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Geometric Moments and Their Invariants

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Abstract

Moments and their invariants have been extensively used in computer vision and pattern recognition. There is an extensive and sometimes confusing literature on the computation of a basis of functionally independent moments up to a given order. Many approaches have been used to solve this problem albeit not entirely successfully. In this paper we present a (purely) matrix algebra approach to compute both orthogonal and affine invariants for planar objects that is ideally suited to both symbolic and numerical computation of the invariants. Furthermore we generate bases for both systems of invariants and, in addition, our approach generalises to higher dimensional cases.

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Acknowledgements

I would like to thank the Galaad group at INRIA Sophia Antipolis for hosting me while this work was completed. In particular, I would like to thank Evelyne Hubert for fruitful discussions on this material and for running her code and comparing its output with the basis given in this paper.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand

    Mark S. Hickman

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  1. Mark S. Hickman
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Correspondence to Mark S. Hickman.

Appendix: Maple code

Appendix: Maple code

The following Maple code allows one to construct orthogonal invariants of any order and in any number of variables.

figure a

The default choice of variables is x, y. For other choices the command Moments must be executed first with the optional argument vars. For example, execution of the command Moments(1,vars=[x,y,z]); will allow orthogonal invariants in ℝ3 to be computed. This code may be easily modified to compute the affine invariants.

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Hickman, M.S. Geometric Moments and Their Invariants. J Math Imaging Vis 44, 223–235 (2012). https://doi.org/10.1007/s10851-011-0323-x

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