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

TL;DR

This paper explores 2D moment invariants through classical invariant theory, establishing algebraic isomorphisms and simplifying calculations by transitioning from group actions to Lie algebra actions.

Contribution

It introduces algebraic frameworks for 2D moment invariants and connects them to classical invariant theory, simplifying their computation.

Findings

Algebras of 2D moment invariants are isomorphic to classical invariant algebras.

Transitioning from group to Lie algebra actions simplifies invariant calculations.

Provides a theoretical foundation linking image invariants to classical invariant theory.

Abstract

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.