Abstract
Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona–Malik equation, one of the first nonlinear PDE’s proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean curvature motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.
Similar content being viewed by others
References
Amann H.: A new approach to quasilinear parabolic problems. In: International Conference on Differential Equations, 2005
Andreu F., Ballester C., Caselles V., Mazon J. M. (2001) Minimizing Total Variation Flow. Differ. Integral Equations 14(3): 321–360
Di Zenzo S. (1986) A note on the gradient of a multi-image. Comput. Vis. Graph. Image Process. 33, 116–125
Esedoglu S. (2001) An analysis of the Perona–Malik scheme. Commun. Pure Appl. Math. 54, 1442–1487
Guichard F., Morel J. M. (2003) A note on two classical enhancement filters and their associated PDE’s. Int. J. Comput. Vis. 52(2–3): 153–160
Harten A., Enquist B., Osher S., Chakravarthy S. (1987) Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231–303
Kichenassamy S. (1997) The Perona–Malik paradox. SIAM J. Appl. Math. 57(2): 1328–1342
Kimia B.B., Tannenbaum A., Zucker S.W. (1992) On the evolution of curves via a function of curvature I the classical case. J. Math. Anal. Appl. 163(2): 438–458
Kimmel R., Malladi R., Sochen N. (2000) Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. Int. J. Comput. Vis. 39(2): 111–129
Kindermann, S., Osher, S., Jones, P.: Deblurring and denoising of images by nonlocal functionals. UCLA Computational and Applied Mathematics Reports 04–75, 2004
Kornprobst, P.: Contributions la Restauration d’Images et l’Analyse de Sequences: Approches Variationnelles et Solutions de Viscosite. PhD thesis, Universite de Nice-Sophia Antipolis, 1998
Kramer H.P., Bruckner J.B. (1975) Iterations of a non-linear transformation for enhancement of digital images. Pattern Recogn. 1–2, 53–58
Lee J.S. (1983) Digital image smoothing and the sigma filter. Comput. Vis. Graph. Image Process. 24, 255–269
Masnou, S.: Filtrage et désocclusion d’images par méthodes d’ensembles de niveau. PhD Dissertation, Université Paris-IX Dauphine, 1998
Osher S., Rudin L. (1990) Feature oriented image enchancement using shock filters. SIAM J Numer. Anal. 27, 919–940
Perona P., Malik J. (1990) Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639
Rudin L., Osher S., Fatemi E. (1992) Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268
Saint-Marc P., Chen J.S., Medioni G. (1991) Adaptive smoothing: a general tool for early vision. IEEE Trans. Pattern Anal. Mach. Intell. 13(6): 514
Sapiro G., Ringach D.L. (1996) Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5(11): 1582–1585
Polzehl, J., Spokoiny, V.: Varying coefficient regression modeling. Preprint, Weierstrass Institute for Applied Analysis and Stochastics pp 818, 2003
Schavemaker J.G.M., Reinders M.J.T., Gerbrands J.J., Backer E. (2000) Image sharpening by morphological filtering. Pattern Recogn. 33, 997–1012
Sethian J. (1985) Curvature and the evolution of fronts. Commun. Math. Phys. 101, 487–499
Smith S.M., Brady J.M. (1997) Susan—a new approach to low level image processing. Int. J. Comput. Vis. 23(1): 45–78
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Sixth International Conference on Computer Vision pp. 839–46, 1998
Weickert J. (1998) Anisotropic Diffusion in Image Processing. Tuebner, Stuttgart
Yaroslavsky L.P. (1985) Digital Picture Processing—An Introduction. Springer, Berlin Heidelberg New York
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been partially financed by the Centre National d’Etudes Spatiales (CNES), the Office of Naval Research under grant N00014-97-1-0839, the Ministerio de Ciencia y Tecnologia under grant MTM2005-08567. During this work, the first author had a fellowship of the Govern de les Illes Balears for the realization of his PhD.
Rights and permissions
About this article
Cite this article
Buades, A., Coll, B. & Morel, JM. Neighborhood filters and PDE’s. Numer. Math. 105, 1–34 (2006). https://doi.org/10.1007/s00211-006-0029-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-006-0029-y