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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 realtime processing. We estimate the disP
^
crete density q = fqu gu=1:::m (with m=1 qu = 1)
^
u^
from the mbin 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 mbin 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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 Fall '10
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 Math, The Land

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