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In: Paparoditis N., Pierrot-Deseilligny M., Mallet C., Tournaire O. (Eds), IAPRS, Vol. XXXVIII, Part ЗА - Saint-Mandé, France, September 1-3, 2010 
x k = Ai-l + B »k-1 + W t -I (D 
with a measurement 
z k= Hx k+ v k (2) 
In these equations the random variables w k and v k represent, 
respectively, the process and measurement noise at the time k. 
In the case of a roundabout, the state equation (1) is non linear 
and the extended Kalman filter must be applied. 
The dynamic state variables are the vectorial position and 
velocity of the vehicles. 
Position and velocity are characterized by the components in x 
and y (or, in a polar system, radial and angular) 
The roundabout studied in this paper, has been monitored 
(Guido et al., 2009) by using virtual detectors (Figure 8); mean 
trajectories and mean velocity profiles has been obtained 
(Figure 9 ). 
By using the obtained trajectories, it is possible to write the 
simplified state equation: 
= ® k X k + n (3) 
where X k+x is the position and velocity vector at the k+1 
time (frame), and O a is the system transition matrix. Given 
the time interval At, the expressions of X k+] and O a are: 
"1 
0 
<1 
0" 
Ум 
Ф î, = 
0 
1 
0 
At 
V x,k+1 
A 
0 
0 
1 
0 
УуМ_ 
0 
0 
0 
1 
The measurement equation is: 
where the measurement matrix is: 
The process noise w k is obtained by the trajectories and 
velocities variances given by Guido et al.. The measurement 
noise v k is obtained by the variance of the barycentres of the 
vehicles detected in the frame. 
For every recognized vehicle, the analysis starts after the first 
two frames in which the vehicle is present. Position and 
velocity are obtained; these parameters are utilized for 
estimating new position and velocity by using equation (3). We 
utilize these predicted positions for recognize the blob (or the 
blobs) in the next frame, corresponding to the vehicle: a simple 
test on the distance between blob and vehicle predicted 
barycentres is applied. The individuated blob is used for 
obtaining the new position and velocity values at k+1 step. In 
this way we can correct errors due to failed vehicle recognition. 
Figure 8. The roundabout with the virtual detectors 
0,00 
Figure 9. Velocity profiles for 8 vehicles 
The Kalman filter is also applied to estimate the a posteriori 
state X k+] and the a posteriori error covariance P k+] . The 
procedure is iterated, and the obtained values can be useful to 
detect overlapping vehicles, when their trajectories diverge. 
3. THE TEST 
A test has been executed on a portion of roundabout in the city 
of Cosenza, Italy. A digital camera “Sony Handycam TRV14E” 
has been used, with 640 x 480 pixels frames. The acquisition 
rate is 25 frames per second. For the elaboration of the images 
and the comparisons the Matlab™ software has been used. The 
comparison have been performed every 10 frames, with a time 
step of 0.4 seconds. The data obtained by Guido et al. have 
been used, along with Kalman filter, to obtain the foreseen 
positions of the vehicles. 
To evaluate the effectiveness of the Kalman filter, we can 
consider the detection and tracking of the red car on the left 
side of figure 2. 
Figure 10 shows the frame 78830, with the bounding boxes of 
the selected regions; figure 11 shows the relevant labelling: it is 
possible to observe that the red car, entering the shadowed 
zone, is recognized. Figure 2 (frame 78880) has been obtained 
two seconds after figure 10; as shown in the figure 4, the red 
car wasn't detected and no regions were identified, compatible 
with its previous position. Due to the prediction of Kalman 
filter (obtained taking into account also frames from 78840 to 
78870), a car should be present in the shadowed zone of figure 
2; for this reason, the threshold used for non maxima 
suppression (0.2) has been reduced to 0.05 for a neighborhood 
of the foreseen position of the red car. The result of this 
operation is shown in figure 12, where the outlines of the 
detected regions are superimposed to the frame 78880. The 
positions foreseen by Kalman filter have been used for all 
frames, and an enhancement of the vehicles detection has been 
obtained.