Introduction to Random Fields and Scale Invariance

Part of the Lecture Notes in Mathematics book series (LNM, volume 2237)


In medical imaging, several authors have proposed to characterize roughness of observed textures by their fractal dimensions. Fractal analysis of 1D signals is mainly based on the stochastic modeling using the famous fractional Brownian motion for which the fractal dimension is determined by its so-called Hurst parameter. Lots of 2D generalizations of this toy model may be defined according to the scope. This lecture intends to present some of them. After an introduction to random fields, the first part will focus on the construction of Gaussian random fields with prescribed invariance properties such as stationarity, self-similarity, or operator scaling property. Sample paths properties such as modulus of continuity and Hausdorff dimension of graphs will be settled in the second part to understand links with fractal analysis. The third part will concern some methods of simulation and estimation for these random fields in a discrete setting. Some applications in medical imaging will be presented. Finally, the last part will be devoted to geometric constructions involving Marked Poisson Point Processes and shot noise processes.



I would like to warmly thanks all my co-authors for the different works partially presented here, especially Clément Chesseboeuf and Olivier Durieu for their careful reading.


  1. 1.
    R.J. Adler, The Geometry of Random Field (Wiley, Hoboken, 1981)zbMATHGoogle Scholar
  2. 2.
    D. Allard, R. Senoussi, E. Porcu, Anisotropy models for spatial data. Math. Geosci. 48(3), 305–328 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Ayache, F. Roueff, A Fourier formulation of the Frostman criterion for random graphs and its applications to wavelet series. Appl. Comput. Harmon. Anal. 14, 75–82 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Benassi, S. Cohen, J. Istas, Local self-similarity and the Hausdorff dimension. C. R. Acad. Sci. 336(3), 267–272 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    C.L. Benhamou, S. Poupon, E. Lespessailles, S. Loiseau, R. Jennane, V. Siroux, W. Ohley, L. Pothuaud, Fractal analysis of radiographic trabecular bone texture and bone mineral density: two complementary parameters related to osteoporotic fractures. J. Bone Miner. Res. 16(4), 697–704 (2001)CrossRefGoogle Scholar
  6. 6.
    C. Berzin, A. Latour, J.R. León, Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion. Lecture Notes in Statistics, vol. 216 (Springer, Cham, 2014). With a foreword by Aline BonamiGoogle Scholar
  7. 7.
    H. Biermé, C. Lacaux, Fast and exact synthesis of some operator scaling Gaussian random fields. Appl. Comput. Harmon. Anal. (2018).
  8. 8.
    H. Biermé, F. Richard, Statistical tests of anisotropy for fractional brownian textures: application to full-field digital mammography. J. Math. Imaging Vision 36(3), 227–240 (2010)CrossRefGoogle Scholar
  9. 9.
    H. Biermé, M.M. Meerschaert, H.P. Scheffler, Operator scaling stable random fields. Stoch. Process. Appl. 117(3), 312–332 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Biermé, C.L. Benhamou, F. Richard, Parametric estimation for gaussian operator scaling random fields and anisotropy analysis of bone radiograph textures, in Proceedings of the International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI’09), Workshop on Probabilistic Models for Medical Imaging, ed. by K. Pohl, London, UK, September 2009, pp. 13–24Google Scholar
  11. 11.
    H. Biermé, A. Estrade, I. Kaj, Self-similar random fields and rescaled random balls models. J. Theor. Probab. 23(4), 1110–1141 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    H. Biermé, A. Bonami, J.R. León, Central limit theorems and quadratic variations in terms of spectral density. Electron. J. Probab. 16(3), 362–395 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    H. Biermé, Y. Demichel, A. Estrade, Fractional Poisson field and fractional Brownian field: why are they resembling but different? Electron. Commun. Probab. 18, 11–13 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Biermé, L. Moisan, F. Richard, A turning-band method for the simulation of anisotropic fractional Brownian fields. J. Comput. Graph. Stat. 24(3), 885–904 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Bilodeau, D. Brenner, Theory of Multivariate Statistics. Springer Texts in Statistics (Springer, New York, 1999)Google Scholar
