A Critical Appraisal of the Box Counting Method to Assess the Fractal Dimension of Tree Crowns

  • D. Da Silva
  • F. Boudon
  • C. Godin
  • O. Puech
  • C. Smith
  • H. Sinoquet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4291)


In this paper, we study the application of the box counting method (BCM) to estimate the fractal dimension of 3D plant foliage. We use artificial crowns with known theoretical fractal dimension to characterize the accuracy of the BCM and we extend the approach to 3D digitized plants. In particular, errors are experimentally characterized for the estimated values of the fractal dimension. Results show that, with careful protocols, the estimated values are quite accurate. Several limits of the BCM are also analyzed in this context. This analysis is used to introduce a new estimator, derived from the BCM estimator, whose behavior is characterized.


Fractal Dimension Voxel Size Leaf Size Iterate Function System Peach Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Godin, C.: Representing and encoding plant architecture: A review. Annals of Forest Science 57, 413–438 (2000)CrossRefGoogle Scholar
  2. 2.
    Mandelbrot, B.B.: The fractal geometry of nature. Freeman, New York (1983)Google Scholar
  3. 3.
    Smith, A.R.: Plants, fractals, and formal languages. In: Siggraph 1984, Computer Graphics Proceedings, vol. 18, pp. 1–10. ACM Press, New York (1984)Google Scholar
  4. 4.
    Barnsley, M.: Fractals Everywhere. Academic Press, Boston (1988)MATHGoogle Scholar
  5. 5.
    Prusinkiewicz, P., Hanan, J.: Lindenmayer systems, fractals, and plants. Lecture Notes in Biomathematics, vol. 75 (1989)Google Scholar
  6. 6.
    Chen, S., Ceulemans, R., Impens, I.: A fractal-based Populus canopy structure model for the calculation of light interception. Forest Ecology and Management 69(1-3), 97–110 (1994)CrossRefGoogle Scholar
  7. 7.
    Prusinkiewicz, P., Mundermann, L., Karwowski, R., Lane, B.: The use of positional information in the modeling of plants. In: Siggraph 2001, Computer Graphics Proceedings, pp. 289–300. ACM Press, New York (2001)Google Scholar
  8. 8.
    Fitter, A.H.: An architectural approach to the comparative ecology of plant root systems. New Phytologist 106(1), 61–77 (1987)Google Scholar
  9. 9.
    Eshel, A.: On the fractal dimensions of a root system. Plant, Cell & Environment 21(2), 247+ (1998)CrossRefGoogle Scholar
  10. 10.
    Oppelt, A.L., Kurth, W., Dzierzon, H., Jentschke, G., Godbold, D.L.: Structure and fractal dimensions of root systems of four co-occurring fruit tree species from Botswana. Annals of Forest Science 57, 463–475 (2000)CrossRefGoogle Scholar
  11. 11.
    Morse, D.R., Lawton, J.H., Dodson, M.M., Williamson, M.H.: Fractal dimension of vegetation and the distribution of arthropod body lengths. Nature 314(6013), 731–733 (1985)CrossRefGoogle Scholar
  12. 12.
    Critten, D.L.: Fractal dimension relationships and values associated with certain plant canopies. Journal of Agricultural Engineering Research 67(1), 61–72 (1997)CrossRefGoogle Scholar
  13. 13.
    Falconer, K.: Fractal geometry: mathematical foundation and applications. John Wiley and Sons, Chichester (1990)Google Scholar
  14. 14.
    Boudon, F., Pradal, C., Nouguier, C., Godin, C.: Geom module manual: I user guide. Technical Report 3, CIRAD (2001)Google Scholar
  15. 15.
    Sonohat, G., Sinoquet, H., Kulandaivelu, V., Combes, D., Lescourret, F.: Three-dimensional reconstruction of partially 3d-digitized peach tree canopies. Tree Physiol 26(3), 337–351 (2006)CrossRefGoogle Scholar
  16. 16.
    Falconer, K.: Techniques in fractal geometry. John Wiley and Sons, Chichester (1997)MATHGoogle Scholar
  17. 17.
    Plotnick, R.E., Gardner, R.H., O’Neill, R.V.: Lacunarity indices as measures of landscape texture. Landscape Ecology 8, 201–211 (1993)CrossRefGoogle Scholar
  18. 18.
    Andres, E., Nehlig, P., Françon, J.: Supercover of straight lines, planes and triangles. In: Proceedings of DGCI 1997, pp. 243–254. Springer, London (1997)Google Scholar
  19. 19.
    Pfister, H., Zwicker, W., van Baar, J., Gross, M.: Surfels: surface elements as rendering primitives. In: Siggraph 2000, Computer Graphics Proceedings, pp. 335–342. ACM Press, Los angeles (2000)Google Scholar
  20. 20.
    Foroutan-Pour, K., Dutilleul, P., Smith, D.L.: Advances in the implementation of the box-counting method of fractal dimension estimation. Applied Mathematics and Computation 105(2), 195–210 (1999)MATHCrossRefGoogle Scholar
  21. 21.
    Halley, J.M., Hartley, S., Kallimanis, A.S., Kunin, W.E., Lennon, J.J., Sgardelis, S.P.: Uses and abuses of fractal methodology in ecology. Ecology 7, 254–271 (2004)Google Scholar
  22. 22.
    Reeve, R.: A warning about standard errors when estimating the fractal dimension. Comput. Geosci. 18(1), 89–91 (1992)CrossRefGoogle Scholar
  23. 23.
    Boudon, F., Godin, C., Pradal, P., Puech, O., Sinoquet, H.: Estimating the fractal dimension of plants using the two-surface method. an analysis based on 3d-digitized tree foliage. Fractals 14(3) (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • D. Da Silva
    • 1
    • 5
  • F. Boudon
    • 2
    • 5
  • C. Godin
    • 2
    • 3
    • 5
  • O. Puech
    • 4
    • 5
  • C. Smith
    • 4
    • 5
  • H. Sinoquet
    • 6
  1. 1.Université de Montpellier II 
  2. 2.CIRAD 
  3. 3.INRIA 
  4. 4.INRA 
  5. 5.Virtual Plants Team, UMR AMAP TA/40EFrance
  6. 6.INRA-UBP, UMR PIAFClermont-FerrandFrance

Personalised recommendations