83. Istanbul 2004
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liscussed below.
OOF FACETS
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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
do not belong to any. For example, at two discontinuous
adjacent roofs with different heights, the laser beam might hit
the wall between those two roofs. In such circumstances, the
best fitting procedure will assign those points to one of the
adjacent roofs even though in reality they do not belong to any
which would result in high slope values.
Moreover, the height intercept also needs to have range limits
as the other two factors. This scaling step is to make the
parameter space homogenous. As in the slope parameters, a few
spikes in the estimated height response were recorded. Unlike
the slope case, limits on the height intercept cannot be predicted
since roof height varies within the same building with a wide
range. First a histogram of the height intercept of the processed
area was constructed. Then values out of the range + ko (k can
take any value from 0 to 2 based on the shape of the histogram
and the outliers values) will be discarded since they don’t seem
to be valid and they are a result of points on edges as discussed
above. This step centers the mean value of the parameter in the
new range and reforms the spread of the data. Then the
resulting values are scaled down to have the range from -1 to
*] as in the other two parameters. The trimming and scaling
procedure are shown in equation (1) and (2). Figure 2 shows
color-coded image of H of the same building before and after
trimming and scaling.
H*tko If H x.y) >u+ ko
Ht (x yy = U ko i H(x y) «gu -ko (1)
H elsewhere
(x,»)
Ht ivy oH
His, py ay (2)
2 20
where H : height intercept
Ht : trimmed value of 7
Hts : trimmed-scaled value of H
H : mean value of H inside a building polygon
6 : standard deviation of H inside a building polygon
Y inm
Figure 2: Height intercept color-coded image before and after
the trimming and rescaling procedure.
3.1.2 Membership criteria (cost function)
The membership criterion between two cells to define whether
they belong to the same roof segment or not is the Euclidean
distance in the parameter space between the two points. If the
cost function between the center of the seed (cell 7) and the
processed cell (cell j) is less than a defined threshold of the
membership criteria, then they belong to the same roof segment.
However, at the beginning as is known in the region growing
segmentation, the candidate cell or pixel should share an
adjacent boundary with the growing region.
3.1.3 2D parameter space
For simple gable roofs, slope in x and slope in y can form a
satisfactory parameter space for the roof features. This is due to
the fact that gable roof pair segments have well defined reverse
slopes as shown in figure 3(a). The 2D search space of the same
building is shown in figure 3(b) where its first axis X is the
slope in x and the second axis Y is the slope in y. Figure 3(c)
shows the raw result of the region growing segmentation
procedure and the labeled roof segments in the parameter space.
As shown in the search space, some pixels are not labeled (red
crosses, figure 3(c)) since they don't belong to any class based
on their parameters. However, those cells will be assigned to
the nearest roof segment in term of position in the object space
not in the search space as shown in figure 3(d). However, in a
complex roof structure, these two parameters are not always
capable of discriminating between all of the segments. Another
parameter may be added as in the following section.
3.1.4 3D parameter space
In more complex roof structures, a third parameter is desirable
to add to the parameter space to increase its information content
and consequently detect a more complete and precise set of roof
segments. Slope in x, slope in y, and height intercept form the
3D parameter space and shape the membership criteria. This
dimensional increment improves the seprablity between classes
(roof segments regions) in the parameter space, which enhances
the possibility to detect roof segments with same slope but with
different heights. Figure 4 shows the procedure and results of
the roof facet segmentation utilizing the estimated surface
parameters resulting from the least squares moving surfaces. As
it shown clearly below, the third vector (height intercept)
enables the system to detect the four elevated rectangular
structures in the lower part of the building; while in the 2D
parameter space (slopes in x and y) the system was not able to
detect them.
f — 398 08 04 42 0 02 04 08 08 1
Slope in x
1 c 1 AM
08! : 08;
» "Ünciassitied Celis (pixels)
06; 9s;
04} x 04; 2
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o4 i 0
06 08
08 08
id 0.8 06 04 0.2 9 02 0.4 06 0.8 1 + JR 06 04 52 n 02 04 08 08 1
Slope in x Slope inx
Figure 3: (a) Estimated slope in y for a simple gable roof
building, (b) 2D search space based on slopes in
