ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
Geometrical descriptors 
The geometrical properties of an object in a point p; can be 
described intrinsically defining an inertial system centered in 
this point. We mustn't forget that we need to know those 
properties or descriptors that simplify object segmentation. In 
order to reach this result, we note that is desirable that these 
descriptors are singular in discontinuities or, in other words, 
that they have a peak value. We can't consider sign change of 
the descriptors, because of the presence of noise in data. 
In the following are described three different descriptors, static 
moment, curvature and junction, but it is possible to define a 
non-limited number of properties, increasing the dimensions of 
the space of descriptors. So, the dimension of the problem is not 
only the dimension of the space of the objects O", but the 
dimension of the union of this space with the space of 
descriptors D": 
S = oO" cb} 
Static moment: the module of the first order moment (or static 
moment) is maximum at the edge and, in a smaller measure, at 
a change of curvature. Static moments are defined as: 
S =) ME) 
S, ES ms tz) 
S, 2S tn) 
The three static moments can be combined in a comprensive 
descriptor that is the total static moment: 
SS 
  
EESEREENEUTE 
un 
jl 
i 
+ | 
i 
  
e 
V 
| hb 
i^ 
* 
MI 
VOLU 
B 
  
  
  
Figure 2 — Total static moments in an airborne laser scanning 
data set by TopoSys. In (A) the most elevated values of static 
moments are in black. In (B) the black points represent a profile 
in the data set, the yellow points the static moment (with sign 
changed). Note the peak value in discontinuities. 
Curvature: in discrete geometry it is possible to define 
curvature in many ways. Fortunately we don't need the exact 
value of curvature; a functional one is sufficient, beacuse we 
are looking for the location of peak values, not for their 
magnitude. In the hypothesis that the point set is a surface, the 
ratio 
=. Can 
ar 
is a functional of the ray of curvature. In fact A, is null for a 
planar surface and A, is not null; that leads to an infinite 
value for p . Increasing curvature, A, Increases and Am, 1s 
A - 291 
constant, while À … is less than 4,,,. When A; becomes 
min 
equal to À the two eigenvalues swap, A, becomes 
max ? 
constant and A4,,, increases with curvature. A functional of 
X 
curvature is the ratio 
A d 
Een 
Junction: the eigenvalues encode the magnitudes of 
orientation (un)certainties, since they indicate the size of the 
corresponding ellipsoid. At curve or point junctions, where 
intersecting surface are present, there is no preferred 
orientation and the eigenvalue A, has peak values. A 
junction map that represents the location of the peak value 
for À … is very similar to the static moments map of the 
figure 2. 
min 
Data distribution anisotropy 
Distribution anisotropy is a problem present in point cloud 
data, such as range data, and leads to a classification of the 
scanning shape as a real shape. PCA simplify anisotropic 
data set processing: principal components are referred to a 
spheric neighbourhood, but it is possible to refer 
aggregation criteria to an ellipsoidic neighbourhood, whose 
semiaxes are defined by RMS coordinates: 
0 
um max . - mid . as 
€x = r $ed = r > Can = r 
c. Oo oO 
max max max 
  
where 9... , 6, and o,,, are the RMS in the directions of 
min ? mid X 
a , P o and P. , 
neighbourhood. 
With reference to figure 3, that represents a laser scanner 
anisotropic data set, we note that the sampling density in the 
first principal direction (~7 points/m) is greater than in the 
second principal direction. (~1 points/m). Using the 
ellipsoidic neighborhood in aggregation, we can take 
advantage of the better definition of the measures in the first 
principal direction. 
and r is the radius of the spheric 
[m] 
spheric 
08 = neighbourhood — 
      
4o 08 08 da 702 Mg Te 04 08 08 18 
{m} 
Figure 3 -  Normalized ellipsoidic neighborood in 
anisotropic case; the arrows rapresent the direction of the 
principal axis (e data points, A centre of mass). The point 
set is from a TopoSys I airborne laser scanner.