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 gray level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. It also excessivelly blurs texture and fine structures when noise dominates the signal.In this chapter, 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 PDEs proposed for image restoration. As a solution to the shock effect, 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 also present a generalization of neighborhood filters, the nonlocal means (NL-means) algorithm, addressing the preservation of structure in a digital image. The NL-means algorithm tries to take advantage of the high degree of redundancy of any natural image. By this, we simply mean that every small window in a natural image has many similar windows in the same image. Now in a very general sense inspired by the neighborhood filters, one can define as “neighborhood of a pixel” any set of pixels with a similar window around. All pixels in that neighborhood can be used for predicting its denoised value.We finally analyze the recently introduced variational formulations of neighborhood filters and their application to segmentation and seed diffusion.
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References
Andreu, F., Ballester, C., Caselles, V., Mazon, J.M.: Minimizing total variation flow. Comptes Rendus de l’Academie des Sciences Series I Mathematics 331(11), 867–872 (2000)
Arias, P., Caselles, V., Sappiro, G.: A variational framework for non-local image inpainting. In: Proceedings of the EMMCVPR, Bonn. Springer, Heidelberg (2009)
Attneave, F.: Some informational aspects of visual perception. Psychol. Rev. 61(3), 183–193 (1954)
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, New York (2006)
Bae, S., Paris, S., Durand, F.: Two-scale tone management for photographic look. ACM Trans. Graph. (TOG) 25(3), 645 (2006)
Barash, D.: A fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Trans. Pattern Anal. Mach. Intell. 24, 844–847 (2002)
Bennett, E.P., Mason, J.L., McMillan, L.: Multispectral bilateral video fusion. IEEE Trans. Image Process. 16(5), 1185 (2007)
Boulanger, J., Sibarita, J.B., Kervrann, C., Bouthemy, P.: Nonparametric regression for patch-based fluorescence microscopy image sequence denoising. In: 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 2008 (ISBI 2008), Paris,pp. 748–751, 2008
Boykov, Y., Jolly, M.P.: Interactive graph cuts for optimal boundary and region segmentation of objects in ND images. Int. Conf. Comput. Vis. 1, 105–112 (2001)
Brailean, J.C., Kleihorst, R.P., Efstratiadis, S., Katsaggelos, A.K., Lagendijk, R.L.: Noise reduction filters for dynamic image sequences: a review. Proc. IEEE 83(9), 1272–1292 (1995)
Buades, A., Coll, B., Lisani, J., Sbert, C.: Conditional image diffusion. Inverse Probl. Imaging 1(4):593 (2007)
Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model Simul. 4(2), 490–530 (2005)
Buades, A., Coll, B., Morel, J.M.: Neighborhood filters and PDE’s. Numer. Math. 105(1), 1–34 (2006)
Buades, A., Coll, B., Morel, J.M.: Nonlocal image and movie denoising. Int. J. Comput. Vision. 76(2), 123–139 (2008)
Buades, A., Coll, B., Morel, J.M., Sbert, C.: Self-similarity driven color demosaicking. IEEE Trans. Image Process. 18(6), 1192–1202 (2009)
Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)
Choudhury, P., Tumblin, J.: The trilateral filter for high contrast images and meshes. In: ACM SIGGRAPH 2005 Courses, Los Angeles, p. 5. ACM (2005)
Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74(368), 829–836 (1979)
Coifman, R.R., Donoho, D.L.: Translation-Invariant De-Noising. Lecture Notes in Statistics, pp. 125–125. Springer, New York (1995)
Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Mach. Intell. 24(5), 603–619 (2002)
Criminisi, A., Pérez, P., Toyama, K.: Region filling and object removal by exemplar-based image inpainting. IEEE Trans. Image Process. 13(9), 1200–1212 (2004)
Danielyan, A., Foi, A., Katkovnik, V., Egiazarian, K.: Image and video super-resolution via spatially adaptive block-matching filtering. In: Proceedings of International Workshop on Local and Non-local Approximation in Image Processing, Lausanne (2008)
De Bonet, J.S.: Multiresolution sampling procedure for analysis and synthesis of texture images. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, Los Angeles, p. 368. ACM Press/Addison-Wesley (1997)
