Analyse multifractale d’images : l’apport des coefficients dominants

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URI: http://hdl.handle.net/2042/28803
Title: Analyse multifractale d’images : l’apport des coefficients dominants
Author: Wendt, Herwig; Abry, Patrice; Roux, Stéphane G.; Jaffard, Stéphane; Vedel, Béatrice
Abstract: Nous nous intéressons à la réalisation pratique d’une procédure permettant d’effectuer une analyse multifractale, c’est-à-dire des fluctuations de régularité locale, de champs scalaires bidimensionnels, d’images notamment. L’originalité de la procédure réside dans la construction, à partir des coefficients d’une transformée discrète bidimensionnelle en ondelettes, de coefficients dominants, impliqués ensuite dans l’estimation des attributs multifractals. Nous donnons des éléments mathématiques relatifs aux problèmes théoriques liés à la validité du formalisme multifractal ainsi construit, et à son application à des images réelles. Nous indiquons comment l’utiliser pour détecter la présence éventuelle de singularités oscillantes. Pour étudier les performances des procédures construites, ces estimateurs sont mis en oeuvre sur un grand nombre de réalisations de processus synthétiques, dont les propriétés multifractales sont connues théoriquement. Nous validons le fait que l’analyse multifractale 2D, construite sur les coefficients dominants, permet une mesure effective et complète des propriétés multifractales des images analysées. De plus, comparant les résultats obtenus d’images mono-fractales à ceux produits sur des images multi-fractales, nous commentons de façon détaillée l’apport des coefficients dominants par rapport à l’usage des coefficients d’ondelettes. Les attributs multifractals ainsi estimés peuvent ensuite être impliqués dans des tâches de classification, par exemple.
Description: 1. Motivation Scale invariance has been observed in numerous applications involving data of various and very different natures. It can be operationally defined as the power law behavior with respect to scale of the structure functions, which are given by the empirical moments of the absolute value of the multiresolution coefficients of the data at a given scale (cf. Eq. (1)). The estimation of the exponents characterizing these power laws – termed scaling exponents – constitutes the ultimate goal of the practical analysis of scale invariance (also called scaling analysis). These scaling exponents are then commonly involved in standard signal processing tasks, such as detection, identification, or classification. In practice, scaling analysis is often conducted within the theoretical framework of multifractal analysis. In a nutshell, multifractal analysis aims at characterizing the fluctuation (in time or space) of the local regularity of the process under analysis through analysis of the (power law) behavior of the structure functions in the limit of fine scales. Though multifractal analysis can theoretically be extended to dimensions higher than 1 without technical difficulties, most practical implementations remain restricted to one dimensional signals. This is mainly due to the fact that multifractal analysis requires the use of a range of both positive and negative empirical moments, hence demanding for multiresolution quantities with adequate properties. To date, the only practically available procedure for the multifractal analysis of 2D signals, hence images, is the so called Wavelet Transform Modulus Maxima (WTMM) procedure (based on the skeleton of a continuous wavelet transform (CWT)). Yet, the WTMM procedure suffers from a number of theoretical and practical drawbacks: It has a high computational cost; The calculation of the 2D CWT skeleton requires involved theoretical definitions as well as a cumbersome practical procedure; It is still lacking a theoretical support. Therefore, in numerous applications where the data are naturally images, multifractal analysis remains restricted to 1D slices of the data and hence incomplete. In the present contribution, elaborating on previous results obtained for (1D) signals, we propose a practical multifractal analysis method for (2D) images based on two key features: The use of a 2D Discrete Wavelet Transform (DWT) (instead of a 2D CWT); The replacement of wavelet coefficients with wavelet Leaders. This yields two major benefits: The computation cost is very low; Wavelet Leaders have been shown to yield a complete and rigorous analysis of the multifractal analysis of bounded functions. This is because wavelet Leaders consist of monotonous increasing quantities that finely account for the irregularities of the analyzed function. The aims of the present contribution are twofold: First, studying the necessary theoretical elements, validity and limitations of a wavelet Leader based multifractal analysis of images, and second, the evaluation of its practical statistical performance....
Subject: Image; analyse multifractale; coefficients dominants; transformée discrète en ondelettes; fonction uniformément höldérienne; intégration fractionnaire; validité du formalisme multifractal; singularité oscillante
Publisher: GRETSI, Saint Martin d'Hères, France
Date: 2009

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