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5. TRACKING OF PARTICLES 
A particle motion in 3D object space can be imaged as a 2D 
path in image space of an observing camera. If corresponding 
particle images were found in at least two cameras the 3D 
particle position can be determined. If in addition the temporal 
assignment over the time is possible, the trajectory can be 
reconstructed. Figure 4 shows the trajectory of a particle in an 
object volume over four time steps and its 2D paths in image 
space recorded with a two-camera arrangement. 
Object 
Space 
      
.| 2D-Track 
/ T- in Image - 
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a... ea 
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Figure 4: Particle trajectory in image and object space 
The goal of the tracking procedure is to select the correct link 
for the same particle from one time step to the next. The 
developed image and object space based tracking technique for 
spatio-temporal matching uses different criteria to find 
corresponding particle positions in object space as well as in 
image space. The following criteria can be used for a reliable 
and effective assignment: 
e The velocity of a particle is limited in all three components 
of the motion vector. 
e The Lagrangian acceleration of a particle (the difference of 
two consecutive velocity vectors of one particle) is limited. 
e In cases of ambiguities the assignment with the smallest 
Lagrangian acceleration is the most probable one. 
The first criterion defines a three-dimensional search volume, 
whose size depends on the minimum and the maximum velocity 
in all three coordinate directions. The limitation of the 
Lagrangian acceleration defines a conic search area. From three 
consecutive time steps the difference of the magnitude and 
direction of predicted and found particle position can be 
calculated. In the case of ambiguities a quality function is used 
to get the final assignment as proposed in the third criteria. 
6. ALGORITHMIC ASPECTS OF SPATIO-TEMPORAL 
MATCHING 
The two most important processing steps of 3D PTV — the 
establishment of spatial and of temporal correspondences 
between particle images in synchronous image sequences of 
multiple CCD cameras — which were strictly separated in the 
existing implementations, have been unified (Willneff and 
Gruen, 2002). In former implementations the trajectories of the 
particles are reconstructed in two ways: 
e The particle positions are determined for each single time 
step; after that, tracking is performed in 3D space. In the 
previous implementation at ETH Zurich multi-camera 
correspondences are established by epipolar line 
intersection techniques in a first step, before the trajectory 
renconstruction is done in 3D space. A similar procedure is 
used in the 3D PTV system of the University of Tokyo 
(Kasagi and Nishino, 1990). 
e Other methods first track the particles in image space. 
Then the spatial correspondences between the 2D 
trajectories are established (e.g. Adamczyk and Rimai, 
1988; Netzsch and Jähne, 1993). 
In the implementation at ETH Zurich the establishment of 
multi-camera correspondences is performed by epipolar line 
intersection technique (Papantoniou and Dracos, 1989; Maas et 
al, 1993; Malik et al, 1993). Although the number of 
ambiguities can be reduced by the use of 3-4 cameras in a 
suitable configuration they cannot be solved completely if only 
one time step is considered. Some detected particle locations 
still remain unmatched (obviously not all detections in image 
space can be matched, e.g. when a particle is visible only in one 
image). Assuming a 4-camera setup a particle at best can be 
detected in four images, which delivers 5 redundant 
observations (4x2 image coordinate observations minus 3 object 
space coordinate unknowns). If a particle can be tracked over 
some consecutive time steps a quite reliable prediction of the 
particle position in the next time step can be made. The 
redundant image coordinate observations combined with the 
prediction for the next particle position allow the establishment 
of spatio-temporal connections even when the velocity field has 
a high density or the movement of the tracer particles is fast. 
The main issue of the new algorithm is the handling of 
redundant matching information. Not only the 3D particle 
coordinates of the single time step but also the detections in 
image space are used to establish the temporal correspondences. 
The tracking procedure works in the following way: 
e From a 3D particle position X p at time step /; a position 
for consecutive time step /j4, is predicted under the 
assumption of a piecewise constant particle velocity u P 
(eq. 1). A simple linear extrapolation is possible if the 
particle of step t; already has a valid link to the previous 
time step /;;, otherwise the position X P from t; itself is 
used as search position. The temporal resolution At is 
given by the frame rate of the sensors used for the image 
acquisition. 
%p(t:1) = 5, (6) +i, (6) 4t 
(1) 
  
e The search of suitable candidates is performed in image 
space. This requires the backprojection of the search 
position into the image space of all cameras, which can 
easily be done when camera orientations and multimedia 
geometry are known. 
—603—