Environmental Engineering Reference
In-Depth Information
are given as discrete 2D functions
g
1
(
x
,
y
)and
g
2
(
x
,
y
), which
might have been derived from analog images.
g
1
(
x
,
y
)and
g
2
(
x
,
y
)
can be defined as conjugate regions of a stereo pair in the 'left'
and the 'right' image respectively.
g
1
(
x
,
y
) is interpreted in the
following as the 'template',
g
2
(
x
,
y
) as the 'picture'. Correlation
is (ideally) established if
g
1
(
x
,
y
)
where,
L
(
i
,
i
+
1)
is perpendicular to
L
(
i
−
1,
i
)
, which can be repre-
sented as
(
x
i
+
1
−
x
i
)(
x
i
−
1
−
x
i
)
+
(
y
i
+
1
−
y
i
)(
y
i
−
1
−
y
i
)
−
l
i
=
0 (6.11)
By linearizing Equation 6.11, we can write
g
2
(
x
,
y
) (6.6)
However, because of the random noise
n
, Equation 6.6 is not
consistent. In other word, there is error
v
between the template
and the picture.
=
(
x
i
+
1
−
x
i
)
·
d
x
i
−
1
−
(
x
i
+
1
+
x
i
−
1
−
2
x
i
)
·
d
x
i
+
(
x
i
−
1
−
x
i
)
·
d
x
i
+
1
+
(
y
i
+
1
−
y
i
)
·
d
y
i
−
1
+
(
y
i
+
1
+
y
i
−
1
−
2
y
i
)
·
d
y
i
+
(
y
i
−
1
−
y
i
)
·
d
y
i
(6.12)
v
=
g
1
(
x
,
y
)
−
g
2
(
x
,
y
)
(6.7)
A combination of Equations 6.8 and 6.12 represents the indirect
adjustment with geometric right-angle constraint by which the
corners coordinate of building will be attained accurately in range
image. This iterative matching processing will go on until the
correction value of the two endpoints of a line meet the criteria
or arrive the given maximal iteration times.
Equation 6.7 can be considered as a non-linear observation
equation, from which the distances between the gray levels in the
template and the picture can be estimated by the least-squares
adjustment.
Depending on the edge shape, the edge template can be
designed in different forms. Figure 6.5 shows the designed edge
template profile and the generated image patch template with the
size of 7
6.2.3.3
Roof surfaces detection
5 pixels, respectively.
If the edge template is defined as
g
1
(
x
,
y
)andtheaerialimage
is defined as
g
2
(
x
,
y
), then the image matching from Equation
6.7, can be given by
×
After regularization of the building boundary, the roof points
of the lidar data in the boundary are available to reconstruct
3D building models. The automated detection of planes is an
essential operation because it can eventually determine the roof
shapes. Currently, there are mainly three proposed methods to
carry out this task such as region growing, Hough transform
and random sample consensus (RANSAC) (Fischler and Bolles,
1981). In a recent work, Tarsha-Kurdi, Landes and Grussen-
meyer (2007) compared the Hough transform method with
the RANSAC method for the automated detection of building
roof planes and concluded that the RANSAC algorithm was
more reliable in detecting the roof planes. In this study, we
use the RANSAC algorithm to search the best plane from 3D
points. The RANSAC algorithm is implemented by the following
four steps:
∂g
∂x
d
x
∂g
∂y
d
y
=
+
−
g
ν
(
x
,
y
)
g
=
g
1
(
x
,
y
)
−
g
2
(
x
,
y
)
⎨
∂g
∂y
d
y
−
g
g
=
g
1
(
x
,
y
)
−
g
2
(
x
,
y
)
∂g
∂x
d
x
+
v
(
x
,
y
)
=
(6.8)
⎩
where
∂g/∂x
is the derivative in the
x
direction,
∂g/∂y
is the
derivative in the
y
direction. They can be approximated by the
first-order difference.
As mentioned above, most of the buildings in the area
have rectangular or near rectangular shape. Therefore, we could
assume that the line segments as shown in Fig. 6.5 (right),
L
(
i
,
i
+
1)
and
L
(
i
−
1,
i
)
are perpendicular each other. Each line segment can
be represented by its two end points.
1
To randomly select three points and calculate the parameters
of the corresponding plane
P
, because the smallest number of
three points is sufficient to determine the parameters of 3D
plane model.
2
To detect every single building point and determine whether
or not belonging to the calculated plane
P
with a predefined
distance threshold
τ
.
y
i
x
i
+
1
−
x
i
(
x
y
i
+
1
−
L
(
i
,
i
+
1)
::
(
y
−
y
i
)
=
−
x
i
)
(6.9)
y
i
x
i
−
1
−
x
i
(
x
−
x
i
)
y
i
−
1
−
L
(
i
−
1,
i
)
::
(
y
−
y
i
)
=
(6.10)
g
P
i
+1
(
x
i
+ 1
,
y
i
+1
)
E
w
/2
L
(
ij
+
1)
x
L
(
i
−
1
j
)
00 13 56 6
p
i
(
x
i
,
y
i
)
P
i
−
1
(
x
i
−
1
,
y
i
−
1
)
(a)
(b)
(c)
FIGURE 6.5
Least-squares template matching with the right-angle constraint: (a) edge profile template, (b) discrete template,
(c) perpendicular condition.
Search WWH ::
Custom Search