he spot in the
ted.
andard devia-
regions, vy of
eshold some-
ny structures.
influenced by
laced by the
ether the line
| of both side
egion and the
ership in the
regions which
ve adopted a
ratio r which
(4.3)
1e number of
oks per pixel.
by the size of
the number of
4 for various
averages and
good approxi-
antly different
(4.4)
om both side
le regions are
where r7 is a
(4.5)
If the side regions are not similar, the ratio response of the
operator is the larger one of the normalized ratios between the
line region and one of the side regions, i.e. the ratio towards
the less contrasting side. If the side regions are similar, a
common median and a normalized ratio between the line and
the united side regions is computed.
probability density
0
0 0.10.2 0.30.4 0.5 0.6 0.7 0.80.9 1
ratio
Fig. 4.2. PDF of the normalized intensity ratio for
various contrasts I;/15.
From the resulting ratio responses the energy values of the
intensity data H,(y; Je.) are derived. Instead of the intensity
of a pixel we utilize the ratio rg to compute Hy, i.e. we set
H,(y1,les) = B,(rle,) (4.6)
where r, is a derived observation.
5
A reasonable way to derive energies from observations is to
assume normally distributed observations. For those the energy
H is computed from (Kóster, 1995)
2
H(x)- eu (4.7)
26,
where pi, and o, are mean and standard deviation of the
normal distribution. From (4.3) it is known that the ratio is not
normally distributed. Nevertheless using (4.3) instead of (4.7)
poses difficulties, as we do not have any reasonable assump-
tion about the line contrast I;/I, a line might have. Further-
more, r, was computed using several tests for region homo-
geneity and similarity which is why the distribution of r, is not
strictly (4.3). Therefore, we propose to compute H, re.) from
(4.7) where p,, —0 for line sites (€, =" line(6,x)"; cf. (3.1)) and
6, is roughly adjusted to the line contrast in the processed
I's
scene.
For no-line sites (€, ="no — line") we propose to use a uniform
distribution instead of a normal distribution
H, (x)= constant (4.8)
This idea is borrowed from maximum-likelihood classification
of multispectral imagery where for the land-use classes normal
distributions are utilized (e.g. Richards, 1993). If the maximum
density Pmax ( ysles) is less than a threshold T, pixel s with the
multispectral data vector y, will not be classified. Such a pixel
corresponds to a no-line site in our case. A reasonable value for
T can be derived from (4.7) when a maximum ratio 7j for
line pixels is assumed. r,,,, can be determined from a sample
image.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
We are now able to formulate the complete energy function of
the intensity ratio:
as if s is a line site
o
H,(,les)= y m (4.9)
—. ifsisano-line site
20,
The PDF of this energy function is illustrated by Fig. 4.3. The
bell-shaped line shows the PDF of a line site based on a
normal distribution, and the horizontal line shows the PDF of a
no-line site. Using Hy(rle,) without a prior energy
H,(e,loe,). which is the case with any maximum-likelihood
classification, is equivalent to thresholding the response of the
ratio operator.
normalized density
e "m 0.4 Sio = o 1
ratio
Fig. 4.3. PDF of the operator output intensity ratio.
The coherence is processed in a similar manner as the inten-
sity. The data is evaluated applying the same detector masks,
but instead of the ratio the difference is computed. The checks
for homogeneity, dissimilarity and similarity of regions are
essentially the same as for intensity data except that the
thresholds are derived much more empirically, as the statistical
properties of coherence data are not as well known as those of
intensity data (but cf. Tough et al, 1994a; 1994b).
Hc, le.) = H,(d,je,) is computed according to (4.9) where
all ratios r are replaced by differences d.
5. ESTIMATION OF THE OBJECT PARAMETERS
Our goal is the estimation of the object parameter vector € (cf.
(3.1)). An optimal € is the ore that results in a global maxi-
mum of p(ely) which is difficult to determine owing to the
overwhelmingly large configuration space of €. Presently, we
apply two methods two approximate the global optimum. As an
approximation to a MAP estimation we use simulated anneal-
ing in combination with Gibbs or Metropolis sampling, and as
a faster deterministic approach which is only guaranteed to
find a local optimum Besag’s ICM estimator. As these methods
have been intensively treated in various publications (e.g.
Geman & Geman, 1984; Busch, 1992; Koch & Schmidt, 1994;
Winkler, 1995; Guyon, 1995; Koster, 1995), we will only give
a brief description.
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