![](bg3.png)
p
(X
n
=1)∩(X
m
=1)
= min(p
n
, p
m
) = 0.7. This exchange
of information can be represented by a noisy channel
with H(X
n
) ≤ H(X
m
).
Figure 4 shows an example of the selection of five
kernels by CMIM based on Figure 1:
[19, 9] XY
[ 8,24] B [18, 4] G [26, 3] XY [ 1, 9] B
Figure 4: Selection of 5 kernels by CMIM.
2.3 k-PPV MCMC Tracking
The goal is to find in each new image the closest
ROI to the original. Information is normalized by
dimension and weighted on UVRGB by the weights
[1/4, 1/4, 1/6, 1/6, 1/6]. Each image can be viewed
as a probability density function (PDF) and compared
with another one by a similarity measure, namely
the Kullback-Leibler divergence. (Boltz et al., 2009)
showed how to estimate it in a k-NN framework and
why it is well aware of the local densities in high di-
mensions.
For a reference population R of n
R
points in di-
mension d and a target population T of n
T
points in
the same dimension (n
R
6= n
T
a priori), with ρ
k
(U, s),
the Euclidean distance between the point s and its k
th
nearest neighbor in U (U ≡ R or T), the Kullback-
Leibler divergence D
KL
(T,R) can be estimated, in an
unbiased way, by:
D
KL
(T,R)
kPPV
= log
n
R
n
T
− 1
+
d
n
T
∑
s∈T
log
ρ
k
(R,s)
ρ
k
(T,s)
An ideal estimation would consider all the pixels
of R and T. However, to avoid the combinatorial ex-
plosion or just speed up the calculations, we try to
downsample these regions. The question is how. We
have compared three types of sub-samples: two regu-
lar, one random and one from the kernels selected by
CMIM.
Trackings use a sequential particle filter (Arulam-
palam et al., 2002) with a Markov chain (Khan et al.,
2005). Each particle represents a region of the im-
age. The upper left corner of the reference ROI R
indicates the first position. From the second we ap-
ply N
p
random transformations ϕ
i
to R. It gener-
ates N
p
particles or ROI targets T
i
whose weight is:
w
i
= e
−µ.D
KL
(T
i
,R)
with µ a fixed constant. The parti-
cle with the maximum weight takes on the new posi-
tion. The others are not forgotten. They will generate
new particles in the subsequent iterations. The repro-
duction rate is governed by the Metropolis-Hastings
rule: min(1,w
new
/w
∗
) < rand(). Few particles of low
weight are also ”burned” at each iteration.
We have also several important assumptions: that
the reference R doesn’t not change, that the object is
far from the camera and therefore that the ROI under-
goes only translations and remains fixed.
3 EXPERIMENTS
A manual tracking provided the ground truth GT. The
particle filter algorithm truth AT was then compared
to GT following different configurations.
Figure 5: Intersection area A
AT∩GT
of the ROI of the algo-
rithm AT and of the ground truth GT in the same image.
To determine the quality η of a tracking, AT and
GT were compared image by image. We calculated
their reports of intersection area and union area and
check whether for a given tolerance τ it makes ”fit”
the tracked ROIs to the ground truth or not. This state,
good or not, is rated β. For N
i
images:
β
i
(τ) =
(
1 if
A
i
AT∩GT
A
i
AT∪GT
≥ τ
0 else
η(τ) =
1
N
i
N
i
∑
i=1
β
i
(τ)
Four significant subsamplings are presented: reg-
ular on raw data (”Raw”), regular on data smoothed
by a Gaussian kernel (”Gauss”), random on data aver-
aged per Voronoi cell (”RandVor”), and finally from
the convolution of kernels selected by CMIM with
their optimized bandwidth (”KerOpt”).
Results are based on the sequence ”Walk-
ByShop1cor” of CAVIAR between images 192 and
309. We tried to follow the head of a man on a win-
dow of size 31× 25 pixels. The curves show the vari-
ation of good tracking η depending on the tolerance
τ on the percentage of area common with the ground
truth. The four curves in green, khaki, brown and red
in Figure 6 represent subsamplings of a pixel of 1, 4,
7 and 10 i.e.: 775, 56, 20 and 12 points for the con-
figurations ”Raw”, ”Gauss” and ”RandVor”, from top
to bottom. Down on the same figure, the three curves
gray, violet and cherry compile the ”KerOpt” config-
uration for respectively: 55, 30 and 5 points filtered
from the selected kernels.
KERNEL SELECTION BY MUTUAL INFORMATION FOR NONPARAMETRIC OBJECT TRACKING
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