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Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging

Author

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  • Bandulasiri, Ananda
  • Bhattacharya, Rabi N.
  • Patrangenaru, Vic

Abstract

For all p>2,k>p, a size-and-reflection-shape space of k-ads in general position in , invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold , a space of orbits of scaled k-ads in general position under the group of isometries of , providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p>2. The Veronese embedding of the planar Kendall shape manifold is extended to an equivariant embedding of the size-and-shape manifold , which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:9:p:1867-1882
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    References listed on IDEAS

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    1. Gordana Derado & Kanti Mardia & Vic Patrangenaru & Hilary Thompson, 2004. "A Shape-based Glaucoma Index for Tomographic Images," Journal of Applied Statistics, Taylor & Francis Journals, vol. 31(10), pages 1241-1248.
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    Cited by:

    1. Crane, M. & Patrangenaru, V., 2011. "Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 225-237, February.
    2. 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.
    3. Kovacev-Nikolic Violeta & Bubenik Peter & Nikolić Dragan & Heo Giseon, 2016. "Using persistent homology and dynamical distances to analyze protein binding," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(1), pages 19-38, March.
    4. Osborne, Daniel & Patrangenaru, Vic & Ellingson, Leif & Groisser, David & Schwartzman, Armin, 2013. "Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 163-175.
    5. 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.

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    1. Crane, M. & Patrangenaru, V., 2011. "Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 225-237, February.

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