Abstract
This paper introduces a probabilistic framework for adaptive morphological dilation and erosion. More precisely our probabilistic formalization is based on using random walk simulations for a stochastic estimation of adaptive and robust morphological operators. Hence, we propose a theoretically sound morphological counterpart of Monte Carlo stochastic filtering. The approach by simulations is inefficient but particularly tailorable for introducing different kinds of adaptability. From a theoretical viewpoint, stochastic morphological operators fit into the framework of Bellman-Maslov chains, the ( max , + )-counterpart of Markov chains, which the basis behind the efficient implementations using sparse matrix products.
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References
Angulo, J.: Morphological Bilateral Filtering and Spatially-Variant Adaptive Structuring Functions. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 212–223. Springer, Heidelberg (2011)
Akian, M., Quadrat, J.P., Viot, M.: Duality between Probability and Optimization. In: Gunawardena (ed.) Idempotency, Cambridge University Press (1998)
Akian, M.: Densities of idempotent measures and large deviations. Trans. of the American Mathematical Society 351(11), 4515–4543 (1999)
Azzabou, N., Paragios, N., Guichard, F.: Random walks, constrained multiple hypothesis testing and image enhancement. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006, Part I. LNCS, vol. 3951, pp. 379–390. Springer, Heidelberg (2006)
Black, M., Sapiro, G., Marimont, D., Heeger, D.: Robust anisotropic diffusion. IEEE Trans. on Image Processing 7, 421–432 (1998)
Bellman, R., Dreyfus, S.: Dynamic Programming and Applications. Dunod, Paris (1965)
Buckley, M., Talbot, H.: Flexible linear openings and closings. In: Mathematical Morphology and its Applications to Image and Signal Processing, pp. 109–118. Springer, New York (2000)
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)
Del Moral, P.: Maslov Optimization Theory: Optimality versus Randomness. In: Kolokoltsov, Maslov (eds.) Idempotency Analysis and its Applications, Kluwer Publishers (1997)
Del Moral, P., Salut, G.: Random Particle Methods in ( max , + )-optimization problems. In: Gunawardena (ed.) Idempotency, Cambridge University Press (1998)
Estrada, F.J., Fleet, D.J., Jepson, A.D.: Stochastic Image Denoising. In: Proc. of BMVC 2009 (2009)
Gondran, M., Minoux, M.: Graphes et Algorithmes. Eyrolles (1995)
Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods. Pattern Recognition 42(10), 2306–2316 (2009)
Heijmans, H.J.A.M.: Morphological image operators. Academic Press, Boston (1994)
Heijmans, H.J.A.M., van den Boomgaard, R.: Algebraic Framework for linear and morphological scale-spaces. Journal of Visual Communication and Image Representation 13(1-2), 269–301 (2002)
Heijmans, H., Buckley, M., Talbot, H.: Path openings and closings. Journal of Mathematical Imaging and Vision 22(2-3), 107–119 (2005)
Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 25(4), 395–404 (2007)
Maslov, V.: Méthodes opératorielles. Editions Mir (1987)
Maragos, P.: Slope Transforms: Theory and Application to Nonlinear Signal Processing. IEEE Trans. on Signal Processing 43(4), 864–877 (1995)
Meyer, F.: Grey-weighted, ultrametric and lexicographic distances. In: Mathematical Morphology: 40 Years On (Proc. of ISMM 2005). Computational Imaging and Vision, vol. 30, pp. 289–298. Springer (2005)
Stawiaski, J.: Optimal Path: Theory and Models for Vessel Segmentation. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 417–428. Springer, Heidelberg (2011)
Velasco-Forero, S., Angulo, J.: On Nonlocal Mathematical Morphology. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 219–230. Springer, Heidelberg (2013)
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Angulo, J., Velasco-Forero, S. (2013). Stochastic Morphological Filtering and Bellman-Maslov Chains. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2013. Lecture Notes in Computer Science, vol 7883. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38294-9_15
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DOI: https://doi.org/10.1007/978-3-642-38294-9_15
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