ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002 
  
  
Figure 2: A plane in 3-D space 
non-linear variations as the surface slopes increase. Mea- 
suring surface consistency may become then more diffi- 
cult. With the implicit representation of a plane 
O=azx+by+cz+p (1) 
with a normalized normal vector || zi ||2 1 (where 5? := 
[a b c] T any surface can be defined uniquely, but the 
non-linearity is still not solved and furthermore the dimen- 
sionality of the feature vector increases. Maintaining the 
three parameter representation and circumventing the non- 
linearity can be achieved by a polar representation of the 
surface normal. As is illustrated in Figure (2), with this 
representation the surface slopes angles can be computed 
as 
ó — cos"! (c) Q) 
0 —tan^! (2) (3) 
a 
The normal direction is then given by, 
cos0cos® 
sin0cos® (4) 
1 
ñ = R, (6) R, (0) |0| — 
0 sino 
leading to the polar representation 
0 = cosûcose x + sinfcospy + singz +p (5) 
In this representation, p, is the distance from the origin to 
the plane. A plane is now defined by the angles, ¢, 0 with 
angular units (radians or degrees), and p with a metric unit. 
2.3 Surface texture analysis 
Surface texture is usually analyzed by measuring the at- 
tributes variation in the neighborhood (usually a window) 
of each point and identifying the point’s class by these 
measures. While point classification implicitly assumes 
homogeneous texture inside the window, that might not al- 
ways be the case. For example, in inhomogeneous cases 
where different processes are covered by the window (e.g., 
around edges and building corners) erroneous surface classes 
will be assigned to the laser points. So, by assigning a 
class to each point this approach becomes rather restrictive 
in forming clusters in the data and relies heavily on the 
neighborhood size that is chosen. 
The approach taken here is different. It is based on direct 
evaluation of the points features in a feature space with 
dimensions similar to the one of the feature vector; the 
values of the feature vector for each laser point determine 
the point's coordinate in the feature space. Data clustering 
is conducted in this space via unsupervised classification. 
Notice that by analyzing all the points simultaneously no 
window based analysis is needed at all, in fact all windows 
are analyzed simultaneously. To accelerate the clustering 
of the data the implementation of the algorithm here parti- 
tions the feature vector into a 4D attribute space consisting 
of the tangent plane parameters and the height differences, 
and the 3D point position in object space. The removal 
of the positional content does not allow for establishing 
proximity measures in the feature space. The clusters in 
this space can only be considered as "surface classes" that 
contain all of the points that share similar features. Ta- 
ble 1 shows that the attributes are sufficient for extracting 
distinct surfaces classes, but a surface class may consist 
of more than one point cluster in object space. Thus, fol- 
lowing the surface class extraction, point clusters are iden- 
tified in object space by proximity measures. The current 
implementation uses a topological neighborhood that is es- 
tablished by the triangulation of the dataset as a measure. 
Smooth surfaces tend to cluster in the attribute space but 
"vegetation" surfaces (categories (i) and (ii) in Table 1) do 
not. Rough or "vegetation" surfaces are defined by their 
lack of consistency, and are identified by analyzing the un- 
clustered points. The surface attributes that are used here, 
in particular the surface normals, enhance the tendency of 
vegetation not cluster. One consequence is that vegetation 
and structured surfaces are unlikely to be grouped together. 
Clustering the "vegetation" points is carried out by analyz- 
ing the “unstructured” points. The separation between high 
vegetation and low vegetation is conducted by analyzing 
the points according to their height difference and graph 
connectivity, although in mixed areas such separation may 
not be possible. 
2.3.1 Relation to other parameter-space based repre- 
sentations As the tangent plane parameters are the key 
feature in identifying surface structure (height differences 
are mostly used to eliminate edge points from the analy- 
sis) one may associate this representation and the Hough- 
transform for planar surfaces. In reality, this similarity 
is rather limited but the comparison enables illuminating 
some properties of the current representation. The Hough 
transform (see details in Vosselman, 2001, for example) is 
optimal for grouping data that have no connectivity, for ex- 
ample the set of all points that form together a line, or in 
this case a plane. However, connectivity between points 
is one of the more important attributes of the sought after 
surface elements. By identifying surface of unconnected 
points the Hough transform generates proposals for many 
spurious planes that do not exist in reality. With the in- 
crease of the size of the dataset that is processed the num- 
ber of spurious surfaces will increase rapidly. The extrac- 
tion of real surface from the Hough space will become 
complex and slow, and identifying physical surfaces with 
a relatively small number points will become more diffi- 
cult. The current, feature based representation, computes 
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