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pattern-box:

We first need to set the elementary box sizes \( B_{x} \) and \( B_{y} \) (in pixels) in which correlations will be calculated along each direction \( x \) and \( y \). To estimate the minimum value required for these parameters, open an image (see section 3.6), and look at the minimum size of a square which always contains at least 5 particles. In practice a value of \( B_{x} \) and \( B_{y} \) between 21 and 31 is chosen. Larger values lead to lower spatial resolution: the velocity obtained is an average in this elementary box. In addition, the actual motion is a pure translation only in a local limit: deformation effect increases in proportion with the box size, leading to deterioration in the quality of the correlation (see section 3.5 for a discussion of this effect).

For a seeding with particles at a uniform mean density \( \sigma \) (in particles per pixel), the probability of finding n point particles in a rectangle \( B_{x}B_{y} \) is the Poisson distribution \( P_{\mu }(n)=exp(-\mu n)/n! \), where \( \mu \) is the mean particle number \( \sigma B_{x}B_{y} \) in the square. For \( \sigma \)=0.05 and \( B_{x}=B_{y} \)=21, we have \( \mu \)=20. Then the Poisson distribution gives a total probability 1.6 10\( ^{-5} \)to get n< 5 particles. For a twice lower density, \( \mu \)=10, this probability rises to 2.9 %, resulting in a significant number of false vectors(3.3) (even for a uniform seeding). We need then to use a larger box, or increase the particle density. Choosing different values of \( B_{x} \) and \( B_{y} \) is justified for rectangular pixels (e.g. with the Pulnix camera) or in the presence of a strong shear. For instance in a boundary layer, we can reduce the size transverse to the wall and increase the streamwise size.


next up previous contents
Next: search-box: Up: Basic algorithm Previous: Basic algorithm   Contents
Joel Sommeria 2003-02-14