Tensor fields and their invariants
One type of multidimensional data is tensor field, where the tensor is defined in each space point. Typically, it is an array 3 x 3 in each voxel. If there is some geometric distortion, it transforms not only the spatial coordinates but also on the field values, which makes analysis of the tensor fields more challenging, even mere translation and scaling invariance is significantly more complicated than that in images. Moment invariants to translation, scaling, rotation and affine transformation will be discussed, visualization will be mentioned and some experiments will be shown.
A description of the tensors and operations with them can be found in [1] or in [2]. A
good explanation can also be found in [3] or in its English translation [4].
References
[1] Bowen, R., Wang, C.: Introduction to Vectors and Tensors. Dover books on mathematics, Dover Publications (2008)
[2] Grinfeld, P.: Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer New York (2013)
[3] Gurevich, G.B.: Osnovy teorii algebraicheskikh invariantov. OGIZ, Moskva, The Union of Soviet Socialist Republics (1937)
[4] Gurevich, G.B.: Foundations of the Theory of Algebraic Invariants. Nordhoff, Groningen, The Netherlands (1964)