3D geometric moment invariants from the point of view of the classical invariant theory
This work addresses a foundational problem in mathematical theory for researchers in image analysis and pattern recognition, offering a theoretical framework but is incremental as it builds on existing invariant theory.
The paper tackles the problem of connecting 3D geometric moment invariants to classical invariant theory by reducing it to the well-known problem of joint SL(2)-invariants of binary forms, using the local isomorphism between SO(3) and SL(2). It provides a precise statement for computing these invariants and proves an isomorphism between algebras of 3D geometric moment invariants and SL(2)-invariants.
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.