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Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours

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  • Ellingson, Leif
  • Patrangenaru, Vic
  • Ruymgaart, Frits

Abstract

Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider the analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert–Schmidt operators, we can define extrinsic mean shapes of random planar contours and their sample analogues. We then apply the general methods to this problem while considering the computational restrictions faced when utilizing digital imaging data. Comparisons of computational cost are provided to another method for analyzing shapes of contours.

Suggested Citation

  • Ellingson, Leif & Patrangenaru, Vic & Ruymgaart, Frits, 2013. "Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 317-333.
    • Handle: RePEc:eee:jmvana:v:122:y:2013:i:c:p:317-333
    DOI: 10.1016/j.jmva.2013.08.010
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    References listed on IDEAS

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    1. Bandulasiri, Ananda & Bhattacharya, Rabi N. & Patrangenaru, Vic, 2009. "Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1867-1882, October.
    2. Munk, A. & Paige, R. & Pang, J. & Patrangenaru, V. & Ruymgaart, F., 2008. "The one- and multi-sample problem for functional data with application to projective shape analysis," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 815-833, May.
    3. Muller, Hans-Georg & Stadtmuller, Ulrich & Yao, Fang, 2006. "Functional Variance Processes," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1007-1018, September.
    4. Kaziska, David & Srivastava, Anuj, 2007. "Gait-Based Human Recognition by Classification of Cyclostationary Processes on Nonlinear Shape Manifolds," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1114-1124, December.
    5. Axel Munk, 2002. "Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 501-533, September.
    6. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
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    Cited by:

    1. Fabian J.E. Telschow & Michael R. Pierrynowski & Stephan F. Huckemann, 2021. "Functional inference on rotational curves under sample‐specific group actions and identification of human gait," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(4), pages 1256-1276, December.
    2. Luis Gutiérrez & Ramsés H. Mena & Carlos Díaz-Avalos, 2020. "Linear models for statistical shape analysis based on parametrized closed curves," Statistical Papers, Springer, vol. 61(3), pages 1213-1229, June.
    3. Ruite Guo & Hwiyoung Lee & Vic Patrangenaru, 2023. "Test for Homogeneity of Random Objects on Manifolds with Applications to Biological Shape Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1178-1204, August.
    4. Vic Patrangenaru & Peter Bubenik & Robert L. Paige & Daniel Osborne, 2019. "Challenges in Topological Object Data Analysis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 244-271, February.
    5. Victor Patrangenaru & Robert Paige & K. David Yao & Mingfei Qiu & David Lester, 2016. "Projective shape analysis of contours and finite 3D configurations from digital camera images," Statistical Papers, Springer, vol. 57(4), pages 1017-1040, December.

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