Morphological Scale-Space Operators for Images Supported on Point Clouds

Conference paper

DOI: 10.1007/978-3-319-18461-6_7

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9087)
Cite this paper as:
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


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.


Mathematical morphology Point clouds image Metric measure space Idempotent measure Hamilton-Jacobi semigroup 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.CMM-Centre de Morphologie Mathématique, MINES ParisTechPSL-Research UniversityParisFrance

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