Abstract
We study a class of mathematical morphology filters to operate conditionally according to a set of pixels marked by a binary mask. The main contribution of this paper is to provide a general framework for several applications including edge enhancement and image denoising, when it is affected by salt-and-pepper noise. We achieve this goal by revisiting shock filters based on erosions and dilations and extending their definition to take into account the prior definition of a mask of pixels that should not be altered. New definitions for conditional erosions and dilations leading to the concept of conditional toggle mapping. We also investigate algebraic properties as well as the convergence of the associate shock filter. Experiments show how the selection of appropriate methods to generate the masks lead to either edge enhancement or salt-and-pepper denoising. A quantitative evaluation of the results demonstrates the effectiveness of the proposed methods. Additionally, we analyse the application of conditional toggle mapping in remote sensing as pre-filtering for hierarchical segmentation.
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Notes
An operator ϕ:𝔼→𝔽 is idempotent if ∀x∈𝔼,ϕ 2(x)=ϕ(ϕ(x))=ϕ(x).
An operator ϕ:𝔼→𝔽 is extensive (resp. anti-extensive) if ∀x∈𝔼,ϕ(x)≥x (resp. ≤).
Flat zones of I are the connected component where the intensity does not change.
References
Alvarez, L., Mazorra, L.: Signal and image restoration using shock filters and anisotropic diffusion. SIAM J. Numer. Anal. 31, 590–605 (1994)
Baudes, A., Coll, B., Morel, J.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)
Boncelet, C.: Image noise models. In: Bovik, A.C. (ed.) Handbook of Image and Video Processing, pp. 325–335. Academic Press, New York (2000)
Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490–530 (2005)
Chan, R., Chung-Wa, H., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans. Image Process. 14(10), 1479–1485 (2005)
Chatterjee, P., Milanfar, P.: Is denoising dead? IEEE Trans. Image Process. 19(4), 895–911 (2010)
Diop, E., Angulo, J.: Spatially adaptive PDEs for robust image sharpening. In: Proc. Int. Conf. Image Processing (ICIP), pp. 949–952 (2012)
Elder, J., Zucker, S.: Local scale control for edge detection and blur estimation. IEEE Trans. Pattern Anal. Mach. Intell. 20(7), 699–716 (1998)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin (1980)
Gilboa, G., Sochen, N., Zeevi, Y.: Image enhancement and denoising by complex diffusion processes. IEEE Trans. Pattern Anal. Mach. Intell. 25(8), 1020–1034 (2004)
Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Regularized shock filters and complex diffusion. In: Proceeding of the 7th European Conference on Computer Vision. Part I, ECCV’02, pp. 399–413. Springer, Berlin (2002)
Gonzalez, R., Woods, R.: Digital Image Processing. Prentice Hall, New York (2008)
Gower, J., Ross, G.: Minimum spanning trees and single linkage cluster analysis. Appl. Stat. 18(1), 54–64 (1969)
Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighborhoods. Pattern Recognit. 42, 2306–2316 (2009)
Heijmans, H.J.A.M.: Morphological Image Operators vol. 10. Academic Press, New York (1994)
Heijmans, H.J.A.M.: Composing morphological filters. IEEE Trans. Image Process. 6(5), 713–724 (1997). doi:10.1109/83.568928
Heijmans, H.J.A.M., Nacken, P., Toet, A., Vincent, L.: Graph morphology. J. Vis. Commun. Image Represent. 3, 24–38 (1992)
Heijmans, H.J.A.M., Ronse, C.: The algebraic basis of mathematical morphology. I. Dilations and erosions. Comput. Vis. Graph. Image Process. 50, 245–295 (1990)
Hwang, H., Haddad, R.A.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4, 499–502 (1995)
Jochems, T.: Morphologie mathématique appliquée au contrôle industriel de pièces coulées. Ph.D. thesis, Ecole National Supérieure des Mines de Paris (1997)
Keshet, R., Heijmans, H.J.A.M.: Adjunctions in pyramids, curve evolution and scale-spaces. Int. J. Comput. Vis. 52, 139–151 (2003)
Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognit. 7(1–2), 53–58 (1975)
Lukac, R., Smolka, B., Martin, K., Plataniotis, K., Venetsanopoulos, A.: Vector filtering for color imaging. IEEE Signal Process. Mag. 22(1), 74–86 (2005)
Mallat, S.: A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edn. Academic Press, New York (2008)
Maragos, P.: Lattice image processing: a unification of morphological and fuzzy algebraic systems. J. Math. Imaging Vis. 22, 333–353 (2005)
Meyer, F.: The levelings. In: Proc. of the ISMM’98, pp. 199–206. Kluwer Academic, Norwell (1998)
