Advertisement

Fractals as Pre-Processing Tool for Computational Intelligence Application

  • Ana M. Tarquis
  • Valeriano Mèndez
  • Juan B. Grau
  • Josè M. Antòn
  • Diego Andina
Chapter

Preprocessing is the process of adapting the input of our Computational Intelligence (CI) problem to the CI technique applied. Images are inputs of many problems, and Fractal processing of the images to extract relevant geometry characteristics is a very important tool. This chapter is dedicated to Fractal Preprocessing. In Pedology, fractal models were fitted to match the structure of soils and techniques of multifractal analysis of soil images were developed as is described in a state-of-the-art panorama. A box-counting method and a gliding box method are presented, both obtaining from images sets of dimension parameters, and are evaluated in a discussed case study from images of samples, and the second seems preferable. Finally, a comprehensive list of references is given

pedology soil structure multifractal soil images box-counting method gliding-box method capacity dimension information dimension correlation dimension multiscale heterogeneity fractal models porous media partition function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aharony, A., 1990, Multifractals in physics – successes, dangers and challenges, Physica A. 168:479–489.CrossRefMathSciNetGoogle Scholar
  2. Ahammer, H., De Vaney, T.T.J. and Tritthart, H.A., 2003, How much resolution is enough? Influence of downscaling the pixel resolution of digital images on the generalised dimensions, Physica D. 181 (3–4):147–156.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Allain, C. and Cloitre, M., 1991, Characterizing the lacunarity of random and deterministic fractal sets, Physical Review A. 44:3552–3558.CrossRefMathSciNetGoogle Scholar
  4. Anderson, A.N., McBratney, A.B. and FitzPatrick, E.A., 1996, Soil Mass, Surface, and Spectral Fractal Dimensions Estimated from Thin Section Photographs, Soil Sci. Soc. Am. J. 60:962–969.CrossRefGoogle Scholar
  5. Anderson, A.N., McBratney, A.B. and Crawford, J.W., Applications of fractals to soil studies. Adv. Agron., 63:1, 1998.CrossRefGoogle Scholar
  6. Barnsley, M.F., Devaney, R.L., Mandelbrot, B.B., Peitgen, H.O., Saupe, D. and Voss, R.F., 1988, The Science of Fractal Images. Edited by H.O. Peitgen and D. Saupe, Springer-Verlag, New York.Google Scholar
  7. Bartoli, F., Philippy, R., Doirisse, S., Niquet, S. and Dubuit, M., 1991, Structure and self-similarity in silty and sandy soils; the fractal approach, J. Soil Sci. 42:167–185.CrossRefGoogle Scholar
  8. Bartoli, F., Bird, N.R., Gomendy, V., Vivier, H. and Niquet, S., 1999, The relation between silty soil structures and their mercury porosimetry curve counterparts: fractals and percolation, Eur. J. Soil Sci., 50(9).Google Scholar
  9. Bartoli, F., Dutartre, P., Gomendy, V., Niquet, S. and Vivier, H., 1998. Fractal and soil structures. In: Fractals in Soil Science, Baveye, Parlange and Stewart, Eds., CRC Press, Boca Raton, 203–232.Google Scholar
  10. Baveye, P. and Boast, C.W. Fractal Geometry, Fragmentation Processes and the Physics of Scale-Invariance: An Introduction. In Fractals in Soil Science, Baveye, Parlange and Stewart, Eds., CRC Press, Boca Raton, 1998, 1.Google Scholar
  11. Baveye, P., Boast, C.W., Ogawa, S., Parlange, J.Y. and Steenhuis, T., 1998. Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils, Water Resour. Res. 34, 2783–2796.Google Scholar
  12. Bird, N., Dìaz, M.C., Saa, A. and Tarquis, A.M., 2006. Fractal and Multifractal Analysis of Pore-Scale Images of Soil. J. Hydrol, 322, 211–219.CrossRefGoogle Scholar
