Multifractal Analysis for Images: The Wavelet Leaders Contribution. Analyse Multifractale d’Images: l’Apport des Coefficients Dominants

Multifractal Analysis for Images: The Wavelet Leaders Contribution

Analyse Multifractale d’Images: l’Apport des Coefficients Dominants

Herwig Wendt Patrice Abry  Stéphane G. Roux  Stéphane Jaffard  Béatrice Vedel 

Laboratoire de Physique, UMR 5672, CNRS, École Normale Supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 7, France

Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050, CNRS, Université Paris XII, 61, Avenue du Général de Gaulle, 94010 Créteil Cedex, France

13 May 2008
28 February 2009
| Citation



Image, Multifractal Analysis, Wavelet Leaders, Discrete Wavelet Transform, Uniform Hölder Function, Fractional Integration, Multifractal Formalism Validity, Oscillating Singularity, Fractional Brownian Motion, Multiplicative Cascade.

Mots clés

Image, analyse multifractale, coefficients dominants, transformée discrète en ondelettes, fonction uniformément höldérienne, intégration fractionnaire, validité du formalisme multifractal, singularité oscillante, mouvement brownien fractionnaire, cascade multiplicative.

1. Motivation
2. Analyse Multifractale: Théorie
3. Mesures et Intégration Fractionnaire
4. Analyse Multifractale Pratique
5. Simulations Numériques
6. Performances d’Estimation
7. Conclusion et Discussion

[1] A. ARNEODO, N. DECOSTER and S. G. ROUX, «A waveletbased method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces, » European Physical Journal B, vol. 15, no. 3, pp. 567-600, 2000.

[2] A. ARNEODO, N. DECOSTER, P. KESTENER and S. ROUX, Advances in Imaging and Electron Physics, ser. Advances in Imaging and Electron Physics, P.W. Hawkes, Eds. Academic Press, 2003, vol. 126, ch. A wavelet-based method for multifractal image analysis: from theoretical concepts to experimental applications, pp. 1-98.

[3] P. ABRY, P. FLANDRIN, M. TAQQU and D. VEITCH, «Wavelets for the analysis, estimation and synthesis of scaling data, » in Selfsimilar Network Trac and Performance Evaluation. New York: Wiley, 2000.

[4] S. JAFFARD, «Wavelet techniques in multifractal analysis, » in Fractal Geometry and Applications : A Jubilee of Benoît Mandelbrot, M. Lapidus et M. van Frankenhuijsen Eds., Proceedings of Symposia in Pure Mathematics, vol. 72(2). AMS, 2004, pp. 91-152.

[5] V. SHARI-SALAMANTIAN, B. PESQUET-POPESCU, J. SIMONILAFONTAINE and J. P. RIGAUT, « A robust index for spatial heterogeneity in breast cancer,» Journal of Microscopy, vol. 216, no. 2, pp. 110-122, 2004.

[6] C. L. BENHAMOU, S. POUPON, E. LESPESSAILLES, S. LOISEAU, R. JENNANE, V. SIROUX, W. J. OHLEY and L. POTHUAUD, « Fractal analysis of radiographic trabecular bone texture and bone mineral density : two complementary parameters related to osteoporotic fractures, » J. Bone Miner. Res., vol. 16, no. 4, pp. 697-704, 2001.

[7] D. SCHERTZER, S. LOVEJOY, F. SCHMITT,Y. GHIGISINSKAYA and D. MARSAN, « Multifractal cascade dynamics and turbulent intermittency, » Fractals, vol. 5, no. 3, pp. 427-471, 1997.

[8] S. G. ROUX, A. ARNEODO and N. DECOSTER, «A waveletbased method for multifractal image analysis. III. Applications to high-resolution satellite images of cloud structure, » European Physical Journal B, vol. 15, no. 4, pp. 765-786, 2000.

[9] R. JENNANE, W. J. OHLEY, S. MAJUMDAR and G. LEMINEUR, « Fractal analysis of bone x-ray computed microscopy projections, » IEEE Transactions on Medical Imaging, vol. 20, no. 5, pp. 443-449, 2001.

[10] S. LOWEN and M. TEICH, Fractal-Based Point Processes. Hoboken, NJ : Wiley, 2005.

[11] J.-L. STARCK, F. MURTAGH and A. BIJAOUI, Image Processing and Data Analysis : The Multiscale Approach. Cambridge: Cambridge University Press, 1998.

[12] N. DECOSTER, S. G. ROUX and A. ARNEODO, «A waveletbased method for multifractal image analysis. II. Applications to synthetic multifractal rough surfaces,» European Physical Journal B, vol. 15, no. 4, pp. 739-764, 2000.

[13] L. PONSON, D. BONAMY, H. AURADOU, G. MOUROT, S.MOREL, E. BOUCHAUD, C. GUILLOT and J. HULIN, « Anisotropic self-ane properties of experimental fracture surface,» International Journal of Fracture, vol. 140, no. 1-4, pp. 27-36, 2006.

[14] S. PELEG, J. NAOR, R. HARTLEY and D. AVNIR, « Multiple resolution texture analysis and classication, » IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 6, no. 4, pp. 518-523, 1984.

[15] Y. XU, J. HUI and C. FERMÜLLER, « A projective invariant for texture,» in IEEE Conference on Computer Vision and Pattern Recognition, New York, 2006, pp. 1932-1939.

