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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