  16. 16.
    A. Bonami, A. Estrade, Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9(3), 215–236 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Breuer, P. Major, Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. Burgess, F. Jacobson, P. Judy, Human observer detection experiments with mammograms and power-law noise. Med. Phys. 28(4), 419–437 (2001)CrossRefGoogle Scholar
  19. 19.
    C. Caldwell, S. Stapleton, D. Holdsworth, et al., On the statistical nature of characterisation of mammographic parenchymal patterns by fractal dimension. Phys. Med. Biol. 35(2), 235–247 (1990)CrossRefGoogle Scholar
  20. 20.
    C.B. Caldwell, J. Rosson, J. Surowiak, T. Hearn, Use of fractal dimension to characterize the structure of cancellous bone in radiographs of the proximal femur, in Fractals in Biology and Medicine (Birkhäuser, Basel, 1994), pp. 300–306CrossRefGoogle Scholar
  21. 21.
    G. Chan, An effective method for simulating Gaussian random fields, in Proceedings of the Statistical Computing Section (American Statistical Association, Boston, 1999), pp. 133–138. Google Scholar
  22. 22.
    S. Cohen, J. Istas, Fractional Fields and Applications. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 73 (Springer, Heidelberg, 2013). With a foreword by Stéphane JaffardGoogle Scholar
  23. 23.
    P.F. Craigmile, Simulating a class of stationary Gaussian processes using the Davies-Harte algorithm, with application to long memory processes. J. Time Ser. Anal. 24(5), 505–511 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    R. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart, Y. Xiao, A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, ed. by D. Khoshnevisan, F. Rassoul-Agha, vol. 1962 (Springer, Berlin, 2009). Held at the University of Utah, Salt Lake City, UT, May 8–19, 2006Google Scholar
  25. 25.
    S. Davies, P. Hall, Fractal analysis of surface roughness by using spatial data. J. R. Stat. Soc. Ser. B 61, 3–37 (1999)MathSciNetCrossRefGoogle Scholar
  26. 26.
    C.R. Dietrich, G.N. Newsam, Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18(4), 1088–1107 (1997)MathSciNetCrossRefGoogle Scholar
  27. 27.
    K.J. Falconer, Fractal Geometry (Wiley, Hoboken, 1990)zbMATHGoogle Scholar
  28. 28.
    W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II. 2nd edn. (Wiley, New York, 1971)Google Scholar
  29. 29.
    T. Gneiting, H. Sevciková, D.B. Percivala, M. Schlather, Y. Jianga, Fast and exact simulation of large gaussian lattice systems in \(\mathbb {R}^2\): exploring the limits. J. Comput. Graph. Stat. 15, 483–501 (1996)Google Scholar
  30. 30.
    B. Grosjean, L. Moisan, A-contrario detectability of spots in textured backgrounds. J. Math. Imaging Vision 33(3), 313–337 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    R. Harba, G. Jacquet, R. Jennane, T. Loussot, C.L. Benhamou, E. Lespessailles, D. Tourlière, Determination of fractal scales on trabecular bone X-ray images. Fractals 2(3), 451–456 (1994)CrossRefGoogle Scholar
  32. 32.
    K. Harrar, R. Jennane, K. Zaouchi, T. Janvier, H. Toumi, E. Lespessailles, Oriented fractal analysis for improved bone microarchitecture characterization. Biomed. Signal Process. Control 39, 474–485 (2018)CrossRefGoogle Scholar
  33. 33.
    J. Heine, R. Velthuizen, Spectral analysis of full field digital mammography data. Med. Phys. 29(5), 647–661 (2002)CrossRefGoogle Scholar
  34. 34.
    E. Herbin, E. Merzbach, The set-indexed Lévy process: stationarity, Markov and sample paths properties. Stoch. Process. Appl. 123(5), 1638–1670 (2013)CrossRefGoogle Scholar
  35. 35.
