In: Stilla U, Rottensteiner F, Paparoditis N (Eds) CMRT09. IAPRS, Vol. XXXVIII, Part 3/W4 — Paris, France, 3-4 September, 2009 
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3.3 Knowledge representation of traffic situations 
In order to evaluate the velocities of the vehicles we need to 
formulate our knowledge and expectations about the typical 
traffic situations such as “free flowing traffic”, “congestions” or 
“traffic jams”. In dependence of the state of traffic and the 
location with respect to intersections or traffic lights, different 
interactions between vehicles occur. 
A well substantiated statistical concept like Bayes’ theorem 
would provide a sound basis for evaluation. However, 
determining the probability density functions is hardly feasible 
because extensive and sufficient samples are missing. Hence, it 
is advisable to avoid a concept that claims statistical integrity. 
In contrast to Bayes’ theorem, fuzzy logic offers an intuitive 
method to represent knowledge of classes by easy 
parameterization (Zadeh, 1965). It is also frequently applied for 
modeling car following behavior (Brackstone and McDonald, 
1999). Therefore, we decided to use fuzzy logic to describe our 
knowledge about traffic. 
3.3.1 3D fuzzy membership function for active vehicles 
Let us define a fuzzy set A that describes vehicles which are 
actively involved in traffic. Besides normally moving cars, 
these may be standing vehicles in traffic jams or waiting at red 
traffic lights or other crossroads. 
Since we are only interested in the possibility of an object 
belonging to A, we neglect the alternative set A of inactive 
objects which may be false alarms of the detection, parking 
vehicles or erroneous tracks. 
For the fuzzy set A, a membership function needs to be defined, 
indicating the possibility ¡i A that a car belongs to A in 
dependence of its velocity v. However, ¡.i A also strongly depends 
on the traffic density D and the distance d from intersections. It 
is quite obvious, that in a free flowing situation in the middle of 
road segment the possibility that a car stands still is 0. In 
contrast to that, zero speed has a rather high possibility near 
intersections or in jam situations. In order to meet these 
different traffic situations, we have to consider the conditional 
possibilities fi A (v\D,d). In the sequel, the units for the measures 
given shall be v [km/h], D [cars/km per lane] and d [m]. 
First, we should outline the ranges of D and d where pi A may 
change significantly. A density of lower or equal to D = 30 
corresponds to free flowing traffic while a density of D = 180 
represents the maximum density of a traffic jam when there is 
almost no motion at all (Hall, 1999). Below 30, ii A (v,D\d) 
remains constant. 
The interesting range for d is approximately between 150 
meters before an intersection because this describes the range 
where drivers start to brake and 50 meters behind the 
intersection where drivers accelerate until they reach their 
desired travel speed. Outside of the range of [-50m; 150m], 
/u A (v,d\D) is constant for all values of d. 
Over the entire space of v, D and d, this results in a 3D 
membership function fj A (v,d,D). Please note that the values also 
depend on the road type, speed limits intersection layout. For 
different road conditions, different functions have to be 
developed. The mentioned example function refers to a major 
city road with multiple lanes with a speed limit of 60 km/h and 
an intersection with a road of equivalent type controlled by 
traffic lights. 
D=30 and d=150 
Figure 3: ID membership functions given a traffic density D = 
0 and D = 180 respectively and distance d = 150 
with support points (black dots) 
Distance d=5 
Velocity v [km/h] 
Figure 4: 2D membership function at distance d=5, with ID 
support functions (black lines) 
To create this 3D function, support points have to be selected. 
I.e., given an open traffic situation (D = 30) and long distance 
from an intersection (d = 150) the possibility of a vehicle 
moving with a speed between 0-20 km/h in shall be 0, while the 
possibility at the same position in the same traffic situation shall 
be 1 for velocities between 50-70 and becoming 0 again at v = 
100.