Journal of Mathematical Imaging and Vision

, Volume 61, Issue 3, pp 310–330 | Cite as

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

  • Julien MilleEmail author
  • Aurélie Leborgne
  • Laure Tougne


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.


Skeleton Distance Flux Ridgeness 



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.

Supplementary material

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Supplementary material 1 (pdf 461 KB)


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LIFAT, EA6300INSA Centre Val de LoireBloisFrance
  2. 2.Institut Pascal, UMR6602Université Clermont Auvergne, CNRSAubièreFrance
  3. 3.Université Lyon 2, LIRIS, UMR5205Université de Lyon, CNRSBronFrance

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