Mathematical Geosciences

, 43:783 | Cite as

Extrapolating the Fractal Characteristics of an Image Using Scale-Invariant Multiple-Point Statistics

  • Grégoire MariethozEmail author
  • Philippe Renard
  • Julien Straubhaar


The resolution of measurement devices can be insufficient for certain purposes. We propose to stochastically simulate spatial features at scales smaller than the measurement resolution. This is accomplished using multiple-point geostatistical simulation (direct sampling in the present case) to interpolate values at the target scale. These structures are inferred using hypothesis of scale invariance and stationarity on the spatial patterns found at the coarse scale. The proposed multiple-point super-resolution mapping method is able to deal with “both continuous and categorical variables,” and can be extended to multivariate problems. The advantages and limitations of the approach are illustrated with examples from satellite imaging.


Geostatistics Multiple-point Zooming Resolution Scale invariance Fractal 


  1. Arpat B, Caers J (2007) Conditional simulations with patterns. Math Geol 39(2):177–203 CrossRefGoogle Scholar
  2. Atkins C, Bouman C, Allebach J (1999) Tree-based resolution synthesis. In: Image processing, image quality, image capture systems conference (PICS-99). IS&T, Savannah, Georgia Google Scholar
  3. Baker S, Kanade T (2002) Limits on super-resolution and how to break them. IEEE Trans Pattern Anal Mach Intell 24(9):1167–1183 CrossRefGoogle Scholar
  4. Barton C, La Pointe P (1991) Fractals in the earth sciences. Plenum, New York Google Scholar
  5. Bertero M, Boccacci P (2003) Super-resolution in computational imaging. Micron 34(6):265–273 CrossRefGoogle Scholar
  6. Boucher A, Kyriakidis C, Cronkite-Ratcliff C (2008) Geostatistical solutions for super-resolution land cover mapping. IEEE Trans Geosci Remote Sens 46(1):272–283 CrossRefGoogle Scholar
  7. Dai S, Han M, Xu W, Wu Y, Gong Y, Katsaggelos A (2009) SoftCuts: a soft edge smoothness prior for color image super-resolution. IEEE Trans Image Process 18(5):969–981 CrossRefGoogle Scholar
  8. Deutsch C, Journel A (1992) GSLIB: geostatistical software library. Oxford Univ Press, Oxford Google Scholar
  9. El Ouassini A, Saucier A, Marcotte D, Favis B (2008) A patchwork approach to stochastic simulation: a route towards the analysis of morphology in multiphase systems. Chaos Solitons Fractals 36(2008):418–436 CrossRefGoogle Scholar
  10. Farsiu S, Robinson D, Elad M, Milanfar P (2004) Advances and challenges in super-resolution. Int J Imaging Syst Technol 14(2):47–57 CrossRefGoogle Scholar
  11. Fattal R (2007) Image upsampling via imposed edge statistics. ACM Trans Graph 26(3): 95 CrossRefGoogle Scholar
  12. Ferraris L, Gabellani S, Rebora N (2003) A comparison of stochastic models for spatial rainfall downscaling. Water Resour Res 39(12) Google Scholar
  13. Freeman WT, Jones TR, Pasztor EC (2002) Example-based super-resolution. IEEE Comput Graph Appl 22(2):56–65 CrossRefGoogle Scholar
  14. Grimstad A, Mannseth T, Nævdal G, Urkedal H (2003) Adaptive multiscale permeability estimation. Comput Geosci 2003(7):1–25 CrossRefGoogle Scholar
  15. Guardiano F, Srivastava M (1993) Multivariate geostatistics: beyond bivariate moments. In: Geostatistics-Troia, pp 133–144. Kluwer Academic, Amsterdam CrossRefGoogle Scholar
  16. Harter T (2005) Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields. Phys Rev E 72:026120 CrossRefGoogle Scholar
  17. Journel A, Zhang T (2006) The necessity of a multiple-point prior model. Math Geol 38(5):591–610 CrossRefGoogle Scholar
  18. Kerrou J, Renard P, Hendricks-Franssen H-J, Lunati I (2008) Issues in characterizing heterogeneity and connectivity in non-multi-Gaussian media. Adv Water Resour 31(1):147–159 CrossRefGoogle Scholar
  19. Kiraly L (1988) Large scale 3-D groundwater flow modelling in highly heterogeneous geologic medium. In: Groundwater flow and quality modelling, pp 761–775 Google Scholar
  20. Liu X, Coulibaly P, Evora N (2008) Comparison of data-driven methods for downscaling ensemble weather forecasts. Hydrol Earth Syst 12(2):615–624 CrossRefGoogle Scholar
  21. Liu Y, Journel A (2008) A package for geostatistical integration of coarse and fine scale data. Comput Geosci 35(3):527–547 Google Scholar
  22. Lovejoy S, Mandelbrot B (1985) Fractal properties of rain and a fractal model. Tellus A 37:209–232 CrossRefGoogle Scholar
  23. Lovejoy S, Schertzer D (1990) Multifractals, universality classes and satellite and radar measurements of cloud and rain fields. J Geophys Res 95(D3):2021–2034 CrossRefGoogle Scholar
  24. Mandelbrot B (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156(3775):636–638 CrossRefGoogle Scholar
  25. Mandelbrot B (1974) Intermittent turbulence in self-similar cascades: divergence of high moments and dimensions of the carrier. J Fluid Mech 62(2):331–358 CrossRefGoogle Scholar
