The density pn 2 4 tracking algorithm can be proved

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Unformatted text preview: roved 19] that, there exists a set of priors for which the error probability for the set is less than the error probability for the set . In addition, starting from (16) upper and lower error bounds can be derived for the probability of error. The derivation of the Bhattacharyya coe cient from sample data involves the estimation of the densities p and q, for which we employ the histogram formulation. Although not the best nonparametric density estimate 25], the histogram satis es the low computational cost imposed by real-time processing. We estimate the disP ^ crete density q = fqu gu=1:::m (with m=1 qu = 1) ^ u^ from the m-bin histogram of the target model, while ^ p(y) = fpu(y)gu=1:::m (with Pm=1 pu = 1) is estimated ^ u^ at a given location y from the m-bin histogram of the target candidate. Hence, the sample estimate of the Bhattacharyya coe cient is given by m Xp ^^ pu (y)^u : ^q (17) ^(y) p(y) q] = We assume in the sequel the support of two modules which should provide (a) detection and localization in the initial frame of the objects to track (targets) 21, 23], and (b) periodic analysis of each object to account for possible updates of the target models due to signi cant changes in color 22]. 4.1 Color Representation Target Model Let fx? gi=1:::n be the pixel locai tions of the target model, centered at 0. We de ne a function b : R2 ! f1 : : : mg which associates to the pixel at location x? the index b(x? ) of the histogram i i bin corresponding to the color of that pixel. The probability of the color u in the target model is derived by employing a convex and monotonic decreasing kernel pro le k which assigns a smaller weight to the locations that are farther from the center of the target. The weighting increases the robustness of the estimation, since the peripheral pixels are the least reliable, being often a ected by occlusions (clutter) or background. The radius of the kernel pro le is taken equal to one, by assuming that the generic coordinates x and y are normalized with hx and hy , respectively. Hence, we can write n X qu = C k(kx? k2 ) b(x? ) ; u]...
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