Morphological Scale-Space Operators for Images Supported on Point Clouds

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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Springer-Verlag Berlin Heidelberg. 5th International Conference on Scale Space and Variational Methods in Computer Vision, Jun 2015, Lège-Cap Ferret, France. LNCS 9087 (78--88), 2015, Proc. of SSVM'15 (5th International Conference on Scale Space and Variational Methods in Computer Vision). 〈10.1007/978-3-319-18461-6_7〉
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Jesus Angulo. Morphological Scale-Space Operators for Images Supported on Point Clouds. Springer-Verlag Berlin Heidelberg. 5th International Conference on Scale Space and Variational Methods in Computer Vision, Jun 2015, Lège-Cap Ferret, France. LNCS 9087 (78--88), 2015, Proc. of SSVM'15 (5th International Conference on Scale Space and Variational Methods in Computer Vision). 〈10.1007/978-3-319-18461-6_7〉. 〈hal-01108141v3〉

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