  26. Mariethoz G (2010) A general parallelization strategy for random path based geostatistical simulation methods. Comput Geosci 37(7):953–958 Google Scholar
  27. Mariethoz G, Renard P (2010) Reconstruction of incomplete data sets or images using direct sampling. Math Geosci 42(3):245–268 CrossRefGoogle Scholar
  28. Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point simulations. Water Resour Res 46:W11536 CrossRefGoogle Scholar
  29. Marsan D, Schertzer D, Lovejoy S (1996) Causal space-time multifractal processes: predictability and forecasting of rain felds. J Geophys Res 101(D21):333–346 CrossRefGoogle Scholar
  30. McCabe M, Wood E (2006) Scale influences on the remote estimation of evapotranspiration using multiple satellite sensors. Remote Sens Environ 105(4):271–285 CrossRefGoogle Scholar
  31. Mosley P (1982) Analysis of the effect of changing discharge on channel morphology and instream uses in a Braided River, Ohau River, New Zealand. Water Resour Res 18(4):800–812 CrossRefGoogle Scholar
  32. Nguyen M, Atkinson P, Lewis H (2006) Superresolution mapping using a Hopfield neural network with fused images. IEEE Trans Geosci Remote Sens 44(3):736–749 CrossRefGoogle Scholar
  33. Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, Cambridge Google Scholar
  34. Renard P, de Marsily G (1997) Calculating equivalent permeability: a review. Adv Water Resour 20(5–6):253–278 CrossRefGoogle Scholar
  35. Ronayne M, Gorelick S (2006) Effective permeability of porous media containing branching channel networks. Phys Rev E 73:026305, 1–10 pp CrossRefGoogle Scholar
  36. Sánchez-Vila X, Carrera J, Girardi JP (1996) Scale effects in transmissivity. J Hydrol 183(1–2):1–22 CrossRefGoogle Scholar
  37. Schulze-Makuch D, Cherkauer D (1998) Variations in hydraulic conductivity with scale of measurement during aquifer tests in heterogeneous, porous carbonate rocks. Hydrogeol J 6:204–215 CrossRefGoogle Scholar
  38. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423 Google Scholar
  39. Shih F (2009) Image processing and mathematical morphology: fundamentals and applications. CRC Press, Boca Raton CrossRefGoogle Scholar
  40. Sroubek F, G C, J F (2007) A unified approach to superresolution and multichannel blind deconvolution. IEEE Trans Image Process 16(9) Google Scholar
  41. Straubhaar J, Walgenwitz A, Renard P, Froidevaux R (2008) Optimization issues in 3D multipoint statistics simulation. In: Geostats 2008, 1–5 Dec 2008, Santiago, Chile Google Scholar
  42. Straubhaar J, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43(3):305–328 CrossRefGoogle Scholar
  43. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–22 CrossRefGoogle Scholar
  44. Tatem A, Lewis H, Atkinson P, Nixon M (2002) Super-resolution land cover pattern prediction using a Hopfield neural network. Remote Sens Environ 79:1–14 CrossRefGoogle Scholar
  45. Tsai RY, Huang TS (1984) Multiframe image restoration and registration. In: Advances in computer vision and image processing. JAI Press, Cambridge, pp 317–339 Google Scholar
  46. Turcotte D (1992) Fractals and chaos in geology and geophysics. Cambridge Press, Cambridge Google Scholar
  47. Von Storch H, Zorita E, Cubasch U (1993) Downscaling of global climate change estimates to regional scales: an application to Iberian rainfall in wintertime. J Climate 6(6):1161–1171 CrossRefGoogle Scholar
  48. Wang J, Gong Y (2008) Fast image super-resolution using connected component enhancement. In: ICME ’08: IEEE international conference on multimedia and expo, June 23–26, Hannover, Germany Google Scholar
  49. Wen X, Gomez-Hernandez J (1998) Numerical modeling of macrodispersion in heterogeneous media: a comparison of multi-Gaussian and non-multi-Gaussian models. J Contam Hydrol 31(1):129–156 CrossRefGoogle Scholar
  50. Wen X, Deutsch C, Cullick A (1997) High resolution reservoir models integrating multiple-well production Data, SPE, SPE 38728 Google Scholar
  51. Wilby R, Wigley T (1997) Downscaling general circulation model output: a review of methods and limitations. Prog Phys Geogr 21(4):530–548 CrossRefGoogle Scholar
  52. Wu J, Boucher A, Zhang T (2008) A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM. Comput Geosci 34(12):1863–1876 CrossRefGoogle Scholar
  53. Zlotnik VA, Zurbuchen BR, Ptak T, Teutsch G (2000) Support volume and scale effect in hydraulic conductivity: experimental aspects. In: Theory, modelling, and field investigations in hydrogeology: A special volume in honor of Shlomo P Neuman’s 60th birthday. Geological Society of America, pp 215–231 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2011

Authors and Affiliations

  • Grégoire Mariethoz
    • 1
    • 2
    Email author
  • Philippe Renard
    • 2
  • Julien Straubhaar
    • 2
  1. 1.National Centre for Groundwater Research and TrainingUniversity of New South WalesSydneyAustralia
  2. 2.Centre for HydrogeologyUniversity of NeuchâtelNeuchâtelSwitzerland

Personalised recommendations