Abstract
Development of spatially-variant filtering is well established in the theory and practice of Gaussian filtering. The aim of the paper is to study how to generalize these linear approaches in order to introduce adaptive nonlinear filters which asymptotically correspond to spatially-variant morphological dilation and erosion. In particular, starting from the bilateral filtering framework and using the notion counter-harmonic mean, our goal is to propose a new low complexity approach to define spatially-variant bilateral structuring functions. Then, the adaptive structuring elements are obtained by thresholding the bilateral structuring functions. The methodological results of the paper are illustrated with various comparative examples.
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References
Angulo, J.: Pseudo-Morphological Image Diffusion using the Counter-Harmonic Paradigm. In: Blanc-Talon, J., Bone, D., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2010, Part I. LNCS, vol. 6474, pp. 426–437. Springer, Heidelberg (2010)
van den Boomgaard, R., Dorst, L.: The morphological equivalent of Gaussian scale-space. In: Proc. of Gaussian Scale-Space Theory, pp. 203–220. Kluwer, Dordrecht (1997)
Bouaynaya, N., Schonfeld, D.: Theoretical foundations of spatially-variant mathematical morphology part ii: Gray-level images. IEEE Trans. Pattern Anal. Mach. Intell. 30, 837–850 (2008)
Bullen, P.S.: Handbook of Means and Their Inequalities, 2nd edn. Springer, Heidelberg (1987)
Burgeth, B., Weickert, J.: An Explanation for the Logarithmic Connection between Linear and Morphological System Theory. International Journal of Computer Vision 64(2-3), 157–169 (2005)
Burgeth, B., Breuß, M., Pizarro, L., Weickert, J.: PDE-driven adaptive morphology for matrix fields. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 247–258. Springer, Heidelberg (2009)
Catte, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis 29(1), 182–193 (1992)
Dorst, L., van den Boomgaard, R.: Morphological Signal Processing and the Slope Transform. Signal Processing 38, 79–98 (1994)
Florack, L., Maas, R., Niessen, W.: Pseudo-Linear Scale-Space Theory. International Journal of Computer Vision 31(2-3), 1–13 (1999)
Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods. Pattern Recognition 42(10), 2306–2316 (2009)
Heijmans, H., Ronse, C.: The Algebraic Basis of Mathematical Morphology. Part I: Dilations and Erosions. CVGIP: Image Understanding 50(3), 245–295 (1990)
Lerallut, R., Decenciere, E., Meyer, F.: Image Filtering Using Morphological Amoebas. Image Vision and Computing 25(4), 395–404 (2007)
Maragos, P.: Slope Transforms: Theory and Application to Nonlinear Signal Processing. IEEE Trans. on Signal Processing 43(4), 864–877 (1995)
Maragos, P., Vachier, C.: A PDE Formulation for Viscous Morphological Operators with Extensions to Intensity- Adaptive Operators. In: Proc. of ICIP 2008 (2008)
Maragos, P., Vachier, C.: Overview of adaptive morphology: trends and perspectives. In: Proc. of ICIP 2009 (2009)
Perona, P., Malik, J.: Scale-Space and Edge Detection Using Anisotropic Diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)
Paris, S., Kornprobst, P., Tumblin, J., Durand, F.: Bilateral Filtering: Theory and Applications. Foundations and Trends in Computer Graphics and Vision 4(1), 1–73 (2008)
Roerdink, J.: Adaptivity and group invariance in mathematical morphology. In: Proc. of ICIP 2009 (2009)
Serra, J.: Image Analysis and Mathematical Morphology. Theoretical Advances, vol. II. Academic Press, London (1988)
Soille, P.: Generalized geodesy via geodesic time. Pattern Recognition Letters 15(12), 1235–1240 (1994)
Soille, P.: Morphological Image Analysis. Springer, Berlin (1999)
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Proc. of 6th Int. Conf. Computer Vision (ICCV 1998), New Delhi, India, pp. 839–846 (1998)
van Vliet, L.J.: Robust Local Max-Min Filters by Normalized Power-Weighted Filtering. In: Proc. of IEEE 17th International Conference of the Pattern Recognition (ICPR 2004), vol. 1, pp. 696–699 (2004)
Vachier, C., Meyer, F.: News from Viscousland. In: Proc. of ISMM 2007 (2007)
Verdú-Monedero, R., Angulo, J., Serra, J.: Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields. IEEE Trans. on Image Processing 20(1), 200–212 (2011)
Weiss, B.: Fast median and bilateral filtering. ACM Transactions on Graphics 25(3), 519–526 (2006)
Welk, M.: Families of generalised morphological scale spaces. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space 2003. LNCS, vol. 2695, pp. 770–784. Springer, Heidelberg (2003)
Welk, M., Breuß, M., Vogel, O.: Morphological amoebas are self-snakes. Journal of Mathematical Imaging and Vision 39(2), 87–99 (2011)
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Angulo, J. (2011). Morphological Bilateral Filtering and Spatially-Variant Adaptive Structuring Functions. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds) Mathematical Morphology and Its Applications to Image and Signal Processing. ISMM 2011. Lecture Notes in Computer Science, vol 6671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21569-8_19
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DOI: https://doi.org/10.1007/978-3-642-21569-8_19
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