Multiscale Representations: Fractals, Self-Similar Random Processes and Wavelets

2 Principles 4 2.1 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Definition and history . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Holder exponent and singularity spectrum . . . . . . . . . . . . . 6 2.2 Self-similar random processes . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Definition and history . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . 9 2.2.4 Multi-fractional Brownian motion . . . . . . . . . . . . . . . . . 10 2.3 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Definition and history . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2 Continuous wavelet transform . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Orthogonal wavelet transform . . . . . . . . . . . . . . . . . . . . 16

[1]  Y. Meyer,et al.  Remarques sur l'analyse de Fourier à fenêtre , 1991 .

[2]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[3]  Lewis F. Richardson,et al.  Weather Prediction by Numerical Process , 1922 .

[4]  Marie Farge,et al.  Transformée en ondelettes pour détecter et analyser les structures cohérentes dans les écoulements turbulents bidimensionnels , 1988 .

[5]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

[6]  Marie Farge,et al.  Wavelet tools to study intermittency: application to vortex bursting , 2009, Journal of Fluid Mechanics.

[7]  O. Vasilyev,et al.  Wavelet Methods in Computational Fluid Dynamics , 2010 .

[8]  Béatrice Pesquet-Popescu,et al.  Ondelettes et applications , 2015, Le traitement du signal et ses applications.

[9]  Tony DeRose,et al.  Wavelet noise , 2005, SIGGRAPH 2005.

[10]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[11]  Prashant Parikh.  A Theory of Communication , 2010 .

[12]  Kai Schneider,et al.  Nonlinear wavelet thresholding: A recursive method to determine the optimal denoising threshold , 2005 .

[13]  R. Kraichnan.  On Kolmogorov's inertial-range theories , 1974, Journal of Fluid Mechanics.

[14]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[15]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[16]  Y. Meyer,et al.  Ondelettes et bases hilbertiennes. , 1986 .

[17]  P. Levy.  Le mouvement brownien , 1955 .

[18]  I. Good,et al.  Fractals: Form, Chance and Dimension , 1978 .

[19]  Ronald R. Coifman,et al.  Entropy-based algorithms for best basis selection , 1992, IEEE Trans. Inf. Theory.

[20]  U. Frisch,et al.  Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade , 1989, Nature.

[21]  Andrew J. Majda,et al.  A Wavelet Monte Carlo Method for Turbulent Diffusion with Many Spatial Scales , 1994 .

[22]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[23]  U. Frisch.  Turbulence: The Legacy of A. N. Kolmogorov , 1996 .

[24]  Kai Schneider,et al.  SPATIAL INTERMITTENCY IN TWO-DIMENSIONAL TURBULENCE: A WAVELET APPROACH , 2004 .

[25]  S. Jaffard.  Construction of Wavelets on Open Sets , 1989 .

[26]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[27]  M. Farge,et al.  Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis , 1999 .

[28]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[29]  R. Balian.  Un principe d'incertitude fort en théorie du signal ou en mécanique quantique , 1981 .

[30]  A. Haar.  Zur Theorie der orthogonalen Funktionensysteme , 1910 .

[31]  A. Kolmogorov,et al.  The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[32]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[33]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[34]  J. Gouyet.  Physics and Fractal Structures , 1996 .

[35]  A. Kolmogorov.  The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[36]  Michael Unser,et al.  Fractional Brownian Vector Fields , 2010, Multiscale Model. Simul..

[37]  M. Holschneider.  On the wavelet transformation of fractal objects , 1988 .

[38]  Dennis Gabor,et al.  Theory of communication , 1946 .

[39]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[40]  I. Daubechies,et al.  PAINLESS NONORTHOGONAL EXPANSIONS , 1986 .

[41]  Richard G. Baraniuk,et al.  Fast, Exact Synthesis of Gaussian and nonGaussian Long-Range-Dependent Processes , 2009 .

[42]  G. A. Hunt.  Random Fourier transforms , 1951 .

[43]  Syed Twareque Ali,et al.  Two-Dimensional Wavelets and their Relatives , 2004 .

[44]  Barbara Hubbard,et al.  The World According to Wavelets , 1996 .

[45]  Andrew J. Majda,et al.  A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales , 1995 .