2D Geometric Moment Invariants from the Point of View of the Classical Invariant Theory
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2D Geometric Moment Invariants from the Point of View of the Classical Invariant Theory Leonid Bedratyuk1 Received: 28 November 2019 / Accepted: 8 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 SO(2)-invariants of several binary forms. Also, to simplify the calculating of the invariants, we proceed from an action of Lie group SO(2) to an action of its Lie algebra so2 . 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. Keywords Pattern recognition · Feature extraction · Geometric moment invariants · Rotation group · Classical invariant theory · Kravchuk polynomials
1 Introduction Since the geometric moment invariants were presented as global image descriptors by Hu [1], they were widely used in the different applications for pattern recognition and image analysis. A problem of description of the 2D geometric moment invariants, in spite of being a typical invariant theory one, was out of focus of the researchers in the field for a long time. In our opinion, it has led to a somewhat confusing and unclear formulation of the problem, and thus, the proposed solutions were not as satisfactory as the invariant theory is concerned. There exists a huge massive of the literature on the 2D geometric moment invariants, but a big amount of it is devoted to the application of the invariants, along with the different ways of their constructions which sometimes are rather elegant and ingenious. Only some of them raise the theoretical foundations of the issue, first of all, the problem of the selection of the minimal generating set. For instance, only after 40 years later on the famous Hu’s article was published, Flusser showed [2] that the set of Hu’s invariants is the functionally dependent and
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Leonid Bedratyuk [email protected] Khmelnytsky National University, Instytuts’ka, 11, Khmelnytsky 29016, Ukraine
incomplete, that is, it does not generate all invariants up to the third order. It implies that the sets of invariants proposed [3,4] are functionally dependent as well and could not generate all the invariants. One of the first attempts to count the number of the invariants in the minimal generating system was made by Li [3], but the obtained result was turned out to be incorrect. The correct formula was firstly obtained in [5]; however, proposed algorithm did not produce a minimal generating set of invariants. Another formula was give
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