Delon, J., Desolneux, A.: Flicker stabilization in image sequences (2009). hal.archives-ouvertes.fr
Di Zenzo, S.: A note on the gradient of a multi-image. Comput. Vis. Graph. 33(1), 116–125 (1986)
Dong, B., Ye, J., Osher, S., Dinov, I.: Level set based nonlocal surface restoration. Multiscale Model. Simul. 7, 589 (2008)
Donoho, D.L.: De-noising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)
Durand, F., Dorsey, J.: Fast bilateral filtering for the display of highdynamic-range images. In: Proceedings of the 29th annual conference on Computer graphics and interactive techniques, ACM New York, pp. 257–266 (2002)
Ebrahimi, M., Vrscay, E.R.: Solving the Inverse Problem of Image Zooming Using “Self-Examples”. In: Kamel, M., Campilho, A. (eds.) Image Analysis and Recognition. Lecture Notes in Computer Science, vol. 4633, p. 117. Springer, Berlin/Heidelberg (2007)
Ebrahimi, M., Vrscay, E.R.: Multi-frame super-resolution with no explicit motion estimation. In: Proceedings of the 2008 International Conference on Image Processing, Computer Vision, and Pattern Recognition, Las Vegas, 2008
Efros, A.A., Leung, T.K.: Texture synthesis by non-parametric sampling. In: International Conference on Computer Vision, Corful, vol. 2, pp. 1033–1038, 1999
Eisemann, E., Durand, F.: Flash photography enhancement via intrinsic relighting. ACM Trans. Graph. (TOG) 23(3), 673–678 (2004)
Elad, M., Datsenko, D.: Example-based regularization deployed to superresolution reconstruction of a single image. Comput. J. 50, 1–16 (2007)
Elmoataz, A., Lezoray, O., Bougleux, S., Ta, V.T.: Unifying local and nonlocal processing with partial difference operators on weighted graphs. In: International Workshop on Local and Non-local Approximation in Image Processing, Lausanne (2008)
Fleishman, S., Drori, I., Cohen-Or, D.: Bilateral mesh denoising. ACM Trans. Graph. (TOG) 22(3), 950–953 (2003)
Gilboa, G., Osher, S.: Nonlocal linear image regularization and supervised segmentation. Multiscale Model. Simul. 6(2), 595–630 (2007)
Grady, L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1 (2006)
Grady, L., Funka-Lea, G.: Multi-label image segmentation for medical applications based on graph-theoretic electrical potentials. In: Computer Vision and Mathematical Methods in Medical and Biomedical Image Analysis (ECCV), pp. 230–245 (2004)
Grady, L.J.: Space-variant computer vision: a graph-theoretic approach. PhD thesis, Boston University (2004)
Guichard, F., Morel, J.M., Ryan, R.: Contrast invariant image analysis and PDEs. Book in preparation
Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71(2), 231–303 (1987)
Huhle, B., Schairer, T., Jenke, P., Straßer, W.: Robust non-local denoising of colored depth data. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, Anchorage, pp. 1–7 (2008)
Jones, T.R., Durand, F., Desbrun, M.: Non-iterative, feature-preserving mesh smoothing. ACM Trans. Graph. 22(3), 943–949 (2003)
Jung, M., Vese, L.A.: Nonlocal variational image deblurring models in the presence of Gaussian or impulse noise. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 5567. Springer, Berlin/Heidelberg (2009)
Kimia, B.B., Tannenbaum, A., Zucker, S.W.: On the evolution of curves via a function of curvature, I: the classical case. J. Math. Anal. Appl. 163(2), 438–458 (1992)
Kimmel, R., Malladi, R., Sochen, N.: Images as embedded maps and minimal surfaces: movies, color, texture, and volumetric medical images. Int. J. Comput. Vis. 39(2), 111–129 (2000)
Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4(4), 1091–1115 (2005)
Lee, J.S.: Digital image smoothing and the sigma filter. Comput. Vis. Graph. 24(2), 255–269 (1983)
Lezoray, O., Ta, V.T., Elmoataz, A.: Nonlocal graph regularization for image colorization. In: International Conference on Pattern Recognition, Tampa (2008)
Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)
Mairal, J., Elad, M., Sapiro, G., et al.: Sparse representation for color image restoration. IEEE Trans. Image Process. 17(1), 53 (2008)
Masnou, S.: Filtrage et désocclusion d’images par méthodes d’ensembles de niveau. PhD thesis, Ceremade, Universit e Paris-Dauphine (1998)
Mignotte, M.: A non-local regularization strategy for image deconvolution. Pattern Recognit. Lett. 29(16), 2206–2212 (2008)
Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM J Numer Anal 27(4), 919–940 (1990)
