Environmental Engineering Reference
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(1) to define the number and nature of the information classes,
and collect sufficient and representative training data for each
class; (2) to estimate the required statistical parameters from the
training data; and (3) to apply an appropriated decision rule.
Given the class
w
i
(
i
=
1, 2,
···
n
), where
n
is the number of the
classes. Features mentioned above yield a
D
dimension feature
space
F
. So, the probability of a region represented by its feature
vector
X
(
X
The conditional probability of the class
w
i
is ascertained
by the choices of the weights (
m
1
,
m
2
,
m
3
,
m
4
)tofeatures
(
X
1
(
f
),
X
2
(
f
),
X
3
(
f
),
X
4
(
f
)) in
P
(
X/w
i
)
=
m
1
×
X
1
(
f
)
+
m
2
×
X
2
(
f
)
+
m
3
×
X
3
(
f
)
+
m
4
×
X
4
(
f
)
(6.5)
For each region, we canget three class values
{
d
1
(
X
),
d
2
(
X
),
d
3
(
X
)
}
according to Equation 6.3. The maximum of three results
d
i
(
X
)
=
max
{
d
1
(
X
),
d
2
(
X
),
d
3
(
X
)
}
labels the region as to which
of classes it belongs. If
d
1
(
X
)
≈
d
2
(
X
)
≈
d
3
(
X
), the region is
temporary to be labeled as the unclassified class.
∈
F
)belongstoclass
w
i
, is defined by Bayes' rule:
P
(
X/w
i
)
•
P
(
w
i
)
P
(
X
)
P
(
w
i
/X
)
=
(6.2)
where
P
(
w
i
) is the prior probability of class
w
i
,
P
(
X/w
i
)is
the conditional probability of class
w
i
has data
X
,
P
(
w
i
/X
)isthe
posterior probability of data
X
belonging to class
w
i
,
P
(
X
)canbe
considered as constant value for class
w
i
. Therefore, Equation 6.2
can be reduced to
6.2.3
Building reconstruction
After having successfully detected the isolated building regions
from lidar data, the next step is building reconstruction.
d
i
(
X
)
=
P
(
X/w
i
)
•
P
(
w
i
)
(6.3)
=
where,
I
1, 2, and 3, defining three distinctive classes: buildings
w
1
,bareground
w
2
, and trees
w
3
. The objects that do not belong
to these three classes are labelled as unclassified ones
w
4
.The
prior probabilities of buildings
P
(
w
1
), bare ground
P
(
w
2
), trees,
P
(
w
3
) and unclassified objects
P
(
w
4
) are obtained according to
specific training data set that can represents the typical features
of urban areas, and meet
6.2.3.1
Simplification of building
boundaries
The majority of buildings in real world are the vertical walls
which form the boundaries of buildings. As shown in Fig. 6.4(a),
the extracted building boundary at first is recorded as an order
sequence of feature points which form a set of ragged small
line segments. Thus, the boundaries of buildings are very noisy.
The Douglas-Peucker algorithm is first employed to generalize
boundary segments so that redundant points can be removed.
Figure 6.4(b) illustrates the simplification of building regions by
the Douglas-Peuker algorithm.
However, the Douglas-Peuker algorithm can only remove
redundant points from those small line segments that are
not perpendicular to each other. Therefore, least-squares tem-
plate matching with right-angle constraint is further applied to
accurately determine the building boundaries with a rectangu-
lar shape.
P
(
w
1
)
+
P
(
w
2
)
+
P
(
w
3
)
+
P
(
w
4
)
=
1
(6.4)
Therefore, the next step is to quantify features before determining
conditional probability of each class.
1
The spatial information of the lidar point using eigen-analysis
X
1
(
f
) is the ratio of the number of points belonging to
''scatter'' and ''edge'' points to the amount points in the
homogenous region. 1
X
1
(
f
) means the percentage of
''plane'' points in the region.
2
The filtering result
X
2
(
f
) is the ratio of the number of
points labelling as on-terrain points to the amount points
in the homogenous region. 1
−
X
2
(
f
) implies the percent of
off-terrain points in region.
3
The height difference of first-echo and last-echo
X
3
(
f
)is
the ratio of the number of points that height differences
are over the given threshold, to the amount points in the
homogenous region.
4
The linear features of the aerial image
X
4
(
f
)aretheration
of the length of line segments near to the region to the
length of region boundary.
−
6.2.3.2
Least-squares template
matching with right-angle constraint
The least-squares matching proposed by Ackermann (1983) uses
plenty of information in image to construct an adjustment model,
by which the matching precision can reach 0.1 pixels, even 0.01
pixels. Gruen (1985) extended the least-squares matching to the
least-squares template matching. Assuming two image regions
(a)
(b)
FIGURE 6.4
An example of boundaries simplified by Douglas-Peuker algorithm.
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