ing all the points the grouping process identified
as candidates for planar surface patches.
plane analysis examines the planar surface patches,
detects regularities, intersects roof planes and an-
alyzes the resulting roof edges.
Fig. 7 shows a perspective view of the detected humps,
obtained from the DEM. In the interest of brevity we con-
centrate on one hump only, located in the lower left cor-
ner of Fig. 7.
Figure 7: Perspective view of the residential area, indicating
the humps detected. The segmentation results of the hump in
the lower left corner are presented in this section.
In the case of non horizontal roofs, grouping by his-
togram thresholding does not work anymore. Since the
roof points have all different elevations, no peak ap-
pears. Also the spatial context is lost in a histogram
which makes it impossible to distinguish points that be-
long to different surfaces of similar elevation extent. In
short, we need another approach.
We employ the Hough technique to find planar surface
patches within the humps. As described in detail in
Schenk (2000b), a parameter space with a,b,c, the
three parameters of the plane equation Eq. 2, is gen-
erated. A closer examination of this equation reveals
that switching from the original to the parameter rep-
resentation simply changes the role of variables and pa-
rameters. Suppose a, b,c are now variables; then x, y,z
become coefficients, but the equation is still defining a
plane. Let us pick a point P in the spatial domain. A
plane passing through P = Exp. Ve. Zp 1° is defined by
its three parameters—hence it corresponds to the point
[Xp, Yp,Zp]! in the parameter space. A second plane
through P creates another point in the parameter space,
and so on. Where are all the points, generated by all
planes passing through P? They are related by Eq. 2,
that is, they define a plane. We have identified the du-
ality of point to plane relationship between spatial and
parameter domain.
The following steps find planes that pass through sur-
face points:
1. Pick a point P; from the hump region.
2. Point P; defines a plane in the parameter space.
Increment all cells in the discrete parameter space
(accumulator array) that are on this plane.
3. Repeat step 1 and 2 until all points are processed.
4. Analyze the accumulator array. Clusters identify
planes; the total number of entries in one cluster
International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 3W14, La Jolla, CA, 9-11 Nov. 1999
corresponds to the number of points that lie on
this particular plane. The spread of the cluster is
a quality measure for how well the plane fits the
points.
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Figure 8: The laser point elevations are grouped into three
planar surface patches. Solid circles and solid squares identify
points that are likely to belong to a roof plane. The crosses
are points on the ground that may lie on a horizontal surface.
Grouping did not classify the points depicted as triangles.
Fig. 8 depicts the result of finding planes within the se-
lected hump region. Solid circles and solid triangles la-
bel candidate points for two different roof planes; circles
refer to ground points that are likely to be on a horizon-
tal surface; crosses mark unclassified points. The final
plane parameters are determined by a least-squares ad-
justment. Table 1 summarizes the results for the three
planes, using the laser points and the DEM points. The
standard deviation for the two roof planes is very simi-
lar to the result obtained from the building analyzed in
the previous section. It confirms again the high ranging
accuracy of the laser points. The standard deviation for
the ground points is considerably higher simply because
Internation:
Table 1: Re:
# pts.
o [m]
# pts.
o [m]
Table 2: Re
azimuth [©
zenith [9]
X
Y
Z
the physical surfac
indicating the rouc
The relatively larg
surfaces computec
sight. Since the DE
pected point accu
fact, the values in
expect from this n
The last quality co
ment concerns the
computed by inter
ridge vector is dir
eters. As shown ir
nents can be used
line, for example k
The azimuth defin
ordinate plane, w
one of the other tv
The ridge was com
data set and from
ments. The edge \
alytical plotter. Ta
ing the angles (azi
building size of 1
gle resulting from
an almost horizon
tions of the three
edge was chosen v
ering the small az
expressed by the
(see Fig. 9).
The numbers in T.
by intersecting tl
fairly close agree
Note that a smal
