IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v124y2014icp398-417.html
   My bibliography  Save this article

Application of second generation wavelets to blind spherical deconvolution

Author

Listed:
  • Vareschi, T.

Abstract

We address the problem of spherical deconvolution in a non-parametric statistical framework, where both the signal and the operator kernel are subject to measurement errors. After a preliminary treatment of the kernel, we apply a thresholding procedure to the signal in a second generation wavelet basis. Under standard assumptions on the kernel, we study the minimax performances of the resulting algorithm in terms of Lp losses (p≥1) on Besov spaces on the sphere. We hereby extend the application of second generation spherical wavelets to the blind spherical deconvolution framework. It is important to stress that the procedure is adaptive with regard to both the target function sparsity and the kernel blurring effect. We end with the study of a concrete example, putting into evidence the improvement of our procedure on the recent blockwise SVD algorithm of Delattre et al. (2012).

Suggested Citation

  • Vareschi, T., 2014. "Application of second generation wavelets to blind spherical deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 398-417.
    • Handle: RePEc:eee:jmvana:v:124:y:2014:i:c:p:398-417
    DOI: 10.1016/j.jmva.2013.11.012
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X13002546
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2013.11.012?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Healy, Dennis M. & Hendriks, Harrie & Kim, Peter T., 1998. "Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 1-22, October.
    2. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    2. 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.
    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 & 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.
    5. 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.
    6. Hendriks, Harrie, 2003. "Application of fast spherical Fourier transform to density estimation," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 209-221, February.
    7. Jammalamadaka, S. Rao & Terdik, György H., 2019. "Harmonic analysis and distribution-free inference for spherical distributions," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 436-451.
    8. Goldenshluger, Alexander, 2002. "Density Deconvolution in the Circular Structural Model," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 360-375, May.
    9. Kim, Yoon Tae & Park, Hyun Suk, 2013. "Geometric structures arising from kernel density estimation on Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 112-126.
    10. Pham Ngoc, Thanh Mai & Rivoirard, Vincent, 2013. "The dictionary approach for spherical deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 138-156.
    11. 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.
    12. Pelletier, Bruno, 2005. "Kernel density estimation on Riemannian manifolds," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 297-304, July.
    13. Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    14. 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;The Institute of Statistical Mathematics, vol. 53(4), pages 781-798, December.
    15. Jeon, Jeong Min & Van Keilegom, Ingrid, 2023. "Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    16. Caponera, Alessia & Durastanti, Claudio & Vidotto, Anna, 2021. "LASSO estimation for spherical autoregressive processes," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 167-199.
    17. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    18. 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.
    19. Durastanti, Claudio & Geller, Daryl & Marinucci, Domenico, 2012. "Adaptive nonparametric regression on spin fiber bundles," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 16-38, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:124:y:2014:i:c:p:398-417. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.