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

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.

2D moment invariants from the point of view of the classical invariant theory

Invariants allow to classify images up to the action of a group of transformations. In this paper we introduce notions of the algebras of simultaneous polynomial and rational 2D moment invariants and prove that they are isomorphic to the algebras of joint polynomial and rational $SO(2)$-invariants of binary forms. Also, to simplify the calculating of invariants we pass from an action of Lie group $SO(2)$ to an action of its Lie algebra $\mathfrak{so}_2$. This allow us to reduce the problem to standard problems of the classical invariant theory.