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Polynomial Eulerian Shape Distributions

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Abstract

In this paper a new approach is derived in the context of shape analysis. The so called polynomial Eulerian shape theory solves an open problem proposed in [27] about the construction of certain shape density involving Euler hypergeometric functions of matrix arguments. The associated distribution is obtained by a connection between the required shape invariants and a known result on canonical correlations published in 1963. As usual in matrix variate statistical shape theory, the densities are expressed in terms of infinite series of zonal polynomials. However, if we consider certain parametric subspace for the parity of the number of landmarks, the computational problem can be solved analytically by deriving the Eulerian matrix relation of two matrix arguments. Under that restriction, the analysis of classical landmark data is based on polynomial distribution with small degree. Finally, a methodology to compare Eulerian shape and landmark discrimination under equivalent classes is proposed and applied in machine vision.

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ACKNOWLEDGMENTS

The authors wish to thank the Editor and the anonymous Reviewers for their constructive comments on the preliminary version of this paper. F. Caro was supported by a University of Medellin, in the context of a join research project with University of Bordeaux and University of Toulouse.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. University of Medellin, Faculty of Basic Sciences, Medellín, Colombia

    Francisco J. Caro-Lopera

  2. Universidad Autónoma de Chihuahua, Facultad de Zootecnia y Ecología, Chihuahua, México

    José A. Díaz-García

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  1. Francisco J. Caro-Lopera
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  2. José A. Díaz-García
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Correspondence to Francisco J. Caro-Lopera or José A. Díaz-García.

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Caro-Lopera, F.J., Díaz-García, J.A. Polynomial Eulerian Shape Distributions. Math. Meth. Stat. 33, 373–391 (2024). http://doi.org/10.3103/S1066530724700194

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