in in optical
approach to
inobservable
neighboring
del. Locally,
2 response of
'riori (MAP)
wal effort the
es intensity,
al evaluation
gs Caves et
ons had the
lited for the
ent imaging
Geiss, 1984;
. Among the
nputing the
lown to give
1990; Caves,
the standard
itself.
thods based
t al, 1986;
inimum-cost
rogramming
ges. In spite
n is limited,
detection of
obal evalua-
(1990) and
and a MAP
to observe
prior prob-
nes are con-
iT,
Our new approach for the extraction of linear structures is
related to these methods, as it is based on Bayesian inference
and formulates prior knowledge about the continuity of lines as
an MRF. To overcome the difficulties in the detection of linear
structures the approach integrates generic knowledge about
lines, given GIS data and the SAR scene data. The generic
knowledge can be subdivided into three parts. The first part is
the knowledge about the physical appearance of lines, i.e.
narrow, elongated areas with approximately constant image
intensity (see above). This type of knowledge is used to evalu-
ate the scene data. In terms of Bayesian approaches it is there-
fore incorporated in the conditional probability density function
(PDF) to observe scene data given a linear structure. The
second part of knowledge about lines says that a line is con-
tinuous over a certain region of the scene. This means that a
line can be assumed in a location where there is not enough
physical evidence, if neighboring locations show sufficient
evidence. This knowledge is derived from a random walk
model and used in the prior PDF modeling the relationships
between pixels of linear structures based on an MRF. In addi-
tion to the generic knowledge about the appearance of linear
structures, the specific knowledge of the presence of a certain
linear structure as given by a GIS is incorporated into the
approach as third part of the knowledge. At pixels located at or
close to where the GIS indicates a linear structure the prob-
ability to detect a linear structure having the corresponding
direction is higher than at pixels at a larger distance.
As SAR data intensity is used optionally complemented by
coherence resulting from an interferometric evaluation of a
SAR scene pair. This feature is a step towards a utilization of
the full information content of the complex SAR data.
In section 2 we explain how Bayes’ theorem provides the
framework to implement an approach to the detection of linear
structures. Section 3 describes the modeling of the prior PDF
of continuous lines based on an MRF and a random walk
model for particles. In section 4 the conditional PDF for the
local evaluation of the scene data is explained. Section 5 is
dedicated to the computation of an optimal interpretation of the
SAR scene by sampling from the posterior PDF. Finally, in
section 6 the results of tests of the algorithm are presented and,
in section 7, conclusions and recommendations are given.
2. BAYESIAN LINE EXTRACTION USING MARKOV
RANDOM FIELDS
The extraction of linear structures can be based on a Bayesian
approach to solve the inverse problem of computing the loca-
tion of lines from the measured scene data (Oliver, 1991; Koch
& Schmidt, 1994; Winkler, 1995). The posterior probability
density of the object parameters given the scene data is derived
according to Bayes’ theorem
ole) = £25.20) e € @1)
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
where £ is the object parameter vector. € contains one element
£, for each site se S, i.e. regularly for each pixel of the scene 5.
Depending on whether the object is described by one or more
parameters at each site, E. can be a scalar or a vector. For the
time being we assume that €, is a scalar taking the state "line
site". or "no-ine site”, ie. the ‘state: space is
E, — [" line site" ," no — line site"). Sometimes we will have to
refer to a site object parameter variable which takes a specific
state, i.e. we will have to make a difference between a variable
and its instantiation. In formulas we will express this as E,=€,-
The scene data vector y also contains one element y, for each
site of the scene. As our goal is the combined evaluation of
intensity and coherence, y, is vectorial. The probability density
of the data vector p(y) can be omitted, because it is independ-
ent of £; then Bayes' theorem becomes
r(eb) « role): »€). (2.2)
The prior probability density p(£) and the conditional probabil-
ity density of the scene data given the object parameters p(ylg)
are to be formulated according to our knowledge about linear
structures and the scene formation process.
To simplify the estimation of the object parameter at a site s
we assume the object parameters as well as the scene data to
be MRF. A random field is Markovian, if for all x
px. * s) = px, ax;) (2.3)
where dx, is a neighborhood of s considerably smaller than
the complete scene. Using this assumption, the conditional
density of an object parameter value at a site s is
pe. |v..2»,.96,) e pov 9v,..): pe. oe,). | 2.
In the case of dy, = {} , i.e. independence of the data from its
neighbors, (2.4) simplifies to
ple,|y,.0e,) « ply,les)- ple on). (2.5)
This is strictly true only for uncorrelated data.
For further reasoning we use the equivalence of MRF and
neighborhood Gibbs fields. In Gibbsian form the probability
density p(x) is expressed as
p(x)- ica wad (2.6)
Y expl —H(z)
zex,
where X, is the configuration space of X, i.e. the set containing
all possible instantiations of X. The energy function H(x) of an
MRF which is equivalent to a neighborhood Gibbs field is
H(x)= X U a(x) (2.7)
AcS
where each clique A is a subset of the scene S containing sites
with a certain geometric configuration, and Uy is a potential of
A. The conditional probability density at a site s results from a
S
summation over the set K of all cliques A containing s
plx,[ox,) oe cts (2.8)
where H,(x,lèx,) BU m . Now we are able to
AeK,
express (2.5) in terms of energies:
Hy(e,|y, dg, ) = H,(v,les) + H,(e,|0e,) ; (2.9)
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