Systematic generation of moment invariant bases for 2D and 3D tensor fields. (March 2022)

• Record Type: Journal Article
• Authors: Bujack, Roxana; Zhang, Xinhua; Suk, Tomáŝ; Rogers, David
• Is Part Of: Pattern recognition. Volume 123(2022)
• Publication Date: 2022-03
• Subjects: Pattern detection -- Rotation invariant -- Moment invariants -- Generator approach -- Basis -- Flexible -- Vector -- Tensor; Pattern perception -- Periodicals; Perception des structures -- Périodiques; Patroonherkenning; 006.4
• DOI: 10.1016/j.patcog.2021.108313
• Languages: English
• ISSNs: 0031-3203

Abstract

Highlights: 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. Abstract: Moment invariants have been successfully applied to pattern detection tasks in 2D and 3D scalar, vector, and matrix valued data. However so far no flexible basis of invariants exists, i.e., no set that is optimal in the sense that it is complete and independent for every input pattern. In this paper, we prove that a basis of moment invariants can be generated that consists of tensor contractions of not more than two different moment tensors each under the conjecture of the set of all possible tensor contractions to be complete. This result allows us to derive the first generator algorithm that produces flexible bases of moment invariants with respect to orthogonal transformations by selecting a single non-zero moment to pair with all others in these two-factor products. Since at least one non-zero moment can be found in every non-zero pattern, this approach always generates a complete set of descriptors.