Journal of Mathematical Imaging and Vision

, Volume 61, Issue 9, pp 1235–1242 | Cite as

Almost Regular Metrics on Groups and Lipschitz-Continuity of Distance Transforms

  • Qi GuoEmail author
  • XunLi Su


In this article, the almost inner (resp. lower, upper) regularity of metrics on a group is proposed. In terms of these regularities, a sufficient and necessary condition for distance transforms to be Lipschitz-1 continuous is given, and relations between balls with different centres and radii are discussed. It turns out that the three almost regularities of metrics introduced and the results obtained here might provide useful tools in discrete analysis, mathematical morphology and image analysis, etc.


Distance transform Lipschitz-continuity Regularity of metrics Discrete analysis 

Mathematics Subject Classification

46B99 68R99 65D18 22A20 



The authors would like to express sincere thanks to the referees for their careful reading of the original and the revised manuscripts, their valuable suggestions and recommendations on the content and the references, and also for their pointing out errors and typos, which improve the article. The study is supported by the National Natural Science Foundation of China (Nos. 11671293, 11271282).


  1. 1.
    Baddeley, A.J., Molchanov, I.S.: Averaging of random sets based on their distance transforms. J. Math. Imaging Vis. 8, 79–92 (1998)CrossRefGoogle Scholar
  2. 2.
    Borgefors, G.: On digital distance transforms in three dimensions. Comput. Vis. Image Underst. 64(3), 368–376 (1996)CrossRefGoogle Scholar
  3. 3.
    Borgefors, G.: Distance transformations in digital images. Comput. Vis. Graph. Image Process. 34, 344–371 (1986)CrossRefGoogle Scholar
  4. 4.
    Delfour, M.C., Zolésio, J.P.: Shape analysis via oriented distance transforms. J. Funct. Anal. 123, 129–201 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Garrido, S., Moreno, L., Blanco, D.: Exploration of a cluttered environment using voronoi transform and fast marching. Robot. Auton. Syst. 56(12), 1069–1081 (2008)CrossRefGoogle Scholar
  6. 6.
    Ilić, V., Lindblad, J., Sladoje, N.: Precise Euclidean distance transforms in 3D from voxel coverage representation. Pattern Recognit. Lett. 65, 184–191 (2015)CrossRefGoogle Scholar
  7. 7.
    Jáuregui, D.A., Horain, P.: Region-based vs. edge-based registration for 3D motion capture by real time monoscopic vision. In: Proceedings of the 4th International Conference on Computing Vision/Computer Graphics Collaboration Techniques, MIRAGE 2009, pp. 344–355. Springer (2009)Google Scholar
  8. 8.
    Kiselman, C.O.: Digital Geometry and Mathematical Morphology. 2004.pdf, item 04-A
  9. 9.
    Kiselman, C.O.: Regularity properties of distance transformations in image analysis. Comput. Vis. Image Underst. 64(3), 390–398 (1996)CrossRefGoogle Scholar
  10. 10.
    Mehnert, A.J.H., Jackway, P.T.: On computing the exact Euclidean distance transform on rectangular and hexagonal grids. J. Math. Imaging Vis. 3(11), 223–230 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Molchnov, I.S., Teran, P.: Distance transforms for real-valued functions. J. Math. Anal. Appl. 278, 472–484 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Rosenfeld, A., Pfalz, J.L.: Distance transforms on digital pictures. Pattern Recognit. 1, 33–61 (1968)CrossRefGoogle Scholar
  13. 13.
    Rosenfeld, A., Pfalz, J.L.: Sequential operations in digital picture processing. J. Assoc. Comput. Mach. 13(4), 471–494 (1966)CrossRefGoogle Scholar
  14. 14.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)zbMATHGoogle Scholar
  15. 15.
    Su, X.L., Guo, J.F., Guo, Q.: The discrete degree of metric spaces and the Lipschitz-continuity of distance transforms. J. Math. Anal. Appl. 389, 863–870 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Strand, R., Nagy, B., Borgefors, G.: Digital distance functions on three-dimensional grids. Theor. Comput. Sci. 412, 1350–1363 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ukil, S., Reinhardt, J.: Anatomy-guided lung lobe segmentation in X-ray CT images. IEEE Trans. Med. Imaging 28(2), 202–214 (2009)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSuzhou University of Science and TechnologySuzhouChina

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