Journal of Mathematical Imaging and Vision

, Volume 36, Issue 3, pp 227–240 | Cite as

Statistical Tests of Anisotropy for Fractional Brownian Textures. Application to Full-field Digital Mammography

  • Frédéric Richard
  • Hermine Bierme


In this paper, we propose a new and generic methodology for the analysis of texture anisotropy. The methodology is based on the stochastic modeling of textures by anisotropic fractional Brownian fields. It includes original statistical tests that permit to determine whether a texture is anisotropic or not. These tests are based on the estimation of directional parameters of the fields by generalized quadratic variations. Their construction is founded on a new theoretical result about the convergence of test statistics, which is proved in the paper. The methodology is applied to simulated data and discussed. We show that on a database composed of 116 full-field digital mammograms, about 60 percent of textures can be considered as anisotropic with a high level of confidence. These empirical results strongly suggest that anisotropic fractional Brownian fields are better-suited than the commonly used fractional Brownian fields to the modeling of mammogram textures.

Anisotropy Anisotropic fractional Brownian field Hurst index Asymptotic test Generalized quadratic variations Texture analysis Mammography Density characterization 


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  1. 1.
    Abry, P., Gonçalves, P., Sellan, F.: Wavelet spectrum analysis and 1/f processes. In: Lecture Notes in Statistics, vol. 103, pp. 15–30. Springer, Berlin (1995) Google Scholar
  2. 2.
    Adler, R.J.: The Geometry of Random Field. Wiley, New York (1981) Google Scholar
  3. 3.
    Astley, S., et al. (eds.): Proc. of the 8th International Workshop on Digital Mammography, Manchester, UK, June 2004. LNCS, vol. 4046. Springer, Berlin (2004) Google Scholar
  4. 4.
    Ayache, A., Bonami, A., Estrade, A.: Identification and series decomposition of anisotropic Gaussian fields. In: Proc. of the Catania ISAAC05 Congress (2005) Google Scholar
  5. 5.
    Bardet, J.M., Lang, G., Oppenheim, G., et al.: Semi-parametric estimation of the long-range dependence parameter: a survey. In: Theory and Applications of Long-range Dependence, pp. 557–577. Birkhauser, Boston (2003) Google Scholar
  6. 6.
    Begyn, A.: Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13(3), 712–753 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Benhamou, C.L., Poupon, S., Lespessailles, E., et al.: Fractal analysis of radiographic trabecular bone texture and bone mineral density. J. Bone Miner. Res. 16(4), 697–703 (2001) CrossRefGoogle Scholar
  8. 8.
    Benson, D., Meerschaert, M.M., Bäumer, B., Scheffler, H.P.: Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resour. Res. 42, 1–18 (2006) CrossRefGoogle Scholar
  9. 9.
    Beran, J.: Statistics for Long-memory Processes. Chapman Hall, London (1994) zbMATHGoogle Scholar
  10. 10.
    Biermé, H., Meerschaert, M.M., Scheffler, H.P.: Operator scaling stable random fields. Stoch. Proc. Appl. 117(3), 312–332 (2007) zbMATHCrossRefGoogle Scholar
  11. 11.
    Biermé, H., Richard, F.: Estimation of anisotropic Gaussian fields through radon transform. ESAIM: Probab. Stat. 12(1), 30–50 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bonami, A., Estrade, A.: Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9, 215–236 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Boyd, N.F., O’Sullivan, B., Campbell, J.E., et al.: Mammographic signs as risk factors for breast cancer. Br. J. Cancer 45, 185–193 (1982) Google Scholar
  14. 14.
    Brisson, J., Merletti, F., Sadowsky, N.L., et al.: Mammographic features of the breast and breast cancer risk. Am. J. Epidemiol. 115(3), 428–437 (1982) Google Scholar
  15. 15.
    Brunet-Imbault, B., Lemineur, G., Chappard, C., et al.: A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform. BMC Med. Imaging 5(4), 4 (2005) CrossRefGoogle Scholar
  16. 16.
    Burgess, A., Jacobson, F., Judy, P.: Human observer detection experiments with mammograms and power-law noise. Med. Phys. 28(4), 419–437 (2001) CrossRefGoogle Scholar
  17. 17.
    Byng, J., Boyd, N.N., Fishell, E.: Automated analysis of mammographic densities. Phys. Med. Biol. 41, 909–923 (1996) CrossRefGoogle Scholar
  18. 18.
    Byng, J., Yaffe, M., Lockwood, G., et al.: Automated analysis of mammographic densities and breast carcinoma risk. Cancer 80(1), 66–74 (1997) CrossRefGoogle Scholar
  19. 19.
    Caldwell, C., Stapleton, S., Holdsworth, D., et al.: Characterisation of mammographic parenchymal patterns by fractal dimension. Phys. Med. Biol. 35(2), 235–247 (1990) CrossRefGoogle Scholar
  20. 20.
    Chen, C.-C., Daponte, J., Fox, M.: Fractal feature analysis and classification in medical imaging. IEEE Trans. Pattern. Anal. Mach. Intell. 8(2), 133–142 (1989) Google Scholar
  21. 21.
    Coeurjolly, J.F.: Inférence statistique pour les mouvements browniens fractionnaires et multifractionnaires. Ph.D. Thesis, University Joseph Fourier (2000) Google Scholar
  22. 22.
    Cross, G., Jain, A.: Markov random field texture models. IEEE Trans. Pattern. Anal. Mach. Intell. 5(1), 25–39 (1983) CrossRefGoogle Scholar
