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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 T )) and describes how well the pixel values along the perpendicular
fit estimated probability distributions S( m T 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 T ) =
e
1
2 π
Σ k,l ·
with
(2.25)
·
k,l
p I k,l |
m k,l 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 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 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 T ) are refined by a single step of a Newton-
Raphson optimisation.
2ln p I |
S( m T T ) ·
X
=−
p( T
|
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