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

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Document pages: 19 pages

Abstract: The aim of this paper is to clear up the problem of the connection betweenthe 3D geometric moments invariants and the invariant theory, considering aproblem of describing of the 3D geometric moments invariants as a problem ofthe classical invariant theory. Using the remarkable fact that the groups$SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving3D geometric moments invariants to the well-known problem of the classicalinvariant theory. We give a precise statement of the 3D geometric invariantmoments computation, introducing the notions of the algebras of simultaneous 3Dgeometric moment invariants, and prove that they are isomorphic to the algebrasof joint $SL(2)$-invariants of several binary forms. To simplify thecalculating of the invariants we proceed from an action of Lie group $SO(3)$ toan action of its Lie algebra $ mathfrak{sl} 2$. The author hopes that theresults will be useful to the researchers in the fields of image analysis andpattern recognition.

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