  13. Bird, N.R.A., Perrier, E. and Rieu, M., 2000. The water retention function for a model of soil structure with pore and solid fractal distributions. Eur. J. Soil Sci. 51, 55–63.CrossRefGoogle Scholar
  14. Bird, N.R.A. and Perrier, E.M.A., 2003. The pore-solid fractal model of soil density scaling. Eur. J. Soil Sci. 54, 467–476.CrossRefGoogle Scholar
  15. Booltink, H.W.G., Hatano, R. and Bouma, J., 1993. Measurement and simulation of bypass flow in a structured clay soil; a physico-morphological approach. J. Hydrol. 148, 149–168.CrossRefGoogle Scholar
  16. Brakensiek, D.L., W.J. Rawls, S.D. Logsdon and Edwards, W.M., 1992. Fractal description of macroporosity. Soil Sci. Soc. Am. J. 56, 1721–1723.CrossRefGoogle Scholar
  17. Buczhowski, S., Hildgen, P. and Cartilier, L. 1998. Measurements of fractal dimension by box-counting: a critical analysis of data scatter. Physica A 252, 23–34.CrossRefGoogle Scholar
  18. Cheng, Q. and Agerberg, F.P. (1996). Comparison between two types of multifractal modeling. Mathematical Geology, 28(8), 1001–1015.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Cheng, Q. (1997a). Discrete multifractals. Mathematical Geology, 29(2), 245–266.Google Scholar
  20. Cheng, Q. (1997b). Multifractal modeling and lacunarity analysis. Mathematical Geology, 29(7), 919–932.CrossRefGoogle Scholar
  21. Crawford, J.W., Baveye, P., Grindrod, P. and Rappoldt, C. Application of Fractals to Soil Properties, Landscape Patterns, and Solute Transport in Porous Media, in Assessment of Non-Point Source Pollution in the Vadose Zone. Geophysical Monograph 108, Corwin, Loague and Ellsworth, Eds., American Geophysical Union, Wahington, DC, 1999, 151.Google Scholar
  22. Crawford, J.W., Ritz, K. and Young, I.M. Quantification of fungal morphology, gaseous transport and microbial dynamics in soil: an integrated framework utilising fractal geometry. Geoderma, 56, 1578, 1993.CrossRefGoogle Scholar
  23. Crawford, J.W., Matsui, N. and Young, I.M. 1995., The relation between the moisture-release curve and the structure of soil. Eur. J. Soil Sci. 46, 369–375.CrossRefGoogle Scholar
  24. Dathe, A., Eins, S., Niemeyer, J. and Gerold, G. The surface fractal dimension of the soil-pore interface as measured by image analysis. Geoderma, 103, 203, 2001.CrossRefGoogle Scholar
  25. Dathe, A., Tarquis, A.M. and Perrier, E., 2006. Multifractal analysis of the pore- and solid-phases in binary two-dimensional images of natural porous structures. Geoderma, doi:10.1016/j.geoderma.2006.03.024, in press.Google Scholar
  26. Dathe, A. and Thullner, M., 2005. The relationship between fractal properties of solid matrix and pore space in porous media. Geoderma, 129, 279–290.CrossRefGoogle Scholar
  27. Feder, J., 1989. Fractals. Plenum Press, New York. 283ppzbMATHGoogle Scholar
  28. Flury, M. and Fluhler, H., 1994. Brilliant blue FCF as a dye tracer for solute transport studies – A toxicological overview. J.Environ. Qual. 23, 1108–1112.CrossRefGoogle Scholar
  29. Flury, M. and Fluhler, H., 1995. Tracer characteristics of brilliant blue. Soil Sci. Soc. Am. J. 59, 22–27.CrossRefGoogle Scholar
  30. Flury, M., Fluhler, H., Jury, W.A. and Leuenberger, J., 1994. Susceptibility of soils to preferential flow of water: A field study, Water Resour. Res. 30, 1945–1954.Google Scholar
  31. Gimènez, D., R.R. Allmaras, E.A. Nater and Huggins, D.R., 1997a. Fractal dimensions for volume and surface of interaggregate pores – scale effects. Geoderma 77, 19–38.CrossRefGoogle Scholar
