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The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology

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Abstract

We address here the problem of irregularity of total orders in metric spaces and their implications for mathematical morphology. We first give a rigorous formulation of the problem. Then, a new approach is proposed to tackle the 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 conflict 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, and images valued in the hyperbolic upper half plane.

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Chevallier, E., Angulo, J. The Irregularity Issue of Total Orders on Metric Spaces and Its Consequences for Mathematical Morphology. J Math Imaging Vis 54, 344–357 (2016). https://doi.org/10.1007/s10851-015-0607-7

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