The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Voi. XXXVII. Part BI. Beijing 2008 
7 
vector. This is done by observation-object-association (Kumar 
et al., 2005; Luo and Bhandarkar, 2005). 
The tracking of every object was realized using a Kalman-filter 
(Anderson and Moor, 1979; Blackmann, 1986). It estimates the 
state of an object for the time stamp of the following picture, 
hence allows to compare the estimated state and the observed 
object data. If both are located within a certain feature space 
distance they can be associated to the same object. A 
considerable problem is initialization of the Kalman-filter. 
The resulting trajectories are submitted to the analysis module 
as soon as they are created for the derivation of traffic 
parameters. 
3. EXTERIOUR ORIENTATION 
The collinearity equations (1) require the parameters of the 
exterior orientation of every camera. The following sections 
present two general approaches to determine these parameters 
based on different input sets of scene knowledge. The first 
algorithms use point correspondences between image points and 
measured points in the surveillance area. A differential GPS can 
be applied to acquire geo-referenced ground control points with 
a standard derivation usually below 2 cm. Other features that 
can be used are straight lines. Lines are a very common feature 
in urban environments. In contrast to ground control points, 
lines have the advantage of being easier to match to their 
correspondences in the image. Furthermore, this implies if these 
features are already geo-referenced on a floor-plane or in an 
orthophoto, the entire process of determining the exterior 
orientation could be automated. 
The approaches that will be presented in the subsections 3.2 and 
3.3 depend on initial values for the adjustment of the exterior 
orientation. With prior normalised images the values can be 
computed in advance by one of the following techniques. 
3.1 Initial values 
The direct linear transformation (DLT) method is based on the 
collinear equations which are extended by an affine 
transformation of the image coordinates (Abdel-Aziz and 
Karara, 1971; Kwon, 1989). Using these equations a system of 
linear equations can be set up and solved via well known 
methods. It results in 11 DLT parameters which define the 
exterior orientation, the focal length and the principal point. 
This method cannot detect erroneous measurements hence it 
relies on well measured image and world coordinate points. 
Another disadvantage is the liability to singularities if all 
control points are in a common plane. At least 6 measured 
points correspondences are needed. 
An alternative approach is the minimum space resection 
(Fischler and Bolles, 1981). Given three points in object space 
and the projection center of the camera, a tetrahedron is defined. 
Knowing the 3 angles (derived from focal length and principal 
point) simple geometric dependencies can be established. By 
solving the resulting quartic equation the length of the three 
sides can be determined. The orientation of the camera is 
deduced by determining the intersection points of three spheres 
constructed using the object points as centers and the edge 
lengths as radius. This method requires 3 control points, the 
focal length and principal point. By taking into account a fourth 
point the ambiguous result is dissolved. 
Using automated methods for determining control points or by 
taking into account the human factor, it is always an adequate 
approach to assume having unreliable ground control data. To 
exclude erroneous control points it is advised to apply the above 
procedures to minimal subsets of points. The final value will be 
the median of randomly chosen subset results. The number of 
subsets used depends on the amount of errors expected. 
3.2 Adjustment using Control Points 
Given the interior orientation and initial values for the exterior 
orientation the following algorithms can be applied to 
determine the exterior orientation (Luhmann et al. 2006, 
McGlone et al. 2004): 
The Newton method is a common mean for retrieving the roots 
of a polynomial function. Thus it can easily be adapted for 
retrieving the parameters of the collinearity equations. After 
having set up the design matrix that is a least-squares estimator 
of a linearized model, singular value decomposition can be 
applied to solve the system of linear equations. This approach 
renders the detection of singular values, i.e. from planar control 
points, possible. 
A general least squares adjustment based on a Gauss - Markov 
method computes the adjusted parameters of the exterior 
orientation. This method uses a system of normal equations. 
The dissolving of this system leads to the cofactor matrix of the 
unknowns as the inverse of the matrix of normal equations 
times the absolute term: 
x =A r PA~ 1 ■ A T PI (2) 
Hereby, the matrix P represents a stochastic model which can 
exclude erroneous points. The usage of trigonometric functions 
for setting up the necessary rotation matrix includes the usual 
ambiguity. Even though the geometric interpretation is difficult 
it is advisable to use quaternions for the definition of rotation. 
Despite the more difficult geometric interpretation quaternions 
are the appropriate mean to disambiguate the rotation. 
3.3 Adjustment using straight lines 
The collinearity equations can easily be extended for using 
straight lines as control data: 
j +S~ y o)5. + (aZ 0 +/- X 0 )r 2l + (q(6 - 7„) - - X 0 ))r lt (3) 
(fZ 0 +8- Y 0 )r n -(aZ 0 +y-X 0 )r 22 -(a(8- Y 0 )-J3(y-X 0 ))r 3l 
b - (M> + 8 - To)*» - ( aZ o +Y- - (a(8 -Y a )-fi(r- X 0 ))r 2i 
(0Z O +8-Y 0 )r l2 -(aZ 0 +y-X 0 )r 22 -(a(8-Y 0 )-f(y-Xj))r n 
g: y = a-x + b (4) 
G:X = a- Z + /AY = f3Z + S 
Having an image line g and an object line G these equations can 
be substituted for the equations of control points. Hence the 
determination of exterior orientation is carried out in the same 
way as it would be done for points as control data.