2D Geometric Moment Invariants from the Point of View of the Classical Invariant Theory

Abstract

The aim of this paper is to clear up the question of the connection between the geometric moment invariants and the invariant theory, considering a problem of describing the 2D geometric moment invariants as a problem of the classical invariant theory. We give a precise statement of the problem of computation of the 2D geometric invariant moments, introducing the notions of the algebras of simultaneous 2D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $$\hbox {SO}(2)$$ -invariants of several binary forms. Also, to simplify the calculating of the invariants, we proceed from an action of Lie group $$\hbox {SO}(2)$$ to an action of its Lie algebra $${{\mathfrak {so}}}_2$$ . Though the 2D geometric moments are not as effective as the orthogonal ones are, the author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.