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Multifractal Analysis in B&W Soil Images

Multifractal Analysis in B&W Soil Images. Ana M. Tarquis anamaria.tarquis@upm.es Dpto. de Matemática Aplicada E.T.S.I. Agrónomos Universidad Politécnica de Madrid. INDEX. Problem: motivation and start point. Fractals and multifractals concepts. Porosity images: resolved?

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Multifractal Analysis in B&W Soil Images

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  1. Multifractal Analysis in B&W Soil Images Ana M. Tarquis anamaria.tarquis@upm.es Dpto. de Matemática Aplicada E.T.S.I. Agrónomos Universidad Politécnica de Madrid International Summer School on Turbulence Diffusion 2006

  2. INDEX • Problem: motivation and start point. • Fractals and multifractals concepts. • Porosity images: resolved? • Configuration Entropy • Griding Methods International Summer School on Turbulence Diffusion 2006

  3. CONSERVATION OF NATURAL RESOURCES • Agriculture : soil degradation and water contamination. • Sustainable agriculture • Quantification of soil quality index? International Summer School on Turbulence Diffusion 2006

  4. Soil structure • Water, solutes and gas transport • Soil resistance • Roots morphology • Microorganism populations PORE AND SOIL MATRIX GEOMETRY International Summer School on Turbulence Diffusion 2006

  5. Fractal structure: structured distribution of pore (and/or soil) in the space such that at any resolution the set is the union of similar subset to the whole. International Summer School on Turbulence Diffusion 2006

  6. Measure techniques • The number-size relation is used normally to measure the fractal dimension of the defined measure (number of white or black pixels), or counting objects: • Or covering the object with regular geometric elements of variable size: International Summer School on Turbulence Diffusion 2006

  7. 1 n “ Box-Counting” -m = fractal dimension, D Black, white or interface International Summer School on Turbulence Diffusion 2006

  8. Multifractal analysis consider the number of black pixels in each box (pore density=m). International Summer School on Turbulence Diffusion 2006

  9. Multifractal: density has an structured distribution in the space such that at any resolution the set is the union of similar subsets to the whole. But the scale factor at different parts of the set is not the same. • More than one dimension is needed => the measure consider (M) is characterized by the union of fractal sets, each one with a fractal dimension. International Summer School on Turbulence Diffusion 2006

  10. 1 n Dq q International Summer School on Turbulence Diffusion 2006

  11. Numerical Analysis of Multifractal Spectrum on 2-D Black and White Images International Summer School on Turbulence Diffusion 2006

  12. RANDOM AND MULTIFRACTAL IMAGES • In this way a hierarchical probability tree was built generating an image of 1024x1024 pixels (ten subdivisions), as the soil images are normally analyzed. • Probabilistic parameters are: { p1, p2, p3, p4 } • Random images : p1= p2 = p3 = p4 = 25% • Multifractal images: p1= 50%, p2= 5%, p3= 25% and p4= 20% (by random arrangements or not). International Summer School on Turbulence Diffusion 2006

  13. Random multifractal International Summer School on Turbulence Diffusion 2006

  14. Generalized dimensions (Dq) obtained for two different distributions based on Stanley and Meakin (1988) formulas with their respective -t(q) curves. International Summer School on Turbulence Diffusion 2006

  15. Most common parameters calculated • D0 q=0  box counting dimension • D1 q=1  entropy dimension • D2 q=2  correlation dimension International Summer School on Turbulence Diffusion 2006

  16. f()  Singularities of the measure (a) For a given a there is a fractal dimension f(a) of the set that support the singularity. At each area the relation number-size is applied: International Summer School on Turbulence Diffusion 2006

  17. f()  Multifractal Spectrum wf wa International Summer School on Turbulence Diffusion 2006

  18. International Summer School on Turbulence Diffusion 2006

  19. INTER DENNY 1500x1000 pixels ABOK MUNCHONG International Summer School on Turbulence Diffusion 2006

  20. ¿How many points? International Summer School on Turbulence Diffusion 2006

  21. ADS BUSO EHV1 International Summer School on Turbulence Diffusion 2006

  22. We have to compare International Summer School on Turbulence Diffusion 2006

  23. International Summer School on Turbulence Diffusion 2006

  24. Obtaining Dq Ehv1, porosity 46,7% International Summer School on Turbulence Diffusion 2006

  25. Calculating Dq ADS, porosity 5,7% International Summer School on Turbulence Diffusion 2006

  26. International Summer School on Turbulence Diffusion 2006

  27. Continuos line = random structure Dashed line = mfract structure Filled Square = values from image soils International Summer School on Turbulence Diffusion 2006

