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error, the more similar the distributions. Therefore,
(contrary to the hypothesis testing), we formulate the
target location estimation problem as the derivation of
the estimate that maximizes the Bayes error associated
with the model and candidate distributions. For the
moment, we assume that the target has equal prior
probability to be present at any location y in the neighborhood of the previously estimated location.
An entity closely related to the Bayes error is the
Bhattacharyya coe cient, whose general form is dened by 19]
Z
p
(y )
p(y) q] =
pz (y)qz dz : (16)
Properties of the Bhattacharyya coe cient such as its
relation to the Fisher measure of information, quality
of the sample estimate, and explicit forms for various
distributions are given in 11, 19].
Our interest in expression (16) is, however, motivated by its near optimality given by the relationship
to the Bayes error. Indeed, let us denote by and
two sets of parameters for the distributions p and q and
by = ( p q ) a set of prior probabilities. If the value
of (16) is smaller for the set than for the set , it (10) ! n
C X g x ; xi 2
(11)
nhd i=1
h
computed with kernel G. Using now (10) and (11), (9) f^G(x) becomes =C
^
rfK (x) = f^G (x) 2h2 Mh G(x) (12) 2 ^f
Mh G(x) = 2h r^ K (x) :
=C fG(x) (13) from where it follows that Expression (13) shows that the sample mean shift vector obtained with kernel G is an estimate of the normalized density gradient obtained with kernel K . This is a
more general formulation of the property rst remarked
by Fukunaga 15, p. 535]. 2.2 A Su cient Convergence Condition The mean shift procedure is de ned recursively by
computing the mean shift vector Mh G(x) and translating the center of kernel G by Mh G(x).
Let us denote by yj j=1 2::: the sequence of successive locations of the kernel G, where yj+1 = Pn i=1 xi g yj ;xi
h 2 j = 1 2 : : : (14)
yj ;xi 2
i=1 g
h
is the weighted mean at yj computed with kernel G
and y1 is the center of the initial kernel. The density
Pn 2 4 Tracking Algorithm can be p...
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 Fall '10
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 Math, The Land, Nonparametric statistics, kernel, Kernel density estimation, Density estimation, Bhattacharyya coe cient

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