Tensor fields are a special kind of multidimensional data. In each pixel or voxel, the field is assigned to a tensor.

The most common tensor fields are Cauchy stress tensor, viscous stress tensor, diffusion tensor, and Maxwell stress tensor. All of them have dimension three and contravariant rank two, i.e. they look like a 3 × 3 matrix in each voxel. They contain informatiom not only about the direction and the magnitude of the quantity, but also about transverse components or curvature.

During affine transformation of the space, both tensor values and space coordinates are transformed, therefore we need special type of invariants.
They can be generated by tensor method as the total contraction of a tensor product of moment tensors and permutation tensors.

The invariants of the symmetric 3D tensor fields to the total affine transformation are in the attachment. The file af3D2_0tenstsinvsymzo5indep_title.pdf contains the invariants from orders 0 to 5. They are generated by the graphs of up to five three-edges. The file af3D2_0tenstsinvsym6indep_title.pdf contains the invariants from orders 2 to 6. They are generated by the graphs of up to six three-edges. Partial invariants are suitable for the normalization of the tensor fields to translation. Unlike the invariants, the partial invariants are vectors. The partial invariants up to the 4th order can be found in af3D2_0tenstsnormzolr4indep_title.pdf.

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