[16] S. JAFFARD, B. LASHERMES and P. ABRY, «Wavelet leaders in multifractal analysis,» in Wavelet Analysis and Applications, T. Qian, M.I. Vai, X. Yuesheng, Eds. Basel, Switzerland : Birkh auser Verlag, 2006, pp. 219-264.

[17] H. WENDT, P. ABRY and S. JAFFARD, « Bootstrap for empirical multifractal analysis,» IEEE Signal Processing Mag., vol. 24, no. 4, pp. 38-48, 2007.

[18] S. JAFFARD, Lois d’échelle, fractales et ondelettes, vol. 2, Éditeurs: P. Abry, P. Goncalvès, J. Lévy Vehel. Lavoisier, 2002, ch. Méthodes d’ondelettes pour l’analyse multifractale de fonctions.

[19] R. RIEDI, Lois d’échelle, fractales et ondelettes, vol. 2, Éditeurs: P. Abry, P. Goncalvès, J. Lévy Vehel. Lavoisier, 2002, ch. Lois d’échelles multifractales : fondements et approche par ondelettes.

[20] R. H. RIEDI, Theory and applications of long range dependence. Birkh auser, 2003, ch. Multifractal Processes, pp. 625-717.

[21] B. MANDELBROT, « Intermittent turbulence in selfsimilar cascades ; divergence of high moments and dimension of the carrier, » J. Fluid Mech., vol. 62, pp. 331-358, 1974.

[22] J. KAHANE and J. PEYRIÈRE, « Sur certaines martingales de Benoit Mandelbrot, » Adv. in Math., vol. 22, 1976.

[23] S. MALLAT, A Wavelet Tour of Signal Processing. San Diego, CA : Academic Press, 1998.

[24] J.-P. ANTOINE, R. MURENZI, P. VANDERGHEYNST and S. T. ALI, Two-Dimensional Wavelets and their Relatives. Cambridge : Cambridge University Press, 2004.

[25] P. ABRY, R. BARANIUK, P. FLANDRIN, R. RIEDI and D. VEITCH, « Multiscale nature of network trac, » IEEE Signal Processing Magazine, vol. 19, no. 3, pp. 28-46, 2002.

[26] E. BACRY, J. MUZY and A. ARNEODO, « Singularity spectrum of fractal signals from wavelet analysis : Exact results, » J. Stat. Phys., vol. 70, no. 3-4, pp. 635-674, 1993.

[27] G. SAMORODNITSKY and M. TAQQU, Stable non-Gaussian random processes. New York : Chapman and Hall, 1994.

[28] J. BARRAL and B. MANDELBROT, « Multiplicative products of cylindrical pulses, » Probab. Theory Relat. Fields, vol. 124, pp. 409-430, 2002.

[29] C. TRICOT and J. L. VEHEL, Lois d’échelle, fractales et ondelettes, vol. 2, Éditeurs : P. Abry, P. Goncalvès, J. Lévy Vehel. Lavoisier, 2002, ch. Analyse fractale et multifractale en traitement des signaux. 

[30] S. JAFFARD, P. ABRY, S. ROUX, B. VEDEL and H. WENDT, Proceedings of the ISFMA summer school on wavelets and their applications,. Zuhai, China, 2007, ch. The contribution of wavelets in multifractal analysis.

[31] P. ABRY, S. JAFFARD, S. G. ROUX, B. VEDEL and H. WENDT, « The contribution of wavelets in multifractal analysis,» vol. submitted, 2008.

[32] S. JAFFARD, H. WENDT, B. VEDEL, S. ROUX, P. ABRY., «Wavelet analysis of multifractal measures,» vol. preprint, 2008.

[33] U. FRISCH, Turbulence. The legacy of A. Kolmogorov. Cambridge, UK : Cambridge University Press, 1995.

[34] B. CASTAING, Y. GAGNE and M. MARCHAND, « Logsimilarity for turbulent ows, » Physica D, vol. 68, nO. 3-4, pp. 387-400, 1993.

[35] J. DELOUR, J. MUZY and A. ARNEODO, « Intermittency of 1d velocity spatial proles in turbulence : A magnitude cumulant analysis,» The Euro. Phys. Jour. B, vol. 23, no. 2, pp. 243-248, 2001.

[36] H. WENDT, Wavelet leaders and Bootstrap : A contribution to empirical multifractal analysis. PhD Thesis, 2008.

[37] M. STEIN, «Fast and exact simulation of fractional brownian surfaces,» J. Comput. Graph. Statist., vol. 11, no. 3, pp. 587-599, 2002.

[38] G. CHAN and A. WOOD, «An algorithm for simulating stationary gaussian random elds,» Applied Statistics, vol. 46, pp. 171-181, 1997.

[39] P. FLANDRIN, «Wavelet analysis and synthesis of fractional Brownian motion, » IEEE Trans. on Info. Theory, vol. IT-38, no. 2, pp. 910-917, 1992.

[40] P. ABRY and P. FLANDRIN, « On the initialization of the discrete wavelet transform, » IEEE Signal Processing Letters, vol. 1, no. 2, pp. 32-34, 1994.

[41] D. VEITCH, P. ABRY and M. S. TAQQU, « On the automatic selection of the onset of scaling, » Fractals, vol. 4, nO. 11, pp. 377-390, 2003.

[42] D. VEITCH and P. ABRY, « A statistical test for the time constancy of scaling exponents, » IEEE Trans. on Sig. Proc., vol. 10, no. 49, pp. 2325-2334, October 2001.