The significance of coherent flow structures for the turbulent mixing ...

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3.2 Evaluation of stereo-scopic image pairs
grid and Ï when a dot pattern is used) in order to determine the coordinates of the line crossing
with sub-pixel accuracy 3 . After this step the imaging function between the image- and object
plane can be changed from a discrete into a continuous representation by means of fitting a
standard least squares surface to each of the image-object point sets, so that the first order
projection matrix can be calculated for each observation direction along with the translation,
rotation and magnification factor, [108].
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In order to take into account aberrations of higher order and other non-linear distortions, the
second order projection in form of equation (3.10) has to be applied. For this purpose the
coefficients of the first order projection matrix have to be used as an initial estimate to a
Levenberg-Marquart non-linear least squares fitting algorithm.
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As both sets of transformation equations describe just a mapping between two planar domains
without any three-dimensional information, the location of the image with respect to the object
has to be known in addition. This can be done either directly, by measuring the exact camera
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positions relative to the centre of the field of view, or indirectly by using the following set of
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equations, see [56] for mathematical details or [95] for the applicability in PIV.
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The appearance of the -coordinate requires that the calibration procedure has to be repeated
for different locations in order to determine all unknown coefficients. This is especially
useful when the position of the cameras is not accessible or for applications in water, where
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the air-glass-water interface has to be taken into E account.
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3.2.2 Image warping
Once the reconstruction coefficients have been properly determined, the transformation equations
can be applied to deform each acquired, single-exposed image in such a way that the
magnification factor is constant across the back-projected image and the field of view is identical
for all acquired images. Using this technique optical parameters such as the focal length
3 As this method is quite time consuming, Hough transformation methods are usually applied in this thesis to
find the coordinates of the line crossings, see [18].
43