Unformatted text preview: ^
(19)
i
i u=1 The geometric interpretation of (17) is the cosine of
the angle p
between thepmdimensional, unit vectors
;
;p
p^
p1 : : : pm > and q1 : : : qm > .
^
^
^
Using now (17) the distance between two distributions can be de ned asp
^^
d(y) = 1 ; p(y) q] :
(18)
The statistical measure (18) is well suited for the
task of target localization since:
1. It is nearly optimal, due to its link to the Bayes
error. Note that the widely used histogram intersection technique 26] has no such theoretical foundation.
2. It imposes a metric structure (see Appendix). The
Bhattacharyya distance 15, p.99] or Kullback divergence 8, p.18] are not metrics since they violate
at least one of the distance axioms.
3. Using discrete densities, (18) is invariant to the
scale of the target (up to quantization e ects). Histogram intersection is scale variant 26].
4. Being valid for arbitrary distributions, the distance (18) is superior to the Fisher linear discriminant, which yields useful results only for distributions that are separated by the meandi erence
15, p.132].
Similar measures were already used in computer vision. The Cherno and Bhattacharyya bounds have
been employed in 20] to determine the e ectiveness of
edge detectors. The Kullback divergence has been used
in 27] for nding the pose of an object in an image.
The next section shows how to minimize (18) as a
function of y in the neighborhood of a given location,
by exploiting the mean shift iterations. Only the distribution of the object colors will be considered, although
the texture distribution can be integrated into the same
framework. i=1 where is the Kronecker delta function. The normalization constant C is derived by imposing the condition
Pm
^
u=1 qu = 1, from where
C = Pn k1kx? k2)
(20)
i=1 ( i
since the summation of delta functions for u = 1 : : : m
is equal to one.
Target Candidates Let fxi gi=1:::nh be the pixel
locations of the target candidate, centered at y in the
current frame. Using the same kernel pro le k, but with
radius h,...
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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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