The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B4. Beijing 2008 
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2.3 Covariance and Normal Vectors of Object Surfaces 
For each point in the data set we consider all points in a small 
spherical neighborhood, calculate the covariance matrix and the 
eigenvalues and eigenvectors (Gross & Thoennessen, 2006). 
These can be used to detect plane surfaces. 
The eigenvalues A, and eigenvectors e- with /' = 1,2,3 of the 
symmetrical matrix deliver additional features for each point. 
The eigenvalues are invariant concerning rotation of the 
coordinate system. For the decision, whether a point belongs to 
a planar surface West et al. (2004) propose the structure tensor 
planarity = ^—^2- and e 3 is the normal vector of the planes. 
Figure 2. Overlay of flight paths on an airborne image. 
3. TEST SCENE 
A measurement campaign was carried out to investigate the 
influences of the incidence angle on the measured intensity. For 
the scene an urban area including buildings, streets, grassland, 
and trees was selected. The data was gathered with the RIEGL 
LMS-Q560. Several flights with different trajectories to gain 
overlapping stripes were performed. The entire scene is covered 
by a high point density of about 13 points per square meter. 
In 
Figure 2 the different flight paths are drawn. Six flight paths are 
parallel and oriented nearly along the valley starting from the 
west side. Flight path seven crosses the other. 
Figure 3a shows each measured point coloured by its height. A 
first impression of the measured intensity is given by Figure 
3b.The influences of the flight path respectively the local 
incidence angle on the intensity values is shown in 
Figure 4. The trajectories 2 to 5 have about 330m distance. If 
the laser beam hits an object only partially, e.g. a roof edge, the 
resulting intensity decreases (Jutzi et al., 2005). Already a small 
offset (116m) between the two flight path trajectories 2 and 3, 
yields essential different incidence angles like presented in 
Figure 4a and b. The square building with four roof planes on 
the left border of the image ( 
Figure 4a-c) demonstrates, that small angles are given, if the 
plane normal vectors point to the sensor. Larger ones can be 
observed, if the normal vector is orientated to the opposite 
direction. 
a b 
Figure 3. Point cloud for an area of 200x350m 2 : a) coloured by 
height values, b) coloured by intensity values. 
3.1 Influence of the flight paths on the incidence angle 
Figure 4. Influence of the incidence angles on the intensity: 
a) flight path 2, b) flight path 3, c) flight path 4, 
d) flight path 5. 
During flight 2 (Figure 4a) we look from the west side onto the 
roof planes (saddle roof) of the buildings in the middle of the 
image. Therefore measured intensity values on the roof planes 
pointing to west are higher than for the roof planes pointing to 
east. For flight 5 (d) we are looking from east onto the same