scholarly journals A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

Fractals ◽  
2021 ◽  
pp. 2240001
Author(s):  
ANOUAR BEN MABROUK ◽  
ADEL FARHAT
Keyword(s):  
Large Class ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Probability Measures ◽  
Original Formulation ◽  
Multifractal Formalism ◽  
Self Similar ◽  
Joint Multifractal Analysis ◽  
Vector Valued ◽  
Special Classes

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.

Conditional and Relative Multifractal Spectra

Fractals ◽  
1997 ◽  
Vol 05 (01) ◽  
pp. 153-168 ◽  
Author(s):  
Rudolf H. Riedi ◽  
Istvan Scheuring
Keyword(s):  
Multifractal Analysis ◽  
Mutual Influence ◽  
Multifractal Spectrum ◽  
Real Data ◽  
The Novel ◽  
Multifractal Formalism ◽  
Geometrical Properties ◽  
Single Mechanism ◽  
Novel Approach ◽  
Self Similar

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.

scholarly journals WAVELET-BASED MULTIFRACTAL FORMALISM TO ASSIST IN DIAGNOSIS IN DIGITIZED MAMMOGRAMS

2011 ◽  
Vol 20 (3) ◽  
pp. 169 ◽  
Author(s):  
Pierre Kestener ◽  
Jean Marc Lina ◽  
Philippe Saint-Jean ◽  
Alain Arneodo
Keyword(s):  
Wavelet Transform ◽  
Hurst Exponent ◽  
Multifractal Analysis ◽  
Computer Assisted ◽  
Scaling Properties ◽  
Multifractal Formalism ◽  
Long Range Correlations ◽  
Multifractal Scaling ◽  
Self Similar ◽  
Assisted Diagnosis

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.

A WAVELET MULTIFRACTAL FORMALISM FOR SIMULTANEOUS SINGULARITIES OF FUNCTIONS

2013 ◽  
Vol 12 (01) ◽  
pp. 1450009 ◽  
Author(s):  
JAMIL AOUIDI ◽  
ANOUAR BEN MABROUK
Keyword(s):  
Multifractal Analysis ◽  
Multifractal Formalism ◽  
Self Similar

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.

MULTIFRACTAL ANALYSIS OF SOME WEIGHTED QUASI-SELF-SIMILAR FUNCTIONS

2011 ◽  
Vol 09 (06) ◽  
pp. 965-987 ◽  
Author(s):  
JAMIL AOUIDI ◽  
ANOUAR BEN MABROUK
Keyword(s):  
Finite Number ◽  
Multifractal Analysis ◽  
Similar Case ◽  
Gibbs Measures ◽  
Box Dimension ◽  
Multifractal Formalism ◽  
Spectrum Of Singularities ◽  
Self Similar

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.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

VARIATIONAL PRINCIPLE FOR SUBADDITIVE SEQUENCE OF POTENTIALS IN BUNDLE RANDOM DYNAMICAL SYSTEMS

2009 ◽  
Vol 09 (02) ◽  
pp. 205-215 ◽  
Author(s):  
XIANFENG MA ◽  
ERCAI CHEN
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Set Up

The topological pressure is defined for subadditive sequence of potentials in bundle random dynamical systems. A variational principle for the topological pressure is set up in a very weak condition. The result may have some applications in the study of multifractal analysis for random version of nonconformal dynamical systems.

Joint Multifractal Analysis of Crop Yield and Terrain Slope

2000 ◽  
Vol 92 (6) ◽  
pp. 1279-1290 ◽  
Author(s):  
Alexandra N. Kravchenko ◽  
Donald G. Bullock ◽  
Charles W. Boast
Keyword(s):  
Crop Yield ◽  
Multifractal Analysis ◽  
Terrain Slope ◽  
Joint Multifractal Analysis

scholarly journals Lattice self-similar sets on the real line are not Minkowski measurable

2018 ◽  
Vol 40 (1) ◽  
pp. 221-232
Author(s):  
SABRINA KOMBRINK ◽  
STEFFEN WINTER
Keyword(s):  
Real Line ◽  
Iterated Function System ◽  
Function System ◽  
Open Set Condition ◽  
The Real ◽  
Open Set ◽  
Puzzle Piece ◽  
Iterated Function ◽  
Self Similar ◽  
Special Classes

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

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