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A Pruning Algorithm for Stable Voronoi Skeletons

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Abstract

We present a skeleton computation algorithm for binary image shape which is stable and efficient. The algorithm follows these steps: first the shape boundary curves are subsampled, then the Voronoi Skeleton is computed from the resulting reduced boundary set of points, and finally, a novel two stage pruning procedure is applied to obtain a simplified skeleton. The first stage removes skeleton edges non fully included in the shape. The second stage applies an enhanced variation of the Discrete Curve Evolution (DCE) for Voronoi skeletons. We obtain improved skeleton stability, complexity reduction and noise robustness. Pruning computing time efficiency is improved thanks to some properties of Voronoi skeletons. Entire skeleton edges can be removed or retained on the basis of conditions tested on the edge endpoints. Pattern recognition experiments and skeleton stability experiments of the algorithm outperform previous approaches in the literature.

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References

  1. Gonzalez, R., Woods, R.: Digital Image Processing, p. 543. Addison-Wesley/Longman, Reading/Harlow (2001), Chap. 9.5.7

    Google Scholar 

  2. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, San Diego (1982)

    MATH  Google Scholar 

  3. Rock, I., Linnett, C.M.: Is a perceived shape based on its retinal image? Perception 22(1), 61–76 (1993)

    Article  Google Scholar 

  4. Hoffman, D., Richards, W.A.: Parts of recognition. Cognition 18, 65–96 (1984)

    Article  Google Scholar 

  5. Blum, H.: Biological shape and visual science. Theor. Biol. 38, 205–287 (1973)

    Article  Google Scholar 

  6. He, L., Han, C., Wee, W.: Object recognition and recovery by skeleton graph matching. In: IEEE International Conference on Multimedia and Expo, July 2006, pp. 993–996 (2006)

    Chapter  Google Scholar 

  7. Eede, M.v., Macrini, D., Telea, A., Sminchisescu, C., Dickinson, S.S.: Canonical skeletons for shape matching. In: ICPR’06: Proceedings of the 18th International Conference on Pattern Recognition, pp. 64–69. IEEE Computer Society, Los Alamitos (2006)

    Google Scholar 

  8. Goh, W.-B.: Strategies for shape matching using skeletons. Comput. Vis. Image Underst. 110(3), 326–345 (2008)

    Article  MathSciNet  Google Scholar 

  9. Bai, X., Latecki, L.J., Liu, W.Y.: Skeleton pruning by contour partitioning with discrete curve evolution. IEEE Trans. Pattern Anal. Mach. Intell. 29(3), 449–462 (2007)

    Article  Google Scholar 

  10. Beristain, A., Graña, M.: A pruning algorithm for Voronoi skeletons. Electron. Lett. 46(1), 39–41 (2010)

    Article  Google Scholar 

  11. Biasotti, S., Attali, D., Boissonnat, J., Edelsbrunner, H., Elber, G., Mortara, M., Sanniti di Baja, G., Spagnuolo, M., Tanase, M., Veltcamp, R.: Skeletal structures. In: Shape Analysis and Structuring, pp. 172–175. Springer, Berlin (2008)

    Google Scholar 

  12. Reeb, G.: Sur les points singuliers d’une forme de pfaff completement integrable ou dune fonction numerique. C. R. Acad. Sci. 222, 847–849 (1946)

    MathSciNet  MATH  Google Scholar 

  13. Shinagawa, Y., Kunii, T.L., Kergosien, Y.L.: Surface coding based on Morse theory. IEEE Comput. Graph. Appl. 11(5), 66–78 (1991)

    Article  Google Scholar 

  14. Blum, H.: A transformation for extracting new descriptors of shape. In: Dunn, W.W. (ed.) Models for the Perception of Speech and Visual Form, pp. 362–380. MIT Press, Cambridge (1967)

    Google Scholar 

  15. Bruce, J., Giblin, P.J., Gibson, C.G.: Symmetry sets. Proc. R. Soc. Edinb. 104(A), 179–204 (1985)

    Google Scholar 

  16. Jain, A.K.: Fundamentals of Digital Image Processing, p. 387. Prentice-Hall, New York (1989), Chap. 9.9

    MATH  Google Scholar 

  17. Mather, J.: Distance from a submanifold in euclidean space. Proc. Symp. Pure Math., Ser. 2 40(2), 199–216 (1983)

    MathSciNet  Google Scholar 

  18. Giblin, P.J., Kimia, B.B.: On the local form and transitions of symmetry sets, medial axes, and shocks. Int. J. Comput. Vis. 54(1–3), 143–156 (2003)

    Article  MATH  Google Scholar 

  19. Giblin, P., Kimia, B.B.: A formal classification of 3d medial axis points and their local geometry. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 238–251 (2004)

    Article  Google Scholar 

  20. Kimia, B.B., Tannenbaum, A.R., Zucker, S.W.: Shapes, shocks, and deformations i: the components of two-dimensional shape and the reaction-diffusion space. Int. J. Comput. Vis. 15(3), 189–224 (1995)

    Article  Google Scholar 

  21. Bai, X., Latecki, L.: Discrete skeleton evolution. In: Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 4679, pp. 362–374. Springer, Berlin (2007)

    Chapter  Google Scholar 

  22. Hesselink, W., Roerdink, J.: Euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform. IEEE Trans. Pattern Anal. Mach. Intell. 30(12), 2204–2217 (2008)

    Article  Google Scholar 

  23. Lam, L., Lee, S., Suen, C.: Thinning methodologies: A comprehensive survey. IEEE Trans. Pattern Anal. Mach. Intell. 14(9), 869–885 (1992)

    Article  Google Scholar 

  24. Siddiqi, K., Pizer, S.: Medial Representations: Mathematics, Algorithms and Applications. Springer, Berlin (2008)

    MATH  Google Scholar 

  25. Torsello, A.: Matching hierarchical structures for shape recognition. Ph.D. dissertation, University of York, July 2004

  26. Giardina, C., Dougherty, E.: Morphological Methods in Image and Signal Processing. Prentice Hall, New York (1988)

    Google Scholar 

  27. Jang, B., Chin, R.: Analysis of thinning algorithms using mathematical morphology. IEEE Trans. Pattern Anal. Mach. Intell. 12(6), 541–551 (1990)

    Article  Google Scholar 

  28. Leymarie, F., Levine, M.: Simulating the grassfire transform using an active contour model. IEEE Trans. Pattern Anal. Mach. Intell. 14(1), 56–75 (1992)

    Article  Google Scholar 

  29. Tek, H., Kimia, B.: Symmetry maps of free-form curve segments via wave propagation. In: The Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999, vol. 1, pp. 362–369 (1999)

    Google Scholar 

  30. Tari, Z.S.G., Shah, J., Pien, H.: Extraction of shape skeletons from grayscale images. Comput. Vis. Image Underst. 66(2), 133–146 (1997)

    Article  Google Scholar 

  31. Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. J. ACM 13(4), 471–494 (1966)

    Article  MATH  Google Scholar 

  32. Rosenfeld, A., Pfaltz, J.: Distance functions on digital pictures. Pattern Recognit. 1(1), 33–61 (1968)

    Article  MathSciNet  Google Scholar 

  33. Ogniewicz, R., Ilg, M.: Voronoi skeletons: theory and applications. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Proceedings CVPR’92, pp. 63–69 (1992)

    Chapter  Google Scholar 

  34. Brandt, J.W., Algazi, V.R.: Continuous skeleton computation by Voronoi diagram. CVGIP, Image Underst. 55(3), 329–338 (1992)

    Article  MATH  Google Scholar 

  35. Choi, S.W., Seidel, H.-P.: Linear onesided stability of mat for weakly injective 3d domain. In: SMA’02: Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications, pp. 344–355. ACM, New York (2002)

    Chapter  Google Scholar 

  36. Attali, D., Lachaud, J.O.: Delaunay conforming iso-surface, skeleton extraction and noise removal. Comput. Geom., Theory Appl. 19, 175–189 (2001)

