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Riemannian Mathematical Morphology

Abstract : This paper introduces mathematical morphology operators for real-valued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance used for the adjoint dilation/erosion. We then extend the canonic case to a most general framework of Riemannian operators based on the notion of admissible Riemannian structuring function. An alternative paradigm of morphological Riemannian operators involves an external structuring function which is parallel transported to each point on the manifold. Besides the definition of the various Riemannian dilation/erosion and Riemannian opening/closing, their main properties are studied. We generalize also some results on Lasry-Lions regularization for non-smooth images on Cartan-Hadamard manifolds. Theoretical connections with previous works on adaptive morphology and manifold shape morphology are also considered. From a practical viewpoint, various useful image embedding into Riemannian manifolds are formalized, with some illustrative examples of morphological processing real-valued 3D surfaces.
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Preprints, Working Papers, ...
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Contributor : Jesus Angulo Connect in order to contact the contributor
Submitted on : Monday, October 28, 2013 - 10:51:41 AM
Last modification on : Wednesday, November 17, 2021 - 12:27:12 PM
Long-term archiving on: : Wednesday, January 29, 2014 - 4:45:26 AM


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  • HAL Id : hal-00877144, version 1


Jesus Angulo, Santiago Velasco-Forero. Riemannian Mathematical Morphology. 2013. ⟨hal-00877144v1⟩



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