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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 Fall '10
 Staff
 Math, The Land, Nonparametric statistics, kernel, Kernel density estimation, Density estimation, Bhattacharyya coe cient

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