    MathSciNet  MATH  Google Scholar 

  37. Dey, T.K., Zhao, W.: Approximate medial axis as a Voronoi subcomplex. In: SMA’02: Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications, pp. 356–366. ACM, New York (2002)

    Chapter  Google Scholar 

  38. Foskey, M., Lin, M.C., Manocha, D.: Efficient computation of a simplified medial axis. In: SM’03: Proceedings of the Eighth ACM Symposium on Solid Modeling and Applications, pp. 96–107. ACM, New York (2003)

    Chapter  Google Scholar 

  39. Mokhtarian, F., Mackworth, A.K.: A theory of multiscale, curvature-based shape representation for planar curves. IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 789–805 (1992)

    Article  Google Scholar 

  40. Shaked, D., Bruckstein, A.: Pruning medial axes. Comput. Vis. Image Underst. 69(2), 156–169 (1998)

    Article  Google Scholar 

  41. Wets, R.J.-B., Tyrrell Rockafellar, R., Wets, M.: Set convergence. In: Variational Analysis, p. 117. Springer, Berlin (2009)

    Google Scholar 

  42. Brandt, J.W.: Convergence and continuity criteria for discrete approximations of the continuous planar skeleton. CVGIP, Image Underst. 59(1), 116–124 (1994)

    Article  Google Scholar 

  43. Schmitt, M.: Some examples of algorithms analysis in computational geometry by means of mathematical morphology techniques. In: Geometry and Robotics, vol. 391, pp. 225–246. Springer, Berlin (1989)

    Google Scholar 

  44. Aronov, B.: A lower bound on Voronoi diagram complexity. Inf. Process. Lett. 83(4), 183–185 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Attali, D.: r-regular shape reconstruction from unorganized points. In: SCG’97: Proceedings of the Thirteenth Annual Symposium on Computational Geometry, pp. 248–253. ACM, New York (1997)

    Chapter  Google Scholar 

  46. Chazal, F., Lieutier, A.: The “$lambda$-medial axis”. Graph. Models 67(4), 304–331 (2005)

    Article  MATH  Google Scholar 

  47. Latecki, L.J., Lakämper, R.: Convexity rule for shape decomposition based on discrete contour evolution. Comput. Vis. Image Underst. 73(3), 441–454 (1999)

    Article  Google Scholar 

  48. Latecki, L.J., Lakämper, R.: Shape similarity measure based on correspondence of visual parts. IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1185–1190 (2000)

    Article  Google Scholar 

  49. Latecki, L.J., Lakämper, R.: Application of planar shape comparison to object retrieval in image databases. Pattern Recognit. 35, 15–29 (2002)

    Article  MATH  Google Scholar 

  50. Available: http://www.cis.temple.edu/~latecki/Programs/skeletonPruning07.htm

  51. Jeannin, S., Bober, M.: Description of core experiments for MPEG-7 motion/shape, MPEG-7 Std. (1999)

  52. Beristain, A.: Conjunto de datos de test mpeg-7 core experiment ce-shape-1 (2009). Available: http://www.ehu.es/ccwintco/index.php/GIC-experimental-databases

  53. Beristain, A.: Colección de imágenes para entrenamiento de un reconocedor de gestos de interacción con un tabletop (2009). Available: http://www.ehu.es/ccwintco/index.php/Synthetic_hand_gesture_images_for_tabletop_interaction

  54. Specht, D.F.: Probabilistic neural networks. Neural Netw. 3(1), 109–118 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. Facultad de Informática de San Sebastian, Computational Intelligence Group, University of the Basque Country, Manuel Lardizabal 1, 20018, Donostia San Sebastian, Gipuzkoa, Spain

    Andoni Beristain, Manuel Graña & Ana I. Gonzalez

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  1. Andoni Beristain
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  2. Manuel Graña
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  3. Ana I. Gonzalez
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Correspondence to Andoni Beristain.

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Beristain, A., Graña, M. & Gonzalez, A.I. A Pruning Algorithm for Stable Voronoi Skeletons. J Math Imaging Vis 42, 225–237 (2012). https://doi.org/10.1007/s10851-011-0291-1

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