Abstract
Multivariate nonparametric smoothers are adversely impacted by the sparseness of data in higher dimension, also known as the curse of dimensionality. Adaptive smoothers, that can exploit the underlying smoothness of the regression function, may partially mitigate this effect. We present an iterative procedure based on traditional kernel smoothers, thin plate spline smoothers or Duchon spline smoother that can be used when the number of covariates is important. However the method is limited to small sample sizes (n < 2, 000) and we will propose some thoughts to circumvent that problem using, for example, pre-clustering of the data. Applications considered here are image denoising.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Boulanger, J., Kervrann, C., Bouthemy, P., Elbau, P., Sibarita, J.B., Salamero, J.: Patch-based non-local functional for denoising fluorescence microscopy image sequences. IEEE Trans. Med. Imaging, 29(2), 442–454 (2009)
Breiman, L., Freiman, J., Olshen, R., Stone, C.: Classification and Regression Trees, 4th edn. CRC Press, Boca Raton (1984)
Bühlmann, P., Yu, B.: Boosting with the l 2 loss: regression and classification. J. Am. Stat. Assoc. 98, 324–339 (2003)
Cornillon, P.A., Hengartner, N., Matzner-Løber, E.: Recursive bias estimation for multivariate regression. arXiv:1105.3430v2 (2011)
Cornillon, P.A., Hengartner, N., Jegou, N., Matzner-Løber, E.: Iterative bias reduction: a comparative study. Stat. Comput. 23, 77–791 (2012)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3d transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)
Di Marzio, M., Taylor, C.: On boosting kernel regression. J. Stat. Plan. Infer. 138, 2483–2498 (2008)
Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Lecture Notes in Mathematics. 571, 85–100 (1977)
Gu, C.: Smoothing Spline ANOVA Models. Springer, Berlin (2002)
Hansen, M., Yu, B.: Model selection and minimal description length principle. J. Am. Stat. Assoc. 96, 746–774 (2001)
Milanfar, P.: A tour of modern image filtering. IEEE Signal Process. Mag. 30, 106–128 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this paper
Cite this paper
Cornillon, PA., Hengartner, N., Matzner-Løber, E., Thieurmel, B. (2014). Nonparametric Regression Based Image Analysis. In: Akritas, M., Lahiri, S., Politis, D. (eds) Topics in Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 74. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0569-0_17
Download citation
DOI: https://doi.org/10.1007/978-1-4939-0569-0_17
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-0568-3
Online ISBN: 978-1-4939-0569-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)