  23. 23.
    Czörgö, M., Révész, P.: Strong Approximation in Probability and Statistics. Academic Press, San Diego (1981) Google Scholar
  24. 24.
    Dacunha-Castelle, D., Duflo, M.: Probabilités et Statistiques, vol. 2. Masson, Paris (1983) zbMATHGoogle Scholar
  25. 25.
    Davies, S., Hall, P.: Fractal analysis of surface roughness by using spatial data. J. R. Stat. Soc. Ser. B 61, 3–37 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Doi, K., et al. (eds.): Proc. of the 3rd International Workshop on Digital Mammography, Chicago, USA, June 1996. Elsevier, Amsterdam (1996) Google Scholar
  27. 27.
    Falconer, K.J.: Fractal Geometry. Wiley, New York (1990) zbMATHGoogle Scholar
  28. 28.
    Gale, A.G., et al. (eds.): Proc. of the 2nd International Workshop on Digital Mammography, York, England, July 1994. Elsevier, Amsterdam (1994) Google Scholar
  29. 29.
    Grosjean, B., Moisan, L.: A-contrario detectability of spots in textured backgrounds. J. Math. Imaging Vis. 33(3), 313–337 (2009) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Heine, J., Deine, S., Velthuizen, R., et al.: On the statistical nature of mammograms. Med. Phys. 26(11), 2254–2269 (1999) CrossRefGoogle Scholar
  31. 31.
    Heine, J., Malhorta, P.: Mammographic tissue, breast cancer risk, serial image analysis, and digital mammography: serial breast tissue change and related temporal influences. Acad. Radiol. 9, 317–335 (2002) CrossRefGoogle Scholar
  32. 32.
    Heine, J., Malhorta, P.: Mammographic tissue, breast cancer risk, serial image analysis, and digital mammography: tissue and related risk factors. Acad. Radiol. 9, 298–316 (2002) CrossRefGoogle Scholar
  33. 33.
    Heine, J., Velthuizen, R.: Spectral analysis of full field digital mammography data. Med. Phys. 29(5), 647–661 (2002) CrossRefGoogle Scholar
  34. 34.
    Istas, J.: Identifying the anisotropical function of a d-dimensional Gaussian self-similar process with stationary increments. Stat. Inference Stoch. Process. 10(1), 97–106 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Istas, J., Lang, G.: Quadratic variations and estimation of the local Holder index of a Gaussian process. Ann. Inst. Henri Poincaré, Probab. Stat. 33(4), 407–436 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Jennane, R., Harba, R., Lemineur, G., et al.: Estimation of the 3D self-similarity parameter of trabecular bone from its projection. Med. Image Anal. 11, 91–98 (2007) CrossRefGoogle Scholar
  37. 37.
    Kamont, A.: On the fractional anisotropic Wiener field. Probab. Math. Stat. 16, 85–98 (1996) zbMATHMathSciNetGoogle Scholar
  38. 38.
    Karssemeijer, N., et al. (eds.): 4th International Workshop on Digital Mammography, Nijmegen, The Netherlands, June 1998. Kluwer Academic, Dordrecht (1998) Google Scholar
  39. 39.
    Kent, J.T., Wood, A.T.A.: Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. R. Stat. Soc. Ser. B 59(3), 679–699 (1997) zbMATHMathSciNetGoogle Scholar
  40. 40.
    Kestener, P., Lina, J.-M., Saint-Jean, P., et al.: Wavelet-based multifractal formalism to assist in diagnosis in digitized mammograms. Image Anal. Stereol. 20, 169–174 (2001) zbMATHGoogle Scholar
  41. 41.
    Kolmogorov, A.N.: Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum. C. R. (Dokl.) Acad. Sci. URSS 26, 115–118 (1940) Google Scholar
  42. 42.
    Leger, S.: Analyse stochastique de signaux multi-fractaux et estimations de paramètres. Ph.D. Thesis, Université d’Orléans (2000) Google Scholar
  43. 43.
    Lundahl, T., Ohley, W.J., Kay, S.M., Siffe, R.: Fractional Brownian motion: a maximum likelihood estimator and its application to image texture. IEEE Trans. Med. Imaging 5(3), 152–161 (1986) CrossRefGoogle Scholar
  44. 44.
    Mandelbrot, B.B., Van Ness, J.: Fractional Brownian motion, fractionnal noises and applications. SIAM Rev. 10, 422–437 (1968) zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Peitgen, H.-O. (ed.): 6th International Workshop on Digital Mammography, Bremen, Germany, June 2002. Springer, Berlin (2002) Google Scholar
  46. 46.
    Pentland, A.: Fractal-based description of natural scenes. IEEE Trans. Pattern. Anal. Mach. Intell. 6, 661–674 (1984) CrossRefGoogle Scholar
  47. 47.
    Stein, M.L.: Fast and exact simulation of fractional Brownian surfaces. J. Comput. Graph. Stat. 11(3), 587–599 (2002) CrossRefGoogle Scholar
  48. 48.
    Wolfe, J.N.: Ducts as a sole indicator of breast carcinoma. Radiology 89, 206–210 (1967) Google Scholar
  49. 49.
    Wolfe, J.N.: A study of breast parenchyma by mammography in the normal woman and those with benign and malignant disease. Radiology 89, 201–205 (1967) Google Scholar
  50. 50.
    Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.): A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math., vol. 1962, pp. 145–212. Springer, New York (2009) CrossRefGoogle Scholar
  51. 51.
    Yaffe, M., et al. (eds.): Proc. of the 5th International Workshop on Digital Mammography, Toronto, Canada, June 2000. Medical Physics Publishing, Toronto (2000) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.University Paris Descartes, Laboratory MAP5CNRS UMR 8145ParisFrance

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