  32. Gimènez D., Perfect E., Rawls W.J. and Pachepsky, Y., 1997b. Fractal models for predicting soil hydraulic properties: a review. Eng. Geol. 48, 161–183.CrossRefGoogle Scholar
  33. Gouyet, J.G. Physics and Fractal Structures. Masson, Paris, 1996.Google Scholar
  34. Grau, J., Mèndez, V., Tarquis, A.M., Dìaz, M.C. and A. Saa, 2006. Comparison of gliding box and box-counting methods in soil image analysis. Geoderma, doi:10.1016/j.geoderma.2006.03.009, in press.Google Scholar
  35. Griffith, D.A.. Advanced Spatial Statistics. Kluwer Academic Publishers, Boston, 1988.Google Scholar
  36. Hallett, P.D., Bird, N.R.A., Dexter, A.R. and Seville, P.K., 1998. Investigation into the fractal scaling of the structure and strength of soil aggregates. Eur. J. Soil Sci. 49, 203–211.Google Scholar
  37. Hatano, R. and Booltink, H.W.G., 1992. Using Fractal Dimensions of Stained Flow Patterns in a Clay Soil to Predict Bypass Flow. J. Hydrol. 135, 121–131.CrossRefGoogle Scholar
  38. Hatano, R., Kawamura, N., Ikeda, J. and Sakuma, T. Evaluation of the effect of morphological features of flow paths on solute transport by using fractal dimensions of methylene blue staining patterns. Geoderma 53, 31, 1992.CrossRefGoogle Scholar
  39. Hentschel, H.G.R. and Procaccia, I. (1983). The infinite number of generalized dimensions of fractals and strange attractors. Physica D, 8, 435, 1983.Google Scholar
  40. Kaye, B.G. A Random Walk through Fractal Dimensions. VCH Verlagsgesellschaft, Weinheim, Germany, 1989, 297.Google Scholar
  41. Mandelbrot, B.B. The Fractal Geometry of Nature. W.H. Freeman, San Francisco, CA, 1982.zbMATHGoogle Scholar
  42. McCauley, J.L. 1992. Models of permeability and conductivity of porous media. Physica A 187, 18–54.CrossRefGoogle Scholar
  43. Moran, C.J., McBratney, A.B. and Koppi, A.J.,1989. A rapid method for analysis of soil macropore structure. I. Specimen preparation and digital binary production. Soil Sci. Soc. Am. J. 53, 921–928.CrossRefGoogle Scholar
  44. Muller, J., 1996. Characterization of pore space in chalk by multifractal analysis. J. Hydrology, 187, 215–222.CrossRefGoogle Scholar
  45. Muller, J., Huseby, O.K. and Saucier, A. Influence of Multifractal Scaling of Pore Geometry on Permeabilities of Sedimentary Rocks. Chaos, Solitons & Fractals, 5, 1485, 1995.CrossRefGoogle Scholar
  46. Muller, J. and McCauley, J.L., 1992. Implication of Fractal Geometry for Fluid Flow Properties of Sedimentary Rocks. Transp. Porous Media 8, 133–147.CrossRefGoogle Scholar
  47. Muller, J., Huseby, O.K. and Saucier, A., 1995. Influence of Multifractal Scaling of Pore Geometry on Permeabilities of Sedimentary Rocks. Chaos, Solitons & Fractals 5, 1485–1492.CrossRefGoogle Scholar
  48. Ogawa, S., Baveye, P., Boast, C.W., Parlange, J.Y. and Steenhuis, T. Surface fractal characteristics of preferential flow patterns in field soils: evaluation and effect of image processing. Geoderma, 88, 109, 1999.CrossRefGoogle Scholar
  49. Oleschko, K., Fuentes, C., Brambila, F. and Alvarez, R. Linear fractal analysis of three Mexican soils in different management systems. Soil Technol., 10, 185, 1997.CrossRefGoogle Scholar
  50. Oleschko, K. Delesse principle and statistical fractal sets: 1. Dimensional equivalents. Soil&Tillage Research, 49, 255, 1998a.Google Scholar
  51. Oleschko, K., Brambila, F., Aceff, F. and Mora, L.P. From fractal analysis along a line to fractals on the plane. Soil&Tillage Research, 45, 389, 1998b.Google Scholar
  52. Orbach, R. Dynamics of fractal networks. Science (Washington, DC) 231, 814, 1986.Google Scholar
