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

  • L. P. Bedratyuk Khmelnytsky National University Khmelnytsky, Ukraine
  • A. I. Bedratyuk Khmelnytsky National University, Khmelnytsky, Ukraine
Keywords: pattern recognition; feature extraction; geometric moment invariants; rotation group; classical invariant theory

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 complex groups $SO(3,\mathbb{C})$ and $SL(2,\mathbb{C})$ 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, intro\-ducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2,\mathbb{C})$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3,\mathbb{C})$ to equivalent action of the complex 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.

Author Biographies

L. P. Bedratyuk, Khmelnytsky National University Khmelnytsky, Ukraine

Khmelnytsky National University,

Khmelnytsky, Ukraine

A. I. Bedratyuk, Khmelnytsky National University, Khmelnytsky, Ukraine

Khmelnytsky National University

Khmelnytsky, Ukraine

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Published
2023-01-16
How to Cite
Bedratyuk, L. P., & Bedratyuk, A. I. (2023). 3D geometric moment invariants from the point of view of the classical invariant theory. Matematychni Studii, 58(2), 115-132. https://doi.org/10.30970/ms.58.2.115-132
Section
Articles