# 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 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 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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https://hal-mines-paristech.archives-ouvertes.fr/hal-01108141
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Submitted on : Thursday, January 22, 2015 - 11:28:56 AM
Last modification on : Wednesday, November 17, 2021 - 12:27:12 PM
Long-term archiving on: : Friday, September 11, 2015 - 8:26:41 AM

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• HAL Id : hal-01108141, version 1

### Citation

Jesus Angulo. Morphological Scale-Space Operators for Images Supported on Point Clouds. 2014. ⟨hal-01108141v1⟩

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