  53. Pachepsky, Y.A.,Yakovchenko, V., Rabenhorst, M.C., Pooley, C. and Sikora, L.J. . Fractal parameters of pore surfaces as derived from micromorphological data: effect of long term management practices. Geoderma, 74, 305, 1996.CrossRefGoogle Scholar
  54. Pachepsky, Y.A., Gimènez, D., Crawford, J.W. and Rawls, W.J. Conventional and fractal geometry in soil science. In Fractals in Soil Science, Pachepsky, Crawford and Rawls, Eds., Elsevier Science, Amsterdam, 2000, 7.Google Scholar
  55. Persson, M., Yasuda, H., Albergel, J., Berndtsson, R., Zante, P., Nasri, S. and Öhrström, P., 2001. Modeling plot scale dye penetration by a diffusion limited aggregation (DLA) model. J. Hydrol. 250, 98–105.CrossRefGoogle Scholar
  56. Peyton, R.L., Gantzer, C.J., Anderson, S.H., Haeffner, B.A. and Pfeifer, P. . Fractal dimension to describe soil macropore structure using X ray computed tomography. Water Resource Research, 30, 691, 1994.CrossRefGoogle Scholar
  57. Posadas, A.N.D., Gimènez, D., Quiroz, R. and Protz, R., 2003. Multifractal Characterization of Soil Pore Spatial Distributions. Soil Sci. Soc. Am. J. 67, 1361–1369CrossRefGoogle Scholar
  58. Protz , R. and VandenBygaart, A.J. 1998. Towards systematic iage analysis in the study of soil micromorphology. Science Soils, 3. (available online at http://link.springer.de/link/service/journals/).Google Scholar
  59. Ripley, B.D. Statistical Inference for Spatial Processes, Cambridge Univ. Press, Cambridge, 1988.Google Scholar
  60. Saucier, A. Effective permeability of multifractal porous media. Physica A, 183, 381, 1992.CrossRefGoogle Scholar
  61. Saucier, A. and Muller, J. Remarks on some properties of multifractals. Physica A, 199, 350, 1993.CrossRefGoogle Scholar
  62. Saucier, A. and Muller, J. Textural analysis of disordered materials with multifractals. Physica A, 267, 221, 1999.CrossRefGoogle Scholar
  63. Saucier, A., Richer, J. and Muller, J., 2002. Statistical mechanics and its applications. Physica A, 311 (1–2): 231–259.zbMATHCrossRefGoogle Scholar
  64. Takayasu, H. Fractals in the Physical Sciences. Manchester University Press, Manchester, 1990.zbMATHGoogle Scholar
  65. Tarquis, A.M., Gimènez, D., Saa, A., Dìaz, M.C. and Gascò, J.M., 2003. Scaling and Multiscaling of Soil Pore Systems Determined by Image Analysis. In: Scaling Methods in Soil Physics, Pachepsky, Radcliffe and Selim Eds., CRC Press, 434 pp.Google Scholar
  66. Tarquis, A.M., McInnes, K.J., Keys, J., Saa, A., Garcìa, M.R. and Dìaz, M.C., 2006. Multiscaling Analysis In A Structured Clay Soil Using 2D Images. J. Hydrol, 322, 236–246.CrossRefGoogle Scholar
  67. Tel, T. and Vicsek, T., 1987. Geometrical multifractality of growing structures, J. Physics A. General, 20, L835–L840.CrossRefMathSciNetGoogle Scholar
  68. VandenBygaart, A.J. and Protz, R., 1999. The representative elementary area (REA) in studies of quantitative soil micromorphology. Geoderma 89, 333–346.CrossRefGoogle Scholar
  69. Vicsek, T. 1990. Mass multifractals. Physica A, 168, 490–497.CrossRefMathSciNetGoogle Scholar
  70. Vogel, H.J. and Kretzschmar, A., 1996. Topological characterization of pore space in soil-sample preparation and digital image-processing. Geoderma 73, 23–38.CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Ana M. Tarquis
    • 1
  • Valeriano Mèndez
    • 1
  • Juan B. Grau
    • 1
  • Josè M. Antòn
    • 1
  • Diego Andina
    • 1
    • 2
  1. 1.Dpto. de Matemàtica Aplicada, E.T.S. de Ingenieros AgrònomosU.P.M., Av. Complutense s.n., Ciudad UniversitariaMadrid 28040Spain
  2. 2.Dpto. De Señales, Sistemas y Radiocomunicaiones, E.T.S. Ingenieros de TelecomunicaciònU.P.M., Av. Complutense s.n., Ciudad UniversitariaMadrid 28040Spain

Personalised recommendations