A kernel g can be de ned as gx cgkxk2 8 1 estimates

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Unformatted text preview: kernel G can be de ned as G(x) = Cg(kxk2 ) (8) 1 estimates computed with kernel K in the points (14) are n o n o f^K = f^K (j ) j=1 2::: f^K (yj ) j=1 2::: : (15) These densities are only implicitly de ned to obtain ^ rfK . However we need them to prove the convergence of the sequences (14) and (15). Theorem 1 If the kernel K has a convex and monotonic decreasing pro le and the kernel G is de ned according to (7) and (8), the sequences (14) and (15) are convergent. The Theorem 1 generalizes the convergence shown in 6], where K was the Epanechnikov kernel, and G the uniform kernel. Its proof is given in the Appendix. Note that Theorem 1 is also valid when we associate to each data point xi a positive weight wi . where C is a normalization constant. Then, by taking the estimate of the density gradient as the gradient of the density estimate we have ! n 2 X (x ; x ) k0 x ; xi 2 ^ rf (x) rf^ (x)= K K n X = nh2+2 (xi ; x) g d " n X i=1 i=1 x ; xi g i nhd+2 i=1 2 h x ; xi !2 Pn # 4 h i=1 xi g Pn i=1 g 2 ! h = nh2+2 d x;xi 2 3 h 5 x;xi 2 ; x (9) h where n=1 g x;xi 2 can be assumed to be i h nonzero. Note that the derivative of the Epanechnikov pro le is the uniform pro le, while the derivative of the normal pro le remains a normal. The last bracket in (9) contains the sample mean shift vector P Mh G(x) x;xi 2 h x;xi 2 ; x h Pn i=1 xi g Pn i=1 g and the density estimate at x 3 Bhattacharyya Coe cient Based Metric for Target Localization The task of nding the target location in the current frame is formulated as follows. The feature z representing the color and/or texture of the target model is assumed to have a density function qz , while the target candidate centered at location y has the feature distributed according to pz (y). The problem is then to nd the discrete location y whose associated density pz (y) is the most similar to the target density qz . To de ne the similarity measure we take into account that the probability of classi cation error in statistical hypothesis testing is directly related to the similarity of the two distributions. The larger the probability...
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This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.

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