Beyond Besov Spaces, Part 2: Oscillation Spaces

Abstract We study several extensions of Besov spaces; these extensions include the oscillation spaces Ops,s’which take into account correlations between the positions of large wavelet coefficients through the scales and, more generally, spaces defined through the distributions of suprema of wavelet coefficients on wavelet subtrees. These spaces are independent of the particular wavelet basis chosen. Examples of applications will be taken from image processing and multifractal analysis.

[1]  E. Candès,et al.  Ridgelets: a key to higher-dimensional intermittency? , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  E. Bacry,et al.  Log-Infinitely Divisible Multifractal Processes , 2002, cond-mat/0207094.

[3]  I. Daubechies,et al.  Tree Approximation and Optimal Encoding , 2001 .

[4]  Eero P. Simoncelli,et al.  Image compression via joint statistical characterization in the wavelet domain , 1999, IEEE Trans. Image Process..

[5]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[6]  R. Jensen,et al.  Direct determination of the f(α) singularity spectrum , 1989 .

[7]  J. Stewart,et al.  Amalgams of $L^p$ and $l^q$ , 1985 .

[8]  Emmanuel Bacry,et al.  Random cascades on wavelet dyadic trees , 1998 .

[9]  Walter Willinger,et al.  Wavelet analysis of conservative cascades , 2003 .

[10]  J. Kahane.  Some Random Series of Functions , 1985 .

[11]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[12]  Richard G. Baraniuk,et al.  Near Best Tree Approximation , 2002, Adv. Comput. Math..

[13]  David Mumford,et al.  Occlusion Models for Natural Images: A Statistical Study of a Scale-Invariant Dead Leaves Model , 2004, International Journal of Computer Vision.

[14]  Stéphane Jaffard,et al.  Oscillation spaces: Properties and applications to fractal and multifractal functions , 1998 .

[15]  H. Feichtinger.  Amalgam spaces and generalized harmonic analysis , 1997 .

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

[17]  William A. Pearlman,et al.  An image multiresolution representation for lossless and lossy compression , 1996, IEEE Trans. Image Process..

[18]  J. Kahane,et al.  Sur certaines martingales de Benoit Mandelbrot , 1976 .

[19]  B. Vidakovic,et al.  Bayesian Inference in Wavelet-Based Models , 1999 .

[20]  Hans G. Feichtinger,et al.  Wiener Amalgams over Euclidean Spaces and Some of Their Applications , 2020 .

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

[22]  I. Daubechies,et al.  Harmonic analysis of the space BV. , 2003 .

[23]  Jacques Lévy Véhel,et al.  Continuous Large Deviation Multifractal Spectrum: Definition and Estimation , 1998 .

[24]  D. Veitch,et al.  Infinitely divisible cascade analysis of network traffic data , 2000, 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100).

[25]  Stéphane Jaffard,et al.  Multifractal formalism for functions part II: self-similar functions , 1997 .

[26]  Stéphane Mallat.  Foveal detection and approximation for singularities , 2003 .

[27]  J. Peinke,et al.  Multiplicative process in turbulent velocity statistics : A simplified analysis , 1996 .

[28]  Stéphane Jaffard,et al.  On the Frisch–Parisi conjecture , 2000 .

[29]  Jensen,et al.  Direct determination of the f( alpha ) singularity spectrum and its application to fully developed turbulence. , 1989, Physical review. A, General physics.

[30]  Charles Meneveau,et al.  Spatial correlations in turbulence: Predictions from the multifractal formalism and comparison with experiments , 1993 .

[31]  R. DeVore,et al.  Nonlinear Approximation and the Space BV(R2) , 1999 .

[32]  Y. Meyer,et al.  Construction of Continuous Functions with Prescribed Local Regularity , 1998 .

[33]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[34]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .