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Robust Computations of Reeb Graphs in 2-D Binary Images

  • Antoine VacavantEmail author
  • Aurélie Leborgne
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9667)

Abstract

In this article, we present a novel approach devoted to robustly compute the Reeb graph of a digital binary image, possibly altered by noise. We first employ a skeletonization algorithm, named DECS (Discrete Euclidean Connected Skeleton), to calculate a discrete structure centered within the object. By means of an iterative process, valid with respect to Morse theory, we finally obtain the Reeb graph of the input object. Our various experiments show that our methodology is capable of computing the Reeb graph of images with a high impact of noise, and is applicable in concrete contexts related to medical image analysis.

Keywords

Skeletonization Reeb graph Topology 

References

  1. 1.
    Arcelli, C., di Baja, G.S., Serino, L.: Distance-driven skeletonization in voxel images. IEEE Trans. Pattern Anal. Mach. Intell. 33(4), 709–720 (2010)CrossRefGoogle Scholar
  2. 2.
    Barra, V., Biasotti, S.: 3D shape retrieval and classification using multiple kernel learning on extended Reeb graphs. Vis. Comput. Int. J. Comput. Graph. 30(11), 1247–1259 (2014)Google Scholar
  3. 3.
    Bertrand, G., Couprie, M.: Powerful parallel and symmetric 3D thinning schemes based on critical kernels. J. Math. Imaging Vis. 48(1), 134–148 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theor. Comput. Sci. 392(1–3), 5–22 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coeurjolly, D., Montanvert, A.: Optimal separable algorithms to compute the reverse Euclidean distance transformation and discrete medial axis in arbitrary dimension. IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 437–448 (2007)CrossRefGoogle Scholar
  6. 6.
    Coeurjolly, D., Vacavant, A.: Separable distance transformation and its applications. In: Brimkov, V., Barneva, R. (eds.) Theoretical Foundations and Applications to Computational Imaging Digital Geometry Algorithms, vol. 2, pp. 189–214. Springer, Heidelberg (2012)Google Scholar
  7. 7.
    Ge, X., Safa, I.I., Belkin, M., Wang, Y.: Data skeletonization via reeb graphs. In: ShaweTaylor, J., Zemel, R.S., Bartlett, P., Pereira, F.C.N., Weinberger, K.Q., (eds.) Advances in Neural Information Processing Systems, vol. 24, pp. 837–845 (2011)Google Scholar
  8. 8.
    Gramain, A.: Topologie des surfaces. Presses Universitaires Françaises, Paris (1971)zbMATHGoogle Scholar
  9. 9.
    Harvey, W., Wang, Y., Wenger, R.: A randomized O(mlogm) time algorithm for computing Reeb graphs of arbitrary simplicial complexes. In: Proceedings of Symposium on Computational Geometry (SCG), pp. 267–276 (2010)Google Scholar
  10. 10.
    Janusch, I., Kropatsch, W.G.: Reeb graphs through local binary patterns. In: Liu, C.-L., Luo, B., Kropatsch, W.G., Cheng, J. (eds.) GbRPR 2015. LNCS, vol. 9069, pp. 54–63. Springer, Heidelberg (2015)Google Scholar
  11. 11.
    Kanungo, T., Haralick, R., Baird, H., Stuezle, W., Madigan, D.: A statistical, nonparametric methodology for document degradation model validation. IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1209–1223 (2000)CrossRefGoogle Scholar
  12. 12.
    Karmakar, N., Biswas, A., Bhowmick, P.: Reeb graph based segmentation of articulated components of 3D digital objects. Theoret. Comput. Sci. 624, 25–40 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Leborgne, A., Mille, J., Tougne, L.: Noise-resistant digital euclidean connected skeleton for graph-based shape matching. J. Vis. Commun. Image Represent. 31, 165–176 (2015)CrossRefGoogle Scholar
  14. 14.
    Morse, M.: The Calculus of Variations in the Large, vol. 18. American Mathematical Society Colloquium Publication, New York (1934)zbMATHGoogle Scholar
  15. 15.
    Reeb, G.: Sur les points singuliers d’une forme de Pfaff complétement intégrable ou d’une fonction numérique. Comptes Rendus de l’Académie des Sciences, Paris 222, 847–849 (1946)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Pascucci, V., Scorzelli, G., Bremer, P.T., Mascarenhas, A.: Robust on-line computation of reeb graphs: Simplicity and speed. ACM Trans. Graph. 26(3), 1–9 (2007). Article number 58CrossRefGoogle Scholar
  17. 17.
    Sherbrooke, E., Patrikalakis, N.M., Wolter, F.E.: Differential and topological properties of medial axis transforms. Graph. Models Image Process. 58, 574–592 (1996)CrossRefGoogle Scholar
  18. 18.
    Thome, N., Vacavant, A., Robinault, L., Miguet, S.: A cognitive and video-based approach for multinational license plate recognition. Mach. Vis. Appl. 22(2), 389–407 (2011)CrossRefGoogle Scholar
  19. 19.
    Tierny, J., Vandeborre, J.P., Daoudi, M.: Invariant high level reeb graphs of 3D polygonal meshes. In: Proceedings of IEEE International Symposium on 3D Data Processing, Visualization and Transmission (3DPVT 2006), pp. 105–112 (2006)Google Scholar
  20. 20.
    Tierny, J., Vandeborre, J.P., Daoudi, M.: Partial 3D shape retrieval by reeb pattern unfolding. Comput. Graph. Forum 28(1), 41–55 (2009). WileyCrossRefGoogle Scholar
  21. 21.
    Vacavant, A., Coeurjolly, D., Tougne, L.: A framework for dynamic implicit curve approximation by an irregular discrete approach. Graph. Models 71(3), 113–124 (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Vacavant, A., Roussillon, T., Kerautret, B., Lachaud, J.O.: A combined multi-scale/irregular algorithm for the vectorization of noisy digital contours. Comput. Vis. Image Underst. 117(4), 438–450 (2013)CrossRefGoogle Scholar
  23. 23.
    Werghi, N., Xiao, Y., Siebert, J.: A functional-based segmentation of human body scans in arbitrary postures. IEEE Trans. Syst. Man. Cybern. Part B Cybern. 36(1), 153–165 (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.ISIT, Université d’Auvergne, UMR/CNRS/6284, BP10448Clermont-FerrandFrance
  2. 2.Université de Lyon, INSA-Lyon, LIRIS, UMR/CNRS/5205VilleurbanneFrance

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