Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents

Let be a domain of . In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of . Classifications based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local Lp regularity of the characteristic function or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of . We develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains.

[1]  S. Havlin,et al.  Fractals and Disordered Systems , 1991 .

[2]  Stéphane Jaffard,et al.  Multifractal formalism for functions part I: results valid for all functions , 1997 .

[3]  B. Mandelbrot.  On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars , 1975, Journal of Fluid Mechanics.

[4]  Alain Arneodo,et al.  Wavelet Based Multifractal Formalism: Applications to DNA Sequences, Satellite Images of the Cloud Structure, and Stock Market Data , 2002 .

[5]  A. Zygmund,et al.  Local properties of solutions of elliptic partial di erential equations , 1961 .

[6]  S. Zucker,et al.  Evaluating the fractal dimension of surfaces , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Yves Meyer,et al.  Wavelets, Vibrations and Scalings , 1997 .

[8]  Yves Meyer,et al.  Wavelet Analysis and Chirps , 1997 .

[9]  Yimin Xiao.  Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields , 1997 .

[10]  Two Results Concerning Chirps and 2-Microlocal Exponents Prescription☆ , 1998 .

[11]  B. Sapoval,et al.  7 Fractal surfaces and interfaces , 1991 .

[12]  Y. Meyer,et al.  Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions , 1996 .

[13]  J. C. Vassilicos.  The Multispiral Model of Turbulence and Intermittency , 1992 .

[14]  S. Jaffard,et al.  Morphology of the mixing layer in the Rayleigh-Taylor instability , 1995 .

[15]  Meneveau,et al.  Interface dimension in intermittent turbulence. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[16]  R. Dalang,et al.  Geography of the level sets of the Brownian sheet , 1993 .

[17]  Emmanuel Bacry,et al.  THE THERMODYNAMICS OF FRACTALS REVISITED WITH WAVELETS , 1995 .

[18]  Y. Meyer.  Principe d'incertitude, bases hilbertiennes et algèbres d'opérateurs , 1986 .

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

[20]  John Christos Vassilicos,et al.  Fractal dimensions and spectra of interfaces with application to turbulence , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[21]  J. Aubry,et al.  Random Wavelet Series , 2002 .

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

[23]  S. Jaffard.  Pointwise smoothness, two-microlocalization and wavelet coefficients , 1991 .

[24]  Sur les singularités oscillantes et le formalisme multifractal , 2002 .

[25]  Yann Gousseau.  Distribution de formes dans les images naturelles , 2000 .

[26]  Paul E. Dimotakis,et al.  Mixing in turbulent jets: scalar measures and isosurface geometry , 1996, Journal of Fluid Mechanics.

[27]  S. Mallat.  VI – Wavelet zoom , 1999 .

[28]  Emmanuel Bacry,et al.  Oscillating singularities on cantor sets: A grand-canonical multifractal formalism , 1997 .

[29]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .

[30]  Yves Meyer,et al.  Wavelets - tools for science and technology , 1987 .