# Morphological Scale-Space Operators for Images Supported on Point Clouds

1 CMM - Centre de Morphologie Mathématique

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.

Keywords : metric measure space point clouds image idempotent measure mathematical morphology Hamilton-Jacobi semigroup

Author: Jesus Angulo -

Source: https://hal.archives-ouvertes.fr/