(d) The points in the rest group are clustered into several 
clusters according to their distances to the nearest neighboring 
points. 
(e) For the points in every cluster, the process is repeated from 
the step (a) to (c). 
(g) The process terminates for a cluster when there are less than 
5 points in the rest group. 
The process descript above is called line-growth method in our 
work, because points on a line segment are found out iteratively. 
After this process, the unorganized points on roof ridges are 
segmented into several line segments. In fact, there are only one 
or two roof ridges for most of residential buildings in European 
cities. For most of public or historic buildings there are 
normally three or more roof ridges. The process is illustrated 
using an example in Figure 2. 
* 999 9.9 099 9 9 99 o? o6 
ee. 
ee... 
e. 
© d, 
| f | . 
Unorganized points on roof ridges 
f* «e«« p * 944.04 = 
: E 
z 
G) ; 
ee DB nn a e 
; TE EG ES e ^ LJ 
Line segment #77 TTT ee, a 
M ET ee A iuc. 
ie! ce”) >, e \ 
le! lef te 
ij x; ne 
2 S000 
tee. 
° 
A ; NES 
The points are classified in Rest group s 9 
Detection of the first ridge 
Le 
e e t 
. . Te, 
© : : e. 
° e >. *. 
. . e x 
e * 
om . * 
z . No 
The points in Rest group are NS - 
segmented in 3 clusters, in which roof 
ridge is detected respectively 
Figure 2. an example of detecting roof ridges 
3.2 Segmentation of points into individual roof plane patch 
In the proposed approach, we assume that the roof of a house 
can be decomposed of a set of planar patches. The process of 
the segmentation is then to fit the planes and find the points on 
their corresponding plane patches. In this section, it is 
introduced how a plane is fitted for a set of points at first. Then 
the process of the segmentation is described. 
3.2.1 Method of plane fitting 
All the points of a roof part should belong to the same planar 
patch. The plane can be defined by using the following equation: 
ax+by+cz+d=0 (1) 
where 7 =[a, b,c] is the normal vector of the plane and d is 
the closest distance of the plane to the origin of the coordinate 
system. 
98 
Assuming the point (x,,y,,z,) is located on the plane, the 
plane of equation (1) can be reformed for all the points of this 
plane as follows: 
a(x X) t b(y - y) *ce(z-z,) 0 Q) 
For the planar patch, the point (x, Vo-Z, ) can be taken from the 
centroid of all the points of the roof part. 
Equation (2) can be represented as: 
X ZN Vin» 27:2) 
ATX Va 7 Vo 227429 
X, — Xo a 7 Va Zn 49 
Where (x;,¥,,2,) . (5, Y4,2,) > --.» and (x,,¥,,2,) are 
points of the roof part. 
The normal vector can be obtained by calculating the 
normalized eigenvectors of N, whereby N= AT. A. 
Finally, the parameter d in Eq. 2 can be calculated by: 
d =—(ax, + by, +cz,) (4) 
3.2.0 Segmentation of points of a roof with one ridge 
As shown in Figure 1, there is only one ridge in three kinds of 
roofs, namely, gabled roof, hipped roof and saltbox roof. For 
the points of the roof ridge, a plane F, is fitted which is 
perpendicularly to the horizon and contains the adjusted line 
segment of this roof ridge. In this work, the points of the roof 
ridge are duplicated and given different heights along the 
vertical direction, in order to ensure that they are located on a 
plane perpendicularly to the horizon. 
Assume that the plane F has the parameters of a,b,c, and 
d_, the points can be easily divided into two clouds by judging 
the vectors of the points to the adjusted plane: if there is 
ax+by+cz+d > (0, the point (x, y, z) is located on the 
same side of the direction of n —[a,,5,,c,] ; while it is 
located on the opposite of the plane 
whena,x t b, y t c,z * d, «O0. 
In the next step, the distances of points in these two clouds to 
the line segment of the roof ridge are calculated respectively. 
For the points in one of the two point clouds, those points are 
selected if their distances to the line segment are shorter than a 
given threshold. These points will be clustered directly as the 
points located on the plane patch which is intersected with 
another plane patch at the roof ridge. Then a plane is fitted 
using these points. For all the points in the current point cloud, 
their distances to the fitted plane are calculated. According to 
the roughness of most of roofs, if the distance is smaller than 
0.2 meter, the point is certainly located on the plane patch and 
identified as a point in the segment of this plane patch. 
Otherwise, it will be classified in the rest group. The same 
process is conducted for the other point cloud as well. 
In case of a gabled or saltbox roof, the rest group is empty, if 
there is no error measurement, while it still contains points of 
two plane patches, if the roof is a hipped roof. In this case, the 
points in the rest group will be clustered into two groups 
representing the hipped roof parts. In this way, the points on a 
gabled roof can be segmented into two (gabled or saltbox roof) 
or f 
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