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

Q3 Mathematics Matematychni Studii Pub Date : 2023-01-16 DOI:10.30970/ms.58.2.115-132
L. P. Bedratyuk, A. I. Bedratyuk
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引用次数: 2

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 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.
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从经典不变量理论的角度研究三维几何矩不变量
本文将三维几何不变量的描述问题看作经典不变量理论的问题,旨在澄清三维几何不变量与不变量理论之间的联系问题。利用群$SO(3)$和$SL(2)$是局部同构的显著事实,我们将三维几何矩不变量的推导问题简化为众所周知的经典不变量理论问题。给出了三维几何不变矩计算的精确表述,引入了同时三维几何不变矩代数的概念,并证明了它们与几种二元形式的联合$SL(2)$-不变量代数同构。为了简化不变量的计算,我们从李群的作用$SO(3)$推进到它的李代数$\mathfrak{sl}_2$的作用。作者希望这些结果对图像分析和模式识别领域的研究人员有所帮助。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
期刊最新文献
On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series Almost periodic distributions and crystalline measures Reflectionless Schrodinger operators and Marchenko parametrization Existence of basic solutions of first order linear homogeneous set-valued differential equations Real univariate polynomials with given signs of coefficients and simple real roots
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