Vector fields are a special kind of multidimensional data. In each pixel, the field is assigned to a vector that shows the direction and the magnitude of the quantity that has been measured. To detect the patterns of interest in the field, special matching methods must be developed. A 2D vector field f(x) can be mathematically described as a pair of scalar fields f(x) = (f_{1}(x,y),f_{2}(x,y)). At each point x = (x,y), the value of f(x) shows the orientation and the magnitude of the measured vector. Scalar field f_{i}(x) can be understood as a graylevel image that may contain also negative values.
The geometric transformations of the vector fields are slightly different from the transformations of the images. The total rotation f → f' acts simultaneously in spatial and function domains
f'(x) = R f(R^{T}x),
where R is a rotation matrix and f is a vector field. The rotation matrix is orthogonal and its determinant equals one, then R^{-1}=R^{T}.
In the total rotation with inner translation, the inner part R^{T}x is replaced with R^{T}(x−t). However, for pattern detection via template matching it is irrelevant to include the translation into the deformation model, because the shift is the key parameter we want to detect.
The invariants of the vector fields to the total rotation can be generated by means of complex moments
The complex moment after the total rotation of the vector field by angle α becomes
Therefore, when we compute a product
where
the phase shift caused by the rotation is canceled and we obtain invariant to the total rotation of the vector field. The simplest possible independent and complete subset (basis) can be obtained by
where
We set by definition
The videos with results of the experiment with Kármán street can be seen here.
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