Abstract
The aim of this paper is to develop the theory, and to propose an algorithm, for morphological processing of images painted on point clouds, viewed as a length metric measure space \((X,d,\mu )\). In order to extend morphological operators to process point cloud supported images, one needs to define dilation and erosion as semigroup operators on \((X,d)\). That corresponds to a supremal convolution (and infimal convolution) using admissible structuring function on \((X,d)\). From a more theoretical perspective, we introduce the notion of abstract structuring functions formulated on length metric Maslov idempotent measurable spaces, which is the appropriate setting for \((X,d)\). In practice, computation of Maslov structuring function is approached by a random walks framework to estimate heat kernel on \((X,d,\mu )\), followed by the logarithmic trick.
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References
Akian, M.: Densities of idempotent measures and large deviations. Trans. of the American Mathematical Society 351(11), 4515–4543 (1999)
Akian, M., Gaubert, S., Walsh, C.: The Max-Plus Martin boundary. Documenta Mathematica 14, 195–240 (2009)
Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Inventiones Mathematicae 195(2), 289–391 (2014)
Angulo, J., Velasco-Forero, S.: Stochastic morphological filtering and bellman-maslov chains. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 171–182. Springer, Heidelberg (2013)
Angulo, J., Velasco-Forero, S.: Riemannian Mathematical Morphology. Pattern Recognition Letters 47, 93–101 (2014)
Belkin, M., Sun, J., Wang, Y.: Constructing laplace operator from point clouds in \(R^d\). In: Proc. of ACM Symp. on Discrete Algorithms, pp. 1031–1040 (2009)
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)
Calderon, S., Boubekeur, T.: Point Morphology. ACM Transactions on Graphics (Proc. SIGGRAPH 2014) (2014)
Crane, K., Weischedel, C., Wardetzky, M.: Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow. ACM Trans. Graph. 32(5) (2013)
Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. American Mathematical Society (2012)
Grigor’yan, A., Hu, J., Lau, K.-S.: Heat kernels on metric measure spaces. In: Geometry and Analysis on Fractals. Springer Proceedings in Mathematics & Statistics, vol. 88, pp. 147–208 (2014)
Heijmans, H.J.A.M.: Morphological image operators. Academic Press, Boston (1994)
Jackway, P.T., Deriche, M.: Scale-Space Properties of the Multiscale Morphological Dilation-Erosion. IEEE Trans. Pattern Anal. Mach. Intell. 18(1), 38–51 (1996)
Liang, J., Zhao, H.: Methods and Algorithms for Scientific Computing Solving Partial Differential Equations on Point Clouds. SIAM J. Sci. Comput. 35(3), A1461–A1486 (2013)
Litvinov, G.L., Maslov, V.P., Shpiz, G.B.: Idempotent Functional Analysis: An Algebraic Approach. Mathematical Notes 69(5–6), 696–729 (2001)
Lott, J., Villani, C.: Hamilton-Jacobi semigroup on length spaces and applications. Journal de Math. Pures et Appliquées 88, 219–229 (2007)
Maragos, P.: Differential morphology and image processing. IEEE Transactions on Image Processing 5(1), 922–937 (1996)
Maslov, V.: Méthodes opératorielles. Editions Mir (1987)
Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics 5(3), 313–347 (2005)
Del Moral, P.: Maslov optimization theory: optimality versus randomness. In: Kolokoltsov, V.N., Maslov, V.P. (eds.) Idempotency Analysis and its Applications. Kluwer Publishers (1997)
Serra, J.: Image Analysis and Mathematical Morphology. Theoretical Advances, vol. II. Academic Press, London (1988)
Sobolevskii, A.N.: Aubry-Mather theory and idempotent eigenfunctions of Bellman operator. Commun. Contemp. Math. 1, 517–533 (1999)
Spira, A., Kimmel, R., Sochen, N.: A short-time Beltrami kernel for smoothing images and manifolds. IEEE Trans. on Image Processing 16(6), 1628–1636 (2007)
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Angulo, J. (2015). Morphological Scale-Space Operators for Images Supported on Point Clouds. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_7
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DOI: https://doi.org/10.1007/978-3-319-18461-6_7
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