Systematic generation of moment invariant bases for 2D and 3D tensor fields

•We propose a systematic approach to find bases of moment invariants with respect to orthogonal transformations using the generator method for scalar, vector, and tensor fields in two and three dimensions.•We show that it is always possible to construct a basis using all homo- 5 geneous invariants and simultaneous invariants with no more than two different moment tensors.•To the best of our knowledge, this results in the first 3D generator ap- proach that produces bases that are complete, independent, and exible, i.e., working for any input pattern.•We reveal so far unknown structural similarity between the 3D generator approach and its 2D counterpart as well as between the 3D generator approach and the 3D normalization approach.