Multifractals

Harte, David

doi:10.1201/9781420036008

Cited by 234 publications

(81 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, M q ∼ V q−1 . However, we consider singular distributions such that their q-moments are non-trivial power laws of V in the continuum limit, namely, distributions such that one can define [19] the exponents…”

Section: Counts In Cells and Coarse Multifractal Analysismentioning

confidence: 99%

See 1 more Smart Citation

Fractal analysis of the dark matter and gas distributions in the Mare-Nostrum universe

Gaite¹

2010

J. Cosmol. Astropart. Phys.

View full text Add to dashboard Cite

We develop a method of multifractal analysis of N -body cosmological simulations that improves on the customary counts-in-cells method by taking special care of the effects of discreteness and large scale homogeneity. The analysis of the Mare-Nostrum simulation with our method provides strong evidence of self-similar multifractal distributions of dark matter and gas, with a halo mass function that is of Press-Schechter type but has a power-law exponent −2, as corresponds to a multifractal. Furthermore, our analysis shows that the dark matter and gas distributions are indistinguishable as multifractals. To determine if there is any gas biasing, we calculate the cross-correlation coefficient, with negative but inconclusive results. Hence, we develop an effective Bayesian analysis connected with information theory, which clearly demonstrates that the gas is biased in a long range of scales, up to the scale of homogeneity. However, entropic measures related to the Bayesian analysis show that this gas bias is small (in a precise sense) and is such that the fractal singularities of both distributions coincide and are identical. We conclude that this common multifractal cosmic web structure is determined by the dynamics and is independent of the initial conditions.

show abstract

Section: Counts In Cells and Coarse Multifractal Analysismentioning

confidence: 99%

“…Besides the multifractal spectrum f (α), it is useful to define the spectrum of Rényi dimensions [19] …”

Section: Counts In Cells and Coarse Multifractal Analysismentioning

confidence: 99%

Fractal analysis of the dark matter and gas distributions in the Mare-Nostrum universe

Gaite¹

2010

J. Cosmol. Astropart. Phys.

View full text Add to dashboard Cite

show abstract

“…One approach to multifractal analysis of a measure on R N involves generalised q-dimensions; see [8,11,12,19,23] for various treatments. The generalised dimensions of a finite Borel measure τ of bounded support may be defined along the lines of box-counting dimension using r-mesh cubes, that is cubes in R N of the form [j 1 r, (j 1 + 1)r) × · · · × [j n r, (j n + 1)r) where j 1 , .…”

Section: Generalised Q-dimensionsmentioning

confidence: 99%

Generalized dimensions of measures on almost self-affine sets

Falconer

2010

Nonlinearity

View full text Add to dashboard Cite

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q > 1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of 'multienergy' integrals which we then bound using induction on families of trees.

show abstract

“…18 One of such advantages is that the technique here presented gathers information from frequency domain inherently, allowing the capturing of details and patterns which escapes from the conventional spatial analysis. Besides, the space used allows the expression of colors and spatial distribution of pixels as being a related entity.…”

Section: Introductionmentioning

confidence: 99%

Fractal descriptors in the Fourier domain applied to