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where Mp denotes the shift of the mask to the point p, in that way that p is localized at the reference point 
of the mask. The dilation of O by the mask M is therefore the set of all points, where the intersecting set 
of O and Mp is not empty. This operation will enlarge objects and typically smooth their border. For more 
details see [8]. To avoid unnecessary clustering of the objects the dilation is not calculated simultaneously 
for all objects in an image but for each object individually [9]. In most cases, in particular for low particle 
concentration(< 300 particles/image), each particle shows only the overlap with corresponding particle in 
the next frame. At higher particle concentration, particles however show overlap with up to typically four 
particles in the next frame. Therefore additional features are required to minimize false correspondences. 
Ideally the sum of gray value for each streak in the image series should be constant, due to the equation of 
continuity for gray values [6]: 
S g(z,y) = const. | (3) 
z,ycO 
This implies a particle at low speed is visualized as a small bright spot. The same particle at higher speed 
is imaged as a fainter object extending over a larger area. The sum of gray value in both cases should be 
identical. The sum of gray value however is erroneous to compute due to ever present segmentation errors. 
Therefore it is more convenient to normalize the sum of gray value by the occupied area. The normalized 
sum of gray value being Gl of the first frame and G2 of the second are required to lie above a threshold of 
the confidence interval C: 
|Gn — Gil 
Le Ti 
Gn + Gal 
A similar expression can be derived for the area of the objects. 
3.4 Calculation of the displacement vector field 
Wierzimok and Hering [11] showed that the center of gray value Z, of an isotropic object represents the 
timely averaged two dimensional location (Z)4,, thus: 
Fe = (Z)A1 : (8) 
Now the knowledge of the location of the same particle in the previous frame (at the time t — 1) enables the 
first-order approximation the velocity field ä(t): 
Ze(t) E — 1) : (6) 
Repeating the described algorithm will automatically track all encountered seeding particles from one frame 
to the next. 
u(t) = 
4 CALIBRATION AND TESTS OF THE PARTICLE TRACKING 
ALGORITHM 
The calibration technique is the same as described earlier by [12]. A calibration grid centered in the light 
sheet is used to calibrate the image coordinate system. The grid points provide points of known location 
in both world an the image coordinate system. Therefore the coefficients of the image transformation can 
be reconstructed. algorithms have been implemented on an i860 board to achieve maximum performance. 
Typical evaluation time of one image including digitization, segmentation and tracking is 10s. Long im- 
age sequences (200-1000) images can therefore be processed. Individual particles can be tracked up to a 
concentration of 800 particles/image at an image size of 480 x 512 pixel. 
For testing the efficiency of the algorithms small particles were attached to a rotating disc of a LP-player [9]. 
The disc rotated at constant frequency of 33 rpm, see Fig. 5. As therefore each vector of a trajectory has 
the same absolute velocity, one can calculate the standard error o for the determination of a displacement 
vector by: 
  
= (sr ley 
oy = AND (7)