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Euclidean Distance-Based Skeletons: A Few Notes on Average Outward Flux and Ridgeness

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Abstract

Among the various existing and mathematically equivalent definitions of the skeleton, we consider the set of critical points of the Euclidean distance transform of the shape. The problem of detecting these points and using them to generate a skeleton that is stable, thin and homotopic to the shape has been the focus of numerous papers. Skeleton branches correspond to ridges of the distance map, i.e., continuous lines of points that are local maxima of the distance in at least one direction. Extracting these ridges is a non-trivial task on a discrete grid. In this context, the average outward flux, used in the Hamilton–Jacobi skeleton (Siddiqi et al. in Int J Comput Vis 48(3):215–231, 2002), and the ridgeness measure (Leborgne et al. in J Vis Commun Image Represent 31:165–176, 2015) have been proposed as ridge detectors. We establish the mathematical relation between these detectors and, extending the work in Dimitrov et al. (Computer vision and pattern recognition, pp 835–841, 2003), we study various local shape configurations, on which closed-form expressions or approximations of the average outward flux and ridgeness can be derived. In addition, we conduct experiments to assess the accuracy of skeletons generated using these measures and study the influence of their respective parameters.

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Notes

  1. Throughout the paper, we refer to detailed mathematical derivations in appendices, which are provided in a supplementary document.

  2. The notion of the object angle is explained in Sect. 3.

  3. A similar result was already stated in [18].

  4. For the CC-AOF, we used the skeleton module by F.-X. Dupé integrated in D. Tschumperlé’s CImg library: https://github.com/dtschump/CImg. For the IMA, we used our own C++ translation of the Java implementation available at http://wimhesselink.nl/imageproc/skeletons.

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Acknowledgements

We thank Moncef Hidane for the fruitful discussions. In particular, he suggested that we study bounds for elliptic integrals and directed us toward several well-known theorems in calculus.

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Authors and Affiliations

  1. LIFAT, EA6300, INSA Centre Val de Loire, 41034, Blois, France

    Julien Mille

  2. Institut Pascal, UMR6602, Université Clermont Auvergne, CNRS, 63178, Aubière, France

    Aurélie Leborgne

  3. Université Lyon 2, LIRIS, UMR5205, Université de Lyon, CNRS, 69676, Bron, France

    Laure Tougne

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  1. Julien Mille
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Correspondence to Julien Mille.

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Mille, J., Leborgne, A. & Tougne, L. Euclidean Distance-Based Skeletons: A Few Notes on Average Outward Flux and Ridgeness. J Math Imaging Vis 61, 310–330 (2019). https://doi.org/10.1007/s10851-018-0836-7

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