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The discontinuity issue of total orders on metric spaces and its consequences for mathematical morphology

Abstract : We address here the problem of discontinuities of total orders in a metric space and its implications for mathematical morphology. We first give a rigorous formulation of the problem. Then, a new approach is proposed to tackle the discontinuity issue by adapting the order to the image to be processed. Given an image and a total order we define a cost that evaluates the importance of the discontinuities for morphological processing. The proposed order is then built as a minimization of this cost function. One of the strength of the proposed framework is its generality: the only ingredient required to build the total order is the graph of distances between values of the image. The adapted order can be computed for any image valued in a metric space where the distance is explicitly known. We present results for color images, diffusion tensor images (DTI) and images valued in the hyperbolic upper half-plane.
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https://hal.archives-ouvertes.fr/hal-00948232
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Submitted on : Thursday, January 15, 2015 - 1:38:04 PM
Last modification on : Wednesday, November 17, 2021 - 12:27:12 PM
Long-term archiving on: : Saturday, April 15, 2017 - 7:11:19 PM

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  • HAL Id : hal-00948232, version 3

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Emmanuel Chevallier, Jesus Angulo. The discontinuity issue of total orders on metric spaces and its consequences for mathematical morphology. 2014. ⟨hal-00948232v3⟩

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