    J. Istas, On fractional stable fields indexed by metric spaces. Electron. Commun. Probab. 11, 242–251 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    J. Istas, G. Lang, Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Stat. 33(4), 407–436 (1997)CrossRefGoogle Scholar
  37. 37.
    O. Kallenberg, Foundations of Modern Probability. Probability and Its Applications (New York) (Springer, New York, 1997)zbMATHGoogle Scholar
  38. 38.
    L.M. Kaplan, C.C.J. Kuo, An improved method for 2-d self-similar image synthesis. IEEE Trans. Image Process. 5(5), 754–761 (1996)CrossRefGoogle Scholar
  39. 39.
    I. Karatzas, E. Shreve, Brownian Motion and Stochastic Calculus (Springer, New York, 1998)CrossRefGoogle Scholar
  40. 40.
    P. Kesterner, J.M. Lina, P. Saint-Jean, A. Arneodo, Waveled-based multifractal formalism to assist in diagnosis in digitized mammograms. Image Anal. Stereol. 20, 169–174 (2001)CrossRefGoogle Scholar
  41. 41.
    A.N. Kolmogorov, The local structure of turbulence in an incompressible viscous fluid for very large reynolds number. Dokl. Akad. Nauk SSSR 30, 301–305 (1941)MathSciNetGoogle Scholar
  42. 42.
    R. Leipus, A. Philippe, D. Puplinskaitė, D. Surgailis, Aggregation and long memory: recent developments. J. Indian Stat. Assoc. 52(1), 81–111 (2014)MathSciNetGoogle Scholar
  43. 43.
    E. Lespessailles, C. Gadois, I. Kousignian, J.P. Neveu, P. Fardellone, S. Kolta, C. Roux, J.P. Do-Huu, C.L. Benhamou, Clinical interest of bone texture analysis in osteoporosis: a case control multicenter study. Osteoporos. Int. 19, 1019–1028 (2008)CrossRefGoogle Scholar
  44. 44.
    Y. Li, W. Wang, Y. Xiao, Exact moduli of continuity for operator-scaling Gaussian random fields. Bernoulli 21(2), 930–956 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    G. Lindgren, Stationary Stochastic Processes: Theory and Applications. Chapman & Hall/CRC Texts in Statistical Science Series (CRC Press, Boca Raton, 2013)Google Scholar
  46. 46.
    R. Lopes, N. Betrouni, Fractal and multifractal analysis: a review. Med. Image Anal. 13, 634–649 (2009)CrossRefGoogle Scholar
  47. 47.
    B.B. Mandelbrot, J. Van Ness, Fractional Brownian motion, fractionnal noises and applications. SIAM Rev. 10, 422–437 (1968)MathSciNetCrossRefGoogle Scholar
  48. 48.
    G. Matheron, The intrinsic random functions and their application. Adv. Appl. Probab. 5, 439–468 (1973)MathSciNetCrossRefGoogle Scholar
  49. 49.
    I. Molchanov, K. Ralchenko, A generalisation of the fractional Brownian field based on non-Euclidean norms. J. Math. Anal. Appl. 430(1), 262–278 (2015)MathSciNetCrossRefGoogle Scholar
  50. 50.
    G. Peccati, C. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, 247–262 (2004)Google Scholar
  51. 51.
    E. Perrin, R. Harba, R. Jennane, I. Iribarren, Fast and exact synthesis for 1-D fractional Brownian motion a nd fractional gaussian noises. IEEE Signal Process. Lett. 9(11), 382–384 (2002)CrossRefGoogle Scholar
  52. 52.
    V. Pilipauskaitė, D. Surgailis, Scaling transition for nonlinear random fields with long-range dependence. Stochastic Process. Appl. 127(8), 2751–2779 (2017)MathSciNetCrossRefGoogle Scholar
  53. 53.
    C.E. Powell, Generating realisations of stationary gaussian random fields by circulant embedding (2014). Technical reportGoogle Scholar
  54. 54.
    N. Privault, Poisson sphere counting processes with random radii. ESAIM Probab. Stat. 20, 417–431 (2016)MathSciNetCrossRefGoogle Scholar
  55. 55.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1986)zbMATHGoogle Scholar
  56. 56.
    G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling (Chapman & Hall, New York, 1994)zbMATHGoogle Scholar
  57. 57.
    M.L. Stein, Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Stat. 11(3), 587–599 (2002)MathSciNetCrossRefGoogle Scholar
  58. 58.
    A.W. van der Vaart, Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3 (Cambridge University Press, Cambridge, 1998)Google Scholar

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Authors and Affiliations

  1. 1.LMA, UMR CNRS 7348Université de PoitiersChasseneuilFrance

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