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d
l
to the curve, where the sign indicates the side of the curve on which a pixel is
located. Hence, the weighting function and the probabilistic assignment need to be
computed only once and can be obtained by a distance-dependent look-up opera-
tion, thus avoiding a large number of nonlinear function evaluations without loss
of accuracy. Based on the weights
w
l,s
we obtain the statistical moments
m
k,s,o
of
order
o
∈{
0
,
1
,
2
}
for every perpendicular
k
and side
s
∈{
1
,
2
}
of the curve, using
L
k
m
k,s,
0
=
w
l,s
(2.19)
l
=
1
L
k
I
k,l
m
k,s,
1
=
w
l,s
·
(2.20)
l
=
1
L
k
I
k,l
·
I
k,l
T
m
k,s,
2
=
w
l,s
·
(2.21)
l
=
1
with
I
k,l
as the pixel value of perpendicular
k
and pixel index
l
. The original CCD
tracker according to Hanek (
2004
) uses a spatial and spatio-temporal smoothing of
the statistical moments to exploit the spatial and spatio-temporal coherence of pixel
values. This improves the robustness of the tracker, since the influence of strong
edges and outliers is reduced. In contrast to the original work of Hanek (
2004
), we
apply only a spatial smoothing of the statistical moments, since our input images
I
∗
according to (
2.11
) are not temporally coherent due to the addition of the absolute
difference image. We rely on a blurring with fixed filter coefficients by applying to
the statistical moments a Gaussian convolution operator of size 1
5 with a standard
deviation of 2 contour points. This leads to the spatially smoothed moments
×
m
k,s,o
of order
o
. The local mean
vectors
m
k,s
(t)
and covariance matrices
Σ
k,s
(t)
for every perpendicular
k
at time
step
t
on both sides
s
∈{
0
,
1
,
2
}
for every perpendicular
k
and side
s
∈{
1
,
2
}
∈{
1
,
2
}
of the curve are computed by
=
m
k,s,
1
(t)
m
k,s
(t)
(2.22)
m
k,s,
0
(t)
=
m
k,s,
2
(t)
m
k,s
(t)
Σ
k,s
(t)
m
k,s,
0
(t)
−
m
k,s
(t)
·
+
κ.
(2.23)
The parameter
κ
avoids singularities. We set it to
κ
=
2
.
5. The local probability
distributions
S(
m
T
,Σ
T
)
of the pixel grey values which are used in the second step
to refine the estimate consist of
(K
+
1
)
different local mean vectors
m
k,s
(t)
and
covariance matrices
Σ
k,s
(t)
. In the following we assume to be at time step
t
and
denote the mean values by
m
k,s
and the covariance matrices by
Σ
k,s
.
Step 2: Refinement of the Estimate (MAP Estimation)
An update of the pa-
rameters
(
m
T
,Σ
T
)
obtained in step 1, describing the probability distribution of the
curve parameters, is performed based on a single Newton-Raphson step which in-
creases the a-posteriori probability according to Eq. (
2.24
).
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