  28. Considerations on Dq calculations • Several authors have shown that the exact value of the generalized dimension is not an easy calculation to do . Vicsek proposed practical methods to compute the generalized dimension • The main difficulty in using the multifractal formalism lies in the fact that the ideal limit cannot be reached in practice International Summer School on Turbulence Diffusion 2006

  29. RESULTS AND DISCUSSION (1) • For all of the soil images with different porosity we obtain convincing straight-line fits to the data having all of them r2 higher than 0.98, International Summer School on Turbulence Diffusion 2006

  30. RESULTS AND DISCUSSION • Finally, a comparison among the different images in each dimension is showed . • In all of them, the points corresponding to porosities higher than 30% lie on the line representing the Dq calculated for the random generated images. • Observing the difference between the fractal dimensions coming from multifractal and random images (discontinue line and continue line respectively) it is obvious that decreases when porosity increases in the images. International Summer School on Turbulence Diffusion 2006

  31. i  w Configuration Entropy H(d) 1 n(d) = boxes of size d from d = 1 to d = w /4 Nj = number of boxes withj black pixels inside The maximum value of j isdxdand the minimum value is 0 (Andraud et al., 1989) International Summer School on Turbulence Diffusion 2006

  32. Configuration Entropy H(d) The probability associated with a case of j black pixels in a box of size d (pj(d)) International Summer School on Turbulence Diffusion 2006

  33. Configuration Entropy H(d) 1 H*(L) H*() 0 1 w/4  (pixels) L International Summer School on Turbulence Diffusion 2006

  34. Methods: gliding, random walks, randomly Box size Jump step length Number of jumps International Summer School on Turbulence Diffusion 2006

  35. Thank you for your attention International Summer School on Turbulence Diffusion 2006

  36. Multifractal Analysis on a Matrix Ana M. Tarquis anamaria.tarquis@upm.es Dpto. de Matemática Aplicada E.T.S.I. Agrónomos Universidad Politécnica de Madrid International Summer School on Turbulence Diffusion 2006

  37. INDEX • Field Percolation • Soil Roughness • Satellite images • Time series International Summer School on Turbulence Diffusion 2006

  38. Z= 10 cm Z = 20 cm Z = 30 cm Z = 40 cm Z = 50 cm Z = 60 cm International Summer School on Turbulence Diffusion 2006

  39. 50% 15 cm % of blue vs. depth International Summer School on Turbulence Diffusion 2006

  40. Z = 25 cm blue staining 28,95% International Summer School on Turbulence Diffusion 2006

  41. Dye Tracer Distribution International Summer School on Turbulence Diffusion 2006

  42. Multifractal Analysis of the Dye Tracer Distribution B) Generalized dimensions A) f(a) spectrum International Summer School on Turbulence Diffusion 2006

  43. Multispectral Satellite Images International Summer School on Turbulence Diffusion 2006

  44. International Summer School on Turbulence Diffusion 2006

  45. Soil Rougness • Roughness indices normally are based on transects data. One of the most used is the Random Roughness (RR). • RR is the standard deviation of the soil heights readings from the transect. This implies that there is not an spatial component. • Several authors have applied fractal dimensions to this type of data. Burrough (1989), Bertuzzi et al. (1990), Huang and Bradford (1992), International Summer School on Turbulence Diffusion 2006

  46. INTRODUCTION • The aim of this work is to study soil height readings with multifractal analysis in the context of soil roughness. • Several soils, with different textures, with different tillage methods have been analysed to compare their multifractal spectrum. International Summer School on Turbulence Diffusion 2006

  47. Soil measurements • Three different soils with different textures. • Three different treatments applying tillage: chisel, moldboard, seedbeds. • Height measures of 2x2 m2 plot area. • Resolution of the measure each 2 cm International Summer School on Turbulence Diffusion 2006

  48. International Summer School on Turbulence Diffusion 2006

  49. Soil texture International Summer School on Turbulence Diffusion 2006

  50. International Summer School on Turbulence Diffusion 2006

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