CMRT09: Object Extraction for 3D City Models, Road Databases and Traffic Monitoring - Concepts, Algorithms, and Evaluation 
44 
• Semi-major axis 
a\ -cos 2 cp + a' 2 •sinço-cos^ + ôj -sin 2 #? 
b 2 = 
a 3 - cos 2 (p-a 2 -sin^ cos^ + a, -sin 2 #? 
(9) 
3. CLASSIFICATION 
3.1 Class definition 
The traffic objects are identified within the image data and 
trajectories are derived from it. The trajectories are fitted to 
curves and their parameters are classified with the corre 
sponding functions. For the classification the same data set is 
used for all function classes. 
The parameter determination is based on the number of ob 
servations n, which are related to the functional model. The 
number of observations n has to be greater than the number 
of unknown parameters. 
a\ • x' 2 + a' 2 ■ x'y' + a\ • y' 2 + a\ • x\ + a' 5 ■ y\ = 1 (10) 
>2 
/2 
t 
t 
r 
*0 
*o.V 0 
Vo 
x o 
To 
, 
(2 
f t 
#2 
X, 
x \ V, 
V,“ 
x ! 
y'\ 
«3 
= 
t2 
r2 
X „-l 
t'-i-V, 
y„-1 
K-i 
A. 
(ID 
A part of the data set is used to train a classifier which in 
tends a class assignment for the trajectory with the parame 
ters. The other part serves for the verification. 
Figure 3. Visualisation of the different traffic lanes (classes) 
of the scene 
This can also be written as follows: 
I = A- x 
(12) 
Seven classes were defined based on the scene (figure 3) and 
the traffic lanes: 
The observation vector 1 is replaced by the measured obser 
vation and a small residuum v. Therefore, the unknown vec 
tor x is replaced by the estimates with the result: 
A T -P- A 
■ A T ■ P-1 
(13) 
This result is known as a least-square adjustment, based on 
the L 2 norm. This approach is not able to decide between 
hyperbola and ellipsoid. (Fitzgibbon et al. 1996) and (Fitz- 
gibbon et al. 1999) describe an attempt for the inclusion of 
additional conditions by integration of a constraint matrix. 
Hence it is possible to reduce the resulting solution space so 
that the type of the object function (ellipse, hyperbole) can be 
steered. (Harlow et al. 2001) and (Harker et al. 2008) enlarge 
Fitzgibbon’s approach by decomposition of the Scattermatrix 
in the square, linear and constant part. The parameter esti 
mate becomes equivalent to the eigenvalue problem. This is a 
direct solution method. The approach determines an ellipse 
as well as two hyperboles. Figure 2 shows examples for the 
hyperbola and ellipse fit. 
Figure 2. Example fits for two tracks and classification 
No 
From 
To 
Class 
Abbreviation 
1 
Wegedom 
Rudower 
right-turn 
WRR 
2 
Wegedom 
Rudower 
left-turn 
WRL 
3 
Rudower 
Rudower 
east- 
direction 
RO 
4 
Rudower 
Rudower 
west- 
direction 
RW 
5 
Rudower 
Wegedom 
right-turn 
RWR 
6 
Rudower 
Wegedom 
left-turn 
RWL 
7 
No class membership 
No Class 
Table 1. Class definition for the observed scene 
The used data set consists of 414 trajectories. Trajectories 
which are part of the classification process need to have a 
minimal length of 10 m and a minimal number of points of at 
least 6 points. The class No_Class consist of trajectories of 
pedestrians, bicyclists and erroneous tracks caused by errors 
in image processing and tracking. It is inadmissible that two 
driving directions are assigned to one trajectory. Relying on 
the shape only, opposite directions is merely to distinguish, 
since their functional parameters are similar. To achieve the 
distinction the approximate same path of the trajectory and 
the fitted function is determined. With this, the direction of 
the trajectory can be determined as an additional feature. 
Hence it direction can be distinguished between close lanes 
trajectories with opposite directions. 
3.2 Classification method 
A classifier determines the class affiliation with the charac 
teristic of item-specific features. These features are repre 
sented as a vector in a multidimensional feature space. The 
features correspond to the parameters which have been de 
termined by the approximation.