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.

中文翻译：

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从经典不变量理论的角度看二维几何矩不变量

本文的目的是解决几何矩不变性与不变性理论之间联系的问题，考虑到将二维几何矩不变性描述为经典不变性理论的问题。我们给出了二维几何不变矩的计算问题的精确说明，介绍了同时二维几何矩不变式的代数概念，并证明它们与联合\（\ hbox {SO}（2 ）\） -几种二进制形式的不变量。另外，为简化不变量的计算，我们从李群\（\ hbox {SO}（2）\）到其李代数\（{{\ mathfrak {so}}}} _ 2 \ ）。尽管二维几何矩不如正交矩有效，但作者希望该结果对图像分析和模式识别领域的研究人员有用。