A WAVELET MULTIFRACTAL FORMALISM FOR SIMULTANEOUS SINGULARITIES OF FUNCTIONS

Author(s):  
JAMIL AOUIDI ◽  
ANOUAR BEN MABROUK

In this paper, a wavelet multifractal analysis is developed which permits to characterize simultaneous singularities for a vector of functions. An associated multifractal formalism is introduced and checked for the case of functions involving self similar aspects.

Fractals ◽  
1997 ◽  
Vol 05 (01) ◽  
pp. 153-168 ◽  
Author(s):  
Rudolf H. Riedi ◽  
Istvan Scheuring

In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relativemultifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.


2011 ◽  
Vol 20 (3) ◽  
pp. 169 ◽  
Author(s):  
Pierre Kestener ◽  
Jean Marc Lina ◽  
Philippe Saint-Jean ◽  
Alain Arneodo

We apply the 2D wavelet transform (WTMM) method to perform a multifractal analysis of digitized mammograms. We show that normal regions display monofractal scaling properties as characterized by the socalled Hurst exponent H =0.3±0.1 in fatty areas which look like antipersistent self-similar random surfaces, while H=0.65±0.1 in dense areas which exibit long-range correlations and possibly multifractal scaling properties. We further demonstrate that the 2D WTMM method provides a very efficient way to detect tumors as well as microcalcifications (MC) which correspond to much stronger singularities than those involved in the background tissue roughness fluctuations. These preliminary results indicate that the texture discriminatory power of the 2D WTMM method may lead to significant improvement in computer-assisted diagnosis in digitized mammograms.


Author(s):  
JAMIL AOUIDI ◽  
ANOUAR BEN MABROUK

In this paper, a multifractal analysis of some non-self-similar functions based on the superposition of finite number of weighted quasi-self-similar ones ∑iωiFi is developed. In general, such superpositions do not yield neither a self-similar nor a quasi-self-similar outcome. Furthermore, there are two main problems that appear. Firstly, a phenomenon of regularity compensation may exist. Secondly, the computation of the spectrum of singularities and therefore the validity of the multifractal formalism based on the possibility of constructing Gibbs measures fail. In this paper, we propose to study such problems by conducting a multifractal analysis of such combinations and to check the validity of the multifractal formalism in the case where there is no compensation of regularity. Furthermore, we compute the box dimension of the associated graphs and provide some examples. The paper in its full subject re-considers the results of Ref. 3 in the quasi-self-similar case.


Fractals ◽  
2021 ◽  
pp. 2240001
Author(s):  
ANOUAR BEN MABROUK ◽  
ADEL FARHAT

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.


2005 ◽  
Vol 12 (2) ◽  
pp. 157-162 ◽  
Author(s):  
Y. Ida ◽  
M. Hayakawa ◽  
A. Adalev ◽  
K. Gotoh

Abstract. In our previous papers we have shown that the fractal (monofractal) dimension (Do) showed a significant increase before the Guam earthquake occurred on 8 August, 1993. In order to have a further support to this precursory effect to the general rupture (earthquake) we have carried out the corresponding multifractal analysis (by means of detrended fluctuation analysis) for the same data to study the statistical self-similar properties in a wide range of scales. We have analyzed the ULF geomagnetic data (the most intense H component) observed at Guam observatory. As the result, we have found that we could observe significant changes in the multifractal parameters at Guam such that αmin showed a meaningful decrease about 25 days before the earthquake and correspondingly Δα increased because αmax exhibited no significant change at all. The most sensitive parameter seems to be non-uniformity factor Δ. Correspondingly, the generalized multifractal dimension Dq (q>1) showed a significant decrease (whereas Dq (q<0) showed no change) and D0 (=Dq (q=0) (as already found in our previous papers) is reconfirmed to increase before the earthquake. These multifractal characteristics seem to be a further support that these changes are closely associated with the earthquake as a precursor to the Guam earthquake, providing us with appreciable information on the pre-rupture evolution of the earthquake.


1998 ◽  
Vol 31 (11) ◽  
pp. 2567-2590 ◽  
Author(s):  
T Huillet ◽  
B Jeannet

Fractals ◽  
2016 ◽  
Vol 24 (04) ◽  
pp. 1650039 ◽  
Author(s):  
MOURAD BEN SLIMANE ◽  
ANOUAR BEN MABROUK ◽  
JAMIL AOUIDI

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector [Formula: see text] of [Formula: see text] functions, with [Formula: see text], we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function [Formula: see text] has exactly a given value [Formula: see text] of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of [Formula: see text]. We will prove an upper bound valid for any vector of uniform Hölder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.


2009 ◽  
Vol 52 (1) ◽  
pp. 179-194 ◽  
Author(s):  
L. OLSEN

AbstractTwo of the main objects of study in multifractal analysis of measures are the coarse multifractal spectra and the Rényi dimensions. In the 1980s it was conjectured in the physics literature that for ‘good’ measures the following result, relating the coarse multifractal spectra to the Legendre transform of the Rényi dimensions, holds, namely This result is known as the multifractal formalism and has now been verified for many classes of measures exhibiting some degree of self-similarity. However, it is also well known that there is an abundance of measures not satisfying the multifractal formalism and that, in general, the Legendre transforms of the Rényi dimensions provide only upper bounds for the coarse multifractal spectra. The purpose of this paper is to prove that even though the multifractal formalism fails in general, it is nevertheless true that all measures (satisfying a mild regularity condition) satisfy the inverse of the multifractal formalism, namely


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