In: Stilla U, Rottensteiner F, Paparoditis N (Eds) CMRT09. IAPRS, Vol, XXXVIII, Part 3/W4 — Paris, France, 3-4 September, 2009 
2.3 Tracking and Trajectories 
The aim of tracking is to map observations of measured ob 
jects to existing trajectories and to update the state vector 
describing those objects, e.g. position or shape. The tracking 
is carried out using a Kalman-filter approach. 
The basic idea is to transfer supplementary information con 
cerning the state into the filter approach in addition to the 
measurement. This forecast of the measuring results (predic 
tion) is derived from earlier results of the filter. Conse 
quently, this approach is recursive. 
The initialization of the state-vector is conducted from two 
consecutive images. The association of a measurement to an 
evaluated track is a statistical based decision-making process. 
Errors are related to clutter, object aggregation and splitting. 
The decision criteria minimize the rejection probability. 
The coordinate projection mentioned in the last paragraph 
and the tracking process provides the possibility to fuse data 
acquired from different sensors. The algorithm is independ 
ent of the sensor as long as the data is referenced in a joint 
coordinate system and they share the same time frame. 
The parametric equation is given by 
x = a-sec(?) v = 6-tan(/) (2) 
Commonly the hyperbola is rotated and shifted: 
fx') ( COS(p sin^cfy fx'-x(3) 
sinCOS(p) yy'-y m ) 
Wherein m x „ m y are the centre coordinates, the angle <p is the 
bearing of the semi-major axis. The implicit form of the hy 
perbola can be written as a general polynomial of second 
degree: 
a[ ■ x 12 + a 2 ■ x'y + a\ • y' 2 + a\ • x' + a' 5 • y' = 1 (4) 
With 
The resulting trajectories are then used for different applica 
tions e.g. for the derivation of traffic parameters (TP). 
2.4 Trajectory analysis 
A deterministic description method for trajectories shall be 
introduced below. The functional descriptions for these tra 
jectories should be as simple as possible and permit a 
straightforward interpretation. Linear movements will be 
described by simple straight lines. 
Numerous suggestions of possible functions for curve tracks 
by functional dependencies have been made in the literature. 
Clothoid (Liscano et al. 1989) or G2-Splines (Forbes 1989) 
are curves whose bend depends of the arc length. Alterna 
tively, closed functions like B-Splines, Cartesian polynomi 
als fifth degree or Polarsplines (Nelson 1989) can be used as 
well. A common approach to approximate vehicle-based 
trajectories is to employ clothoids. Those functions derived 
from the fresnel integral are highly non linear. They are fun 
damental in road and railroad construction. Due to urban 
constraints the tracks of intersections and curves cannot fol 
low the curve of a clothoid whose shape is regarded as a 
trajectory that is especially comfortable to drive. Because 
there are only partial approximations of clothoids, they do 
not fit into the set of elementary functions that shall be re 
garded in this work. Moreover, the given trajectory has to be 
subdivided into parts in order to apply a clothoidal approxi 
mation. (Anderson et al. 1979) have proposed a description 
of tracks by hyperbolas. The great advantage is that the de 
rived parameters clarify directly geometric connections and 
permit a categorization and derivation of important features 
of the trajectories. A hyperbola is able to replicate straight 
lines as well as turning trajectories. 
The hyperbola fit serves as an example and is described next. 
The approach is based on least-square fitting of geometric 
elements. The equation for a hyperbola with semi-major axis 
parallel to the x-axis and semi-minor axis b parallel to the y- 
axis is given by 
, fl, , 
a, = — a, 
a. 
°4 
and 
, 2 • xa, + ya 7 
a 4 = 
a 4 
, _ x ,„ a 2 + 2 -T,A 
U 5 ~ 
a 4 
= - (2 • x m a[ + y m a 2 ) 
= -(x m a' 2 + 2 ■ y m a[) 
a i 
°4 
COS' (p sin' (p 
a' 
„ . , 1 1 
= 2-sm<p-cosiiH —- + — 
sin~^> cos" (p 
b 2 
= 1 - (x^a, + x m y m a 2 + y 2 m a 2 ) 
(5) 
(6) 
The following equations describe the conversion of the im 
plicit to the hyperbola parametric form: 
■ Bearing of the semi-major axis 
(p = a tan — 
a\ - a\ 
■ Center coordinates 
x _ 2 • a\a\ - a\d 2 (8) 
4 • a\a\ - a' 2 
2 • a'.al - d,a\ 
y — — —=- 
4 • a x a\ - a' 2