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) = (f₁(x,y),f₂(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^Tx), 

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^Tx 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

Publications:

1. Flusser Jan, Suk Tomáš, Zitová Barbara: 2D and 3D Image Analysis by Moments, Wiley & Sons Ltd., 2016.
2. Yang Bo , Kostková Jitka, Flusser Jan, Suk Tomáš, Bujack Roxana: Rotation invariants of vector fields from orthogonal moments, Pattern Recognition vol.74, 1 (2018), p. 110-121.

Relevant publications by other authors:

1. M. Schlemmer, M. Heringer, F. Morr, I. Hotz, M.-H. Bertram, C. Garth, W. Kollmann, B. Hamann, and H. Hagen, “Moment invariants for the analysis of 2D flow fields,” IEEE Transactions on Visualization and Computer Graphics, vol. 13, no. 6, pp. 1743–1750, 2007.
2. R. Bujack, M. Hlawitschka, G. Scheuermann, and E. Hitzer, “Customized TRS invariants for 2D vector fields via moment normalization,” Pattern Recognition Letters, vol. 46, no. 1, pp. 46–59, 2014.