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# Stochastic model for a line network

Bayesian approach

Notations :
• D : the observed image
• T : finite grid on the image
• : the gray level of a pixel

Hypothesis : the line network, S, we want to extract is composed of a finite number of segments si

 (1)

We have :
• si=(pi,mi) : a segment in the configuration
• pi=(xi,yi) : the coordinates of its center
• : the vector containing the width, the length and the orientation of the segment

Problem : to detect the number of the segments, their location and their parameters

Solution : Bayes rule

 (2)

MAP estimation :

 (3)

Considering we have a Gibbs point process :

 (4)

The estimator of the line network is :

 (5)

Candy model

The parameters of a segment are independent random variables :
• : the coordinates of the center of the segment
• : the width of the segment
• : the length of the segment
• : the orientation of the segment
is a uniform law.

State of segment : a segment has two extremities to be connected with
• : no connection
• s=st||q : one connected extremity
• s=stq : two connected extremities

Probability density :

 (6)

State penalties : the short segments and the free segments are penalized

 (7)

 (8)

 (9)

We have : .

Rejection interactions : we penalize the overlaping segments, but we enable the crossing segments.

Figure: Rejection region for a segment

Figure 2: Rejection interactions btw segments

Attraction interactions : to form a network, the segments attract each other. We penalize the segments which are not well aligned.

Figure 3: Attraction region for a segment

Figure 4: Attraction interactions btw segments

If there is a rejection interaction btw two segments :

 (10)

If there is an attraction interaction btw two segments, the function h is penalizing the orientation between segments :

 (11)

For each segment, the local prior energy is:

 (12)

hence for the configuration S :

 UP(S) = = (13)

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