The scene data y is a vectorial MRF y. = (vi, ve,) where 
Yr. and Yc, are the intensity and the coherence, respectively. If 
I, and ye, are considered as independent, (2.9) can be written 
as (Schistad Solberg & Taxt, 1994) 
PAG SC. 3e, = 
Hy le) + H,(yc Je.) + H,(e,|0e,) 
The components of (2.10) are explained in detail in chapters 3 
and 4. 
(2.10) 
3. PRIOR KNOWLEDGE ABOUT LINEAR 
STRUCTURES 
Two types of prior knowledge are expressed by the prior PDF: 
the generic knowledge about continuous, elongated linear 
structures, and the specific knowledge about certain linear 
structures given by a GIS. 
3.1 Generic Knowledge About Continuous Curvilinear 
Structures 
The model of continuous curvilinear structures was inspired by 
the work of Williams & Jacobs (1995) about stochastic com- 
pletion fields. They describe occluded, but perceptually salient 
contours with random walks of particles having its source at 
unoccluded points of the contours. The path most probably 
taken by the particles is assumed to be the location of the 
illusory contour. We use a similar random walk model to 
derive the potentials of two-pixel cliques of a neighborhood 
Gibbs field. A neighboring line site ¢ is treated as a source of 
random walks whereas the site s, i.e. in terms of MRF the site 
for which the energy is computed, serves as a sink. The more 
particles pass through s the higher is the probability that s is a 
line site. 
In section 2 we assumed a site has the object parameter values 
€, line site" or "no-line site". We will now refine this rather 
general model. A line or a migrating particle passing a site has 
more properties than only its quality of being a line or a line 
particle. Its path has a certain direction and curvature which 
can be estimated as well. Thus the state space E, of the object 
parameters €, becomes 
E, ={"no—line","line(0,x ;)':ie {1.1}, je{L...}} 6.) 
where 0; are / discrete directions equally spaced in the interval 
[0], and Kj are J discrete curvatures equally spaced in the 
interval [Ka Nas. and Kmax 1S the magnitude of a maxi- 
mum curvature. Note that £s is still a scalar. 
The particles of the random walk originate at a certain position 
(Xo, yg) in the x, y-coordinate plane and possess a direction 09 
and a curvature K, (Fig. 3.1). During each step of the random 
walks x, y, 0, and x are updated according to 
314 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
JUNE SE SUM K 
ÿ=T-si(o +15) (3.2) 
6 =Ix +6(0,0g) 
K =K(0,0, ) 
= À K oe . : ; 
where / ~ Lain i. X, y,0 andK specify the change in 
K 
position, direction and curvature, / is the step size, and 6 and 
K are normally distributed, zero-mean random variables with 
standard deviations og and o,. With each step a certain frac- 
1 
tion of the particles decays. The probability of decay is /—e * 
at each step where 1 is a decay parameter being large for strong 
or long lines and small for weak or short lines. Figs. 3.2 and 
3.3 show examples of simulated random walks. At each grid 
point particles have been counted differentiated by state ac- 
cording to (3.1). For reasons of better presentation the curva- 
ture-state counts have been combined. The resulting count for 
each direction state is shown by a line in the appropriate di- 
rection with a length proportional to the logarithm of the count. 
ME 
  
  
«Y 
P. Va 
Fig. 3.1. One step of a random walk. 
  
  
  
  
e 1 2 3 4 5 6 7 8 9 
Fig. 3.2. Random walk simulation with k,=-0.1, 1-30, 6970.02, 
0,=0.02. 
   
  
Fig.3.3. 
For the 
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1994). 
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for €, 
neighb: 
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Fig. 3.4. 
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