Ozkan, M.K., Sezan, M.I., Tekalp, A.M.: Adaptive motion-compensated filtering of noisy image sequences. IEEE Trans. Circuits Syst. Video Technol. 3(4), 277–290 (1993)
Peng, H., Rao, R., Messinger, D.W.: Spatio-spectral bilateral filters for hyperspectral imaging. In: SPIE Defense and Security Symposium. International Society for Optics and Photonics (2008)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. PAMI 12(7), 629–639 (1990)
Petschnigg, G., Szeliski, R., Agrawala, M., Cohen, M., Hoppe, H., Toyama, K.: Digital photography with flash and no-flash image pairs. ACM Trans. Graph. (TOG) 23(2), 664–672 (2004)
Peyré, G.: Manifold models for signals and images. Comput. Vis. Image Underst. 113(2),249–260 (2009)
Peyré, G.: Sparse modeling of textures. J. Math. Imaging Vis. 34(1), 17–31 (2009)
Polzehl, J., Spokoiny, V.: Varying coefficient regression modeling by adaptive weights smoothing. WIAS preprint no. 818 (2003)
Protter, M., Elad, M., Takeda, H., Milanfar, P.: Generalizing the nonlocal-means to super-resolution reconstruction. IEEE Trans. Image Process. 18(1), 35–51 (2009)
Ramanath, R., Snyder, W.E.: Adaptive demosaicking. J. Electron. Imaging 12, 633 (2003)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)
Sapiro, G., Ringach, D.L.: Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5(11), 1582–1586 (1996)
Sethian, J.A.: Curvature and the evolution of fronts. Commun. Math. Phys. 101(4), 487–499 (1985)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)
Smith, S.M., Brady, J.M.: SUSAN: a new approach to low level image processing. Int. J. Comput. Vis. 23(1), 45–78 (1997)
Szlam, A.D., Maggioni, M., Coifman, R.R.: A general framework for adaptive regularization based on diffusion processes on graphs. Yale technical report (2006)
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proceedings of the Sixth International Conference on Computer Vision, Bombay, pp. 839–846, 1998
van den Boomgaard, R., van de Weijer, J.: On the equivalence of localmode finding, robust estimation and mean-shift analysis as used in early vision tasks. In: International conference on pattern recognition, Quebec, vol. 16, Citeseer, pp. 927–930, 2002
Weickert, J.: Anisotropic Diffusion in Image Processing. B.G. Teubner, Stuttgart (1998). Citeseer
Winnemoller, H., Olsen, S.C., Gooch, B.: Real-time video abstraction. ACM Trans. Graph. (TOG) 25(3), 1226 (2006)
Wong, A., Orchard, J.: A nonlocal-means approach to exemplar-based inpainting. In: 15th IEEE International Conference on Image Processing, San Diego, 2008, pp. 2600–2603
Yaroslavsky, L.P.: Digital Picture Processing. Springer Secaucus. Springer, Berlin/New York (1985)
Yaroslavsky, L.P.: Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window. In: Unser, M.A., Aldroubi, A., Laine, A.F. (eds.) Wavelet Applications in Signal and Image Processing IV. Proceedings of SPIE, vol. 2825, p. 2. SPIE International Society for Optics and Photonics, Bellingham (1996)
Yaroslavsky, L.P., Egiazarian, K.O., Astola, J.T.: Transform domain image restoration methods: review, comparison, and interpretation. In: Dougherty, E.R., Astola, J.T. (eds.) Nonlinear Image Processing and Pattern Analysis XII. Proceedings of SPIE, vol. 4304, p. 155. SPIE International Society for Optics and Photonics, Bellingham (2001)
Yatziv, L., Sapiro, G.: Fast image and video colorization using chrominance blending. IEEE Trans. Image Process. 15(5), 1120–1129 (2006)
Yoshizawa, S., Belyaev, A., Seidel, H.P.: Smoothing by example: Mesh denoising by averaging with similarity-based weights. In: IEEE International Conference on Shape Modeling and Applications, Matsushima, pp. 38–44, 2006
Zhang, D., Wang, Z.: Image information restoration based on longrange correlation. IEEE Trans. Circuits Syst. Video Technol. 12(5), 331–341 (2002)
Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. UCLA CAM report 09-03 (2009)
Zhu, S.C., Yuille, A.: Region competition: unifying snakes, region growing, and bayes/MDL for multiband image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18(9), 884–900 (1996)
Zhu, X., Lafferty, J., Ghahramani, Z.: Semi-supervised learning: from Gaussian fields to gaussian processes. School of Computer Science, Carnegie Mellon University (2003)
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Morel, JM., Buades, A., Coll, T. (2015). Local Smoothing Neighborhood Filters. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0790-8_26
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