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Unformatted text preview: n shift was
used for tracking human faces 4], by projecting the APPENDIX Proof of Theorem 1 ^
Since n is nite the sequence fK is bounded, therefore, it is su cient to show that f^K is strictly monotonic
^
increasing, i.e., if yj 6= yj+1 then f^K (j ) < fK (j + 1),
for all j = 1 2 : : :.
By assuming without loss of generality that yj = 0
we can write
f^K (j + 1) ; f^K (j ) =
!
#
n"
1 X k yj+1 ; xi 2 ; k xi 2 (A.1)
=
: nhd i=1 6 h h 0.7 0.6 0.5 d 0.4 0.3 0.2 0.1 0
3000 3500 4000 4500 5000
5500
Frame Index 6000 6500 7000 Figure 6: The detected minimum value of distance d
function of the frame index for the 2 minute Subway2
sequence. The peaks in the graph correspond to occlusions or rotations in depth of the target. For example,
the peak of value d 0:6 corresponds to the partial
occlusion in frame 3697, shown in Figure 5. At the end
of the sequence, the person being tracked gets on the
train, which produces a complete occlusion.
and by employing (14) it results that
n
1 ky k2 X g
f^K (j + 1) ; f^K (j )
j +1
d+2 nh The convexity of the pro le k implies that
k(x2 ) k(x1 ) + k0 (x1 )(x2 ; x1 )
(A.2)
for all x1 x2 2 0 1), x1 6= x2 , and since k0 = ;g, the
inequality (A.2) becomes
k(x2 ) ; k(x1 ) g(x1 )(x1 ; x2 ):
(A.3)
Using now (A.1) and (A.3) we obtain
f^K (j + 1) ; f^K (j )
n
1 X g xi 2 kx k2 ; ky ; x k2
1 = nhd+2
" 2y>+1
j n
X
i=1
n
X
i=1 i h g xi
h xi g j +1 h 2 : (A.5)
Since k is monotonic decreasing we have ;k0(x)
P
g(x) 0 for all x 2 0 1). The sum n=1 g xi 2
i
h
is strictly positive, since it was assumed to be nonzero
in the de nition of the mean shift vector (10). Thus, as
long as yj+1 6= yj = 0, the right term of (A.5) is strictly
^
^
positive, i.e., fK (j + 1) ; fK (j ) > 0. Consequently, the
^K is convergent.
sequence f
To prove the convergence of the sequence yj j=1 2:::
we rewrite (A.5) but without assuming that yj = 0.
After some algebra we have
!
n
1 ky ;y k2X g yj ;xi 2
f^K (j +1);f^K (j )
d+2 j +1 j Figure 5: Subway2 sequence: The frames 3140, 3516,...
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This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.
 Fall '10
 Staff
 Math, The Land

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