3D geometric moment invariants from the point of view of the classical invariant theory

TL;DR

This paper connects 3D geometric moment invariants with classical invariant theory by leveraging the isomorphism between groups SO(3) and SL(2), providing a new algebraic framework for their computation.

Contribution

It establishes a precise algebraic correspondence between 3D geometric moments invariants and classical invariant theory, simplifying their calculation via Lie algebra methods.

Findings

Invariants are isomorphic to algebras of joint SL(2)-invariants of binary forms.

Reduced the problem from group action to Lie algebra action for easier computation.

Provides a theoretical foundation for improved image analysis techniques.

Abstract

The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.