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Spherical Deconvolution

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  • Healy, Dennis M.
  • Hendriks, Harrie
  • Kim, Peter T.
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    Abstract

    This paper proposes nonparametric deconvolution density estimation overS2. Here we would think of theS2elements of interest being corrupted by randomSO(3) elements (rotations). The resulting density on the observations would be a convolution of theSO(3) density with the trueS2density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis onSO(3) andS2, involving rotational and spherical harmonics, respectively. We especially consider the case where the deconvolution operator is a bounded operator lowering the Sobolev order by a finite amount. Consistency results are obtained with rates of convergence calculated under the expectedS2and Sobolev square norms that are proportionally inverse to some power of the sample size. As an example we introduce the rotational version of the Laplace distribution.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 67 (1998)
    Issue (Month): 1 (October)
    Pages: 1-22

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    Handle: RePEc:eee:jmvana:v:67:y:1998:i:1:p:1-22

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    Related research

    Keywords: consistency density estimation deconvolution rotational harmonics rotational Laplace distribution Sobolev spaces spherical harmonics;

    References

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    1. Bai, Z. D. & Rao, C. Radhakrishna & Zhao, L. C., 1988. "Kernel estimators of density function of directional data," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 24-39, October.
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    Citations

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    Cited by:
    1. Kim, Peter T. & Koo, Ja-Yong & Luo, Zhi-Ming, 2009. "Weyl eigenvalue asymptotics and sharp adaptation on vector bundles," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1962-1978, October.
    2. Vareschi, T., 2014. "Application of second generation wavelets to blind spherical deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 398-417.
    3. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    4. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    5. Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
    6. Arnoud van Rooij & Frits Ruymgaart, 2001. "Abstract Inverse Estimation with Application to Deconvolution on Locally Compact Abelian Groups," Annals of the Institute of Statistical Mathematics, Springer, vol. 53(4), pages 781-798, December.
    7. Hendriks, Harrie, 2003. "Application of fast spherical Fourier transform to density estimation," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 209-221, February.
    8. Kim, Peter T. & Koo, Ja-Yong & Park, Heon Jin, 2004. "Sharp minimaxity and spherical deconvolution for super-smooth error distributions," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 384-392, August.
    9. Koo, Ja-Yong & Kim, Peter T., 2008. "Sharp adaptation for spherical inverse problems with applications to medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 165-190, February.
    10. Goldenshluger, Alexander, 2002. "Density Deconvolution in the Circular Structural Model," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 360-375, May.
    11. Hielscher, Ralf, 2013. "Kernel density estimation on the rotation group and its application to crystallographic texture analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 119-143.

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