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A Geometric Dissimilarity Criterion Between Jordan Spatial Mosaics. Theoretical Aspects and Application to Segmentation Evaluation

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Abstract

An image segmentation process often results in a special spatial set, called a mosaic, as the subdivision of a domain S within the n-dimensional Euclidean space. In this paper, S will be a compact domain and the study will be focused on finite Jordan mosaics, that is to say mosaics with a finite number of regions and where the boundary of each region is a Jordan hypersurface.

The first part of this paper addresses the problem of comparing a Jordan mosaic to a given reference Jordan mosaic and introduces the ε dissimilarity criterion. The second part will show that the ε dissimilarity criterion can be used to perform the evaluation of image segmentation processes. It will be compared to classical criterions in regard to several geometric transformations. The pros and cons of these criterions are presented and discussed, showing that the ε dissimilarity criterion outperforms the other ones.

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  1. CIS and LPMG, UMR CNRS 5148, IFRESIS, IFR INSERM 143, École Nationale Supérieure des Mines de Saint-Etienne, 158 Cours Fauriel, 42023, Saint-Etienne cedex, France

    Yann Gavet & Jean-Charles Pinoli

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Gavet, Y., Pinoli, JC. A Geometric Dissimilarity Criterion Between Jordan Spatial Mosaics. Theoretical Aspects and Application to Segmentation Evaluation. J Math Imaging Vis 42, 25–49 (2012). https://doi.org/10.1007/s10851-011-0272-4

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