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Characterizing 3D Shapes Using Fractal Dimension

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Part of the Lecture Notes in Computer Science book series (LNIP,volume 6419)


Developments in techniques for modeling and digitizing have made the use of 3D models popular to a large number of new applications. With the diffusion and spreading of 3D models employment, the demand for efficient search and retrieval methods is high. Researchers have dedicated effort to investigate and overcome the problem of 3D shape retrieval. In this work, we propose a new way to employ shape complexity analysis methods, such as the fractal dimension, to perform the 3D shape characterization for those purposes. This approach is described and experimental results are performed on a 3D models data set. We also compare the technique to two other known methods for 3D model description, reported in literature, namely shape histograms and shape distributions. The technique presented here has performed considerably better than any of the others in the experiments.


  • Fractal dimension
  • complexity
  • 3D shape descriptor


  1. Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Matching 3D models with shape distributions. In: SMI 2001: Proceedings of the International Conference on Shape Modeling & Applications, Washington, DC, USA, p. 154. IEEE Computer Society, Los Alamitos (2001)

    CrossRef  Google Scholar 

  2. Yang, Y., Lin, H., Zhang, Y.: Content-based 3-D model retrieval: A survey. IEEE Transactions on Systems, Man, and Cybernetics 37(6), 1081–1098 (2007)

    CrossRef  Google Scholar 

  3. Bimbo, A.D., Pala, P.: Content-based retrieval of 3d models. ACM Trans. Multimedia Comput. Commun. Appl. 2(1), 20–43 (2006)

    CrossRef  Google Scholar 

  4. Tangelder, J.W., Veltkamp, R.C.: A survey of content based 3d shape retrieval methods. Multimedia Tools Appl. 39(3), 441–471 (2008)

    CrossRef  Google Scholar 

  5. Tricot, C.: Curves and Fractal Dimension. Springer, Heidelberg (1995)

    CrossRef  MATH  Google Scholar 

  6. Backes, A.R., Casanova, D., Bruno, O.M.: Plant leaf identification based on volumetric fractal dimension. IJPRAI 23(6), 1145–1160 (2009)

    Google Scholar 

  7. da Costa, L.F., Cesar Jr., R.M.: Shape Analysis and Classification: Theory and Practice. CRC Press, Boca Raton (2000)

    CrossRef  MATH  Google Scholar 

  8. Carlin, M.: Measuring the complexity of non-fractal shapes by a fractal method. PRL: Pattern Recognition Letters 21(11), 1013–1017 (2000)

    CrossRef  MATH  Google Scholar 

  9. Chen, X., Golovinskiy, A., Funkhouser, T.: A benchmark for 3D mesh segmentation. ACM Transactions on Graphics (Proc. SIGGRAPH) 28(3) (2009)

    Google Scholar 

  10. Sarkar, N., Chaudhuri, B.B.: An efficient approach to estimate fractal dimension of textural images. Pattern Recognition 25(9), 1035–1041 (1992)

    CrossRef  Google Scholar 

  11. de Plotze, R.O., Falvo, M., Pádua, J.G., Bernacci, L.C., Vieira, M.L.C., Oliveira, G.C.X., Bruno, O.M.: Leaf shape analysis using the multiscale Minkowski fractal dimension, a new morphometric method: a study with passiflora (passifloraceae). Canadian Journal of Botany 83(3), 287–301 (2005)

    CrossRef  Google Scholar 

  12. Bruno, O.M., de Plotze, R.O., Falvo, M., de Castro, M.: Fractal dimension applied to plant identification. Information Sciences 178, 2722–2733 (2008)

    MathSciNet  CrossRef  Google Scholar 

  13. Backes, A.R., de Sa Jr., J.J.M., Kolb, R.M., Bruno, O.M.: Plant species identification using multi-scale fractal dimension applied to images of adaxial surface epidermis. In: Jiang, X., Petkov, N. (eds.) CAIP 2009. LNCS, vol. 5702, pp. 680–688. Springer, Heidelberg (2009)

    Google Scholar 

  14. Emerson, C.W., Lam, N.N., Quattrochi, D.A.: Multi-scale fractal analysis of image texture and patterns. Photogrammetric Engineering and Remote Sensing 65(1), 51–62 (1999)

    Google Scholar 

  15. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 2nd edn. Prentic-Hall, New Jersey (2002)

    Google Scholar 

  16. Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edn., Oxford (1986)

    Google Scholar 

  17. Everitt, B.S., Dunn, G.: Applied Multivariate Analysis, 2nd edn. Arnold, London (2001)

    MATH  Google Scholar 

  18. Fukunaga, K.: Introduction to Statistical Pattern Recognition, 2nd edn. Academic Press, London (1990)

    MATH  Google Scholar 

  19. Ankerst, M., Kastenmüller, G., Kriegel, H.P., Seidl, T.: 3D shape histograms for similarity search and classification in spatial databases. In: Güting, R.H., Papadias, D., Lochovsky, F.H. (eds.) SSD 1999. LNCS, vol. 1651, pp. 207–226. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  20. Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Shape distributions. ACM Transactions on Graphics 21(4), 807–832 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Bruno, O.M., da Fontoura Costa, L.: A parallel implementation of exact Euclidean distance transform based on exact dilations. Microprocessors and Microsystems 28(3), 107–113 (2004)

    CrossRef  Google Scholar 

  22. Fabbri, R., da Fontoura Costa, L., Torelli, J.C., Bruno, O.M.: 2D Euclidean distance transform algorithms: A comparative survey. ACM Computing Surveys 40(1), 1–44 (2008)

    CrossRef  Google Scholar 

  23. Shilane, P., Min, P., Kazhdan, M., Funkhouser, T.: The Princeton shape benchmark. In: SMI 2004: Proceedings of the Shape Modeling International 2004, Washington, DC, USA, pp. 167–178. IEEE Computer Society, Los Alamitos (2004)

    Google Scholar 

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Backes, A.R., Eler, D.M., Minghim, R., Bruno, O.M. (2010). Characterizing 3D Shapes Using Fractal Dimension. In: Bloch, I., Cesar, R.M. (eds) Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications. CIARP 2010. Lecture Notes in Computer Science, vol 6419. Springer, Berlin, Heidelberg.

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  • Print ISBN: 978-3-642-16686-0

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