Meyer, F., Serra, J.: Contrasts and activity lattice. Signal Process. 16(4), 303–317 (1989). doi:10.1016/0165-1684(89)90028-5
Motta, G., Ordentlich, E., Ramírez, I., Seroussi, G., Weinberger, M.J.: The iDUDE framework for grayscale image denoising. IEEE Trans. Image Process. 20(1), 1–21 (2011)
Osher, S., Rudin, L.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27, 919–940 (1990)
Osher, S., Rudin, L.: Shocks and other nonlinear filtering applied to image processing. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 1567, pp. 414–431 (1991)
Rivest, J.F., Soille, P., Beucher, S.: Morphological gradients. J. Electron. Imaging 2(4), 326–336 (1993). doi:10.1117/12.159642
Roerdink, J.B.T.M.: Adaptivity and group invariance in mathematical morphology. In: International Conference of Image Processing, pp. 2253–2256. IEEE Press, New York (2009). doi:10.1109/ICIP.2009.5413983
Roerdink, J.B.T.M.: Personal communication (2012)
Schavemaker, J., Reinders, M.J.T., Gerbrands, J.J., Backer, E.: Image sharpening by morphological filtering. Pattern Recognit. 33(6), 997–1012 (2000)
Serra, J.: Image Analysis and Mathematical Morphology. Volume 2: Theoretical Advances. Academic Press, New York (1988)
Serra, J.: Toggle mappings. In: Simon, J.C. (ed.) From Pixels to Features, pp. 61–72. North Holland, Amsterdam (1989)
Serra, J., Vincent, L.: An overview of morphological filtering. Circuits Syst. Signal Process. 11(1), 47–108 (1992)
Soille, P.: Beyond self-duality in morphological image analysis. Image Vis. Comput. 23(2), 249–257 (2005)
Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008)
Soille, P.: Constrained connectivity for the processing of very-high-resolution satellite images. Int. J. Remote Sens. 31(22), 5879–5893 (2010)
Soille, P.: Preventing chaining through transitions while favouring it within homogeneous regions. In: Mathematical Morphology and Its Applications to Image and Signal Processing. Lecture Notes in Computer Science, vol. 6671, pp. 96–107. Springer, Berlin (2011)
Soille, P., Grazzini, J.: Constrained Connectivity and Transition Regions. In: Proc. of the ISMM’09, pp. 59–69. Springer, Berlin (2009)
Srinivasan, K., Ebenezer, D.: A new fast and efficient decision-based algorithm for removal of high density impulse noises. IEEE Signal Process. Lett. 14(3), 189–192 (2007)
Taguchi, A., Matsumoto, T.: Removal of impulse noise from highly corrupted images by using noise position information and directional information of image. In: Proc. SPIE, Nonlinear Image Processing and Pattern Analysis XII, vol. 4304, pp. 188–196 (2001)
Takeda, H., Farsiu, S., Milanfar, P.: Robust kernel regression for restoration and reconstruction of images from sparse, noisy data. In: Proc. Int. Conf. Image Processing (ICIP), pp. 1257–1260 (2006)
Takeda, H., Farsiu, S., Milanfar, P.: Kernel regression for image processing and reconstruction. IEEE Trans. Image Process. 16(2), 349–366 (2007)
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Sixth International Conference on Computer Vision, pp. 839–846 (1998)
Velasco-Forero, S., Angulo, J.: Supervised ordering in ℝp: application to morphological processing of hyperspectral images. IEEE Trans. Image Process. 20(11), 3301–3308 (2011)
Velasco-Forero, S., Angulo, J.: Random projection depth for multivariate mathematical morphology. IEEE J. Sel. Top. Signal Process. 6(7), 753–763 (2012)
van Vliet, L., Young, I., Beckers, G.: A nonlinear Laplace operator as edge detector in noisy images. Comput. Vis. Graph. Image Process. 45(2), 167–195 (1989). doi:10.1016/0734-189X(89)90131-X
Wang, Z., Zhang, D.: Progressive switching median filter for the removal of impulse noise from highly corrupted images. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 46(1), 78–80 (1999)
Weickert, J.: Coherence-enhancing shock filters. In: Lecture Notes in Computer Science, pp. 1–8. Springer, Berlin (2003)
Welk, M., Weickert, J., Galć, I.: Theoretical foundations for spatially discrete 1-d shock filtering. Image Vis. Comput. 25, 455–463 (2007)
Xu, H., Zhu, G., Peng, H., Wang, D.: Adaptive fuzzy switching filter for images corrupted by impulse noise. Pattern Recognit. Lett. 25(15), 1657–1663 (2004)
Zhang, S., Karim, M.: A new impulse detector for switching median filters. IEEE Signal Process. Lett. 9(11), 360–363 (2002)
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Velasco-Forero, S., Angulo, J. & Soille, P. Conditional Toggle Mappings: Principles and Applications. J Math Imaging Vis 48, 544–565 (2014). https://doi.org/10.1007/s10851-013-0429-4
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DOI: https://doi.org/10.1007/s10851-013-0429-4