting the mean shift iterations. Only the distribution of the object colors will be considered, although the texture distribution can be integrated into the same framework. i=1 where is the Kronecker delta function. The normalization constant C is derived by imposing the condition Pm ^ u=1 qu = 1, from where C = Pn k1kx? k2) (20) i=1 ( i since the summation of delta functions for u = 1 : : : m is equal to one. Target Candidates Let fxi gi=1:::nh be the pixel locations of the target candidate, centered at y in the current frame. Using the same kernel pro le k, but with radius h,...
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