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Fig. 2.9
The CCD algorithm determines the parameters of the curve (
black line
) by computing the
probability distributions of the pixel grey values in a local neighbourhood of the curve (indicated
by the
grey lines
) and by maximising the a posteriori probability of the curve model. Non-optimal
(
left
) and optimal (
right
) configuration
The principle of the CCD algorithm is depicted in Fig.
2.9
. The CCD algorithm
estimates a model pose by computing the maximum of the a posteriori probability
according to
p
T
I
∗
=
p
I
∗
|
T
·
|
p(
T
).
(2.24)
Since the probability distributions of
p(
T
)
and
p(I
∗
|
T
)
are unknown, they are
approximated. The Gaussian a priori density
p(
T
)
≈
p(
T
m
T
, Σ
T
)
of the model
|
m
T
and the covariance matrix
Σ
T
.The
parameters
T
is defined by the mean
likelihood function
p(I
∗
|
T
)
is approximated by the Gaussian density function
p(I
∗
|
S(
m
T
,Σ
T
))
and describes how well the pixel values along the perpendicular
fit estimated probability distributions
S(
m
T
,Σ
T
)
close to the curve. The observation
model, i.e. the likelihood function, is computed by
h
k
2
Σ
k,l
p
I
∗
|
S(
m
T
,Σ
T
)
=
e
−
1
√
2
π
Σ
k,l
·
with
(2.25)
·
k,l
p
I
k,l
|
m
k,l
,Σ
k,l
−
h
k
=
m
k,l
(2.26)
where
l
defines the pixel index on the perpendicular
k
and
p(I
k,l
|
m
k,l
,Σ
k,l
)
a Gaus-
sian probability density with mean
m
k,l
and covariance
Σ
k,l
according to
m
k,l
=
a
l,
1
·
m
k,
1
+
a
l,
2
·
m
k,
2
(2.27)
Σ
k,
2
.
(2.28)
These values depend on the probabilistic assignment
a
l,s
of the pixel with index
l
and the two-sided probability distributions
S(
m
T
,Σ
T
)
of the pixel grey values with
the mean vector
m
k,s
and covariance matrix
Σ
k,s
for perpendicular
k
.
To increase the numerical stability of the optimisation procedure, the log-
likelihood
Σ
k,l
=
a
l,
1
·
Σ
k,
1
+
a
l,
2
·
m
T
, Σ
T
)
,
(2.29)
rather than the a posteriori probability according to Eq. (
2.24
) is computed, and
the curve density parameters
(
m
T
,Σ
T
)
are refined by a single step of a Newton-
Raphson optimisation.
2ln
p
I
∗
|
S(
m
T
,Σ
T
)
·
X
=−
p(
T
|
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