V. HYPOTHESIS VERIFICATION 
The hypothesis verification algorithm checks if the as- 
sumed hypothesis of a second occluded object becomes 
consistent over some images by using a linearized Kal- 
man filtering technique to estimate the state variables 
ofthe partially occluded object. Otherwise the hypothe- 
sis of a second occluded object will be canceled and the 
generation mechanism goes on searching at the left and 
right boundary of the tracked object ’Ot’. The results 
are analysed by fuzzy setsin a knowledge base exploiting 
the possible constraints in motion parameters (figure 
6). The state variables of each object tracked are eval- 
uated by an recursive estimation algorithm. The Kalman 
filter for the optimal estimate of x(k) is divided into two 
steps [Brammer, Siffling 77] and [Maybeck 79]: 
1. prediction (extrapolation) of x(k) 
2. innovation (correction) by measurement update. 
In order to improve the performance and the handling 
of the recursive estimation process some additional 
features as described in the sequel had been added. The 
order of the system matrix modeling the dynamical 
model is reduced from 4 (6) state variables ( distance, 
velocity, lateral offset, lateral velocity, ev. yaw angle and 
yaw velocity) to two (three) matrices with the order of 
two by estimating position and velocity of each degree 
of freedom separately. That way the efficiency could be 
improved without loosing much performance. For the 
estimation of distance the width respectively the height 
of the object measured in pixel is the input to the 
estimation process. In this case the evaluation of the 
observation matrix C(k) which has to be done every 
system cycle becomes rather complex if analytically 
done, but by using numerical differencing techniques 
this task can be solved in an easier manner. In order to 
improve the initialisation phase of the estimation 
process the error of the system modelis represented by 
an exponentially decreasing function Q(k) (figure 7). So 
the system variance Q is about an order of magnitude 
higher in the beginning of each estimation than later on. 
The dynamical model for all estimated state variables 
and parameters is 
xk 
  
  
Figure 7 Recursive estimation by a Kalman filter scheme 
Xs ET dx, 
Y 0 1 . (4) 
Xs 9x, 
k+1 k k 
Abstand in [m] 
Abstand in [m] 
The state vector can be substituted by the desired value 
to be estimated 
Xs XR Yo Vo 
x = XR 9 Yo 9 [s 9: tese (5) 
k 
Q: covariance matrix of system error 
A: system transition matrix 
C: observation matrix 
K: Kalman gain matrix 
For the partially occluded object an estimation process 
is instantiated also. But the results are worse in general 
because the measurement vector is reduced due to 
occluded features. Nevertheless, the state variables can 
be estimated but the estimation error has increased 
(figure 8a and 8b). The standard deviation for the esti- 
mated distance of the occluded object is about two 
times larger than for the not occluded object. In the case 
that the estimation process for the second occluded 
object becomes consistent the results were analysed by 
comparing them with those of the original tracked ob- 
ject’Ot’. 
Abstand x: xs{rot--], real[blau-], PS_Dx{gruen-.] * 100 
  
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50 100 
kafil 35 
Figure8a Estimated distance with a complete measurement vector 
  
Abstand x: xs[rot--], real[blau-], PS Dx[gruen-.] * 10 
[o € EE tree os - - 
  
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kafi2 35 
Figure8b Estimated distance with a reduced measurement vector 
due to occlusion