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Shape Interpretation of Second-Order Moment Invariants

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Abstract

This paper deals with the following problem: What can be said about the shape of an object if a certain invariant of it is known? Such a, herein called, shape invariant interpretation problem has not been studied/solved for the most of invariants, but also it is not known to which extent the shape interpretation of certain invariants does exist. In this paper, we consider a well-known second-order affine moment invariant. This invariant has been expressed recently Xu and Li (2008) as the average square area of triangles whose one vertex is the shape centroid while the remaining two vertices vary through the shape considered. The main results of the paper are (i) the ellipses are shapes which minimize such an average square triangle area, i.e., which minimize the affine invariant considered; (ii) this minimum is \(1/(16\pi ^2)\) and is reached by the ellipses only. As by-products, we obtain several results including the expression of the second Hu moment invariant in terms of one shape compactness measure and one shape ellipticity measure. This expression further leads to the shape interpretation of the second Hu moment invariant, which is also given in the paper.

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Notes

  1. The measure \(\mathcal{E}_\mathcal{P}(S),\) defined as in [15], is constructed to be maximized by an ellipse. But [15] does not discover that the ellipses are the only shapes which maximize \(\mathcal{E}_\mathcal{P}(S).\)

  2. There is an overlap with results in a recent work [13], based on arguments coming from the statistics, as mentioned earlier.

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Acknowledgments

This work is partially supported by the Serbian Ministry of Science—projects OI174008/OI174026.

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Authors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

    Joviša Žunić

  2. Faculty of Computer Science, Belgrade, Serbia

    Dragiša Žunić

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  1. Joviša Žunić
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  2. Dragiša Žunić
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Correspondence to Joviša Žunić.

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Žunić, J., Žunić, D. Shape Interpretation of Second-Order Moment Invariants. J Math Imaging Vis 56, 125–136 (2016). https://doi.org/10.1007/s10851-016-0638-8

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