Unformatted text preview: the probability of the color u in the target
candidate is given by pu (y) = Ch
^ nh
X
i=1 k y ; xi
h 2! b(xi ) ; u] (21) where Ch is the normalization constant. The radius of
the kernel pro le determines the number of pixels (i.e.,
the scale) of the m
P target candidate. By imposing the
condition that u=1 pu = 1 we obtain
^
Ch = Pnh 1 y;xi 2 :
(22)
i=1 k (k h k )
Note that Ch does not depend on y, since the pixel locations xi are organized in a regular lattice, y being one
of the lattice nodes. Therefore, Ch can be precalculated
for a given kernel and di erent values of h.
3 4.2 Distance Minimization Update fpu (^ 1 )gu=1:::m , and evaluate
^y
p
^y ^
^yq
p(^ 1) q] = Pm=1 pu(^ 1)^u :
u According to Section 3, the most probable location y of the target in the current frame is obtained by minimizing the distance (18), which is equivalent to maximizing the Bhattacharyya coe cient ^(y). The search
for the new target location in the current frame starts at
^
the estimated location y0 of the target in the previous
frame. Thus, the color probabilities fpu (^ 0 )gu=1:::m
^y
^
of the target candidate at location y0 in the current
frame have to be computed rst. Using Taylor expansion around the values pu (^ 0 ), the Bhattacharyya co^y ^y ^
^y ^
4. While p(^ 1 ) q] < p(^ 0 ) q]
^
Do
y1 1 (^ 0 + y1).
y^
2
^^
5. If ky1 ; y0 k < Stop.
^
^
Otherwise
Set y0 y1 and go to Step 1.
The proposed optimization employs the mean shift vector in Step 3 to increase the value of the approximated
Bhattacharyya coe cient expressed by (24). Since this
operation does not necessarily increase the value of
^^
p(y) q], the test included in Step 4 is needed to validate the new location of the target. However, practical
experiments (tracking di erent objects, for long periods of time) showed that the Bhattacharyya coe cient
computed at the location de ned by equation (26) was
almost always larger than the coe cient corresponding
^
to y0 . Less than 0:1% of the performed maximizations
yielded cases where the Step 4 iterations were necessary.
The termination threshold used in Step 5 is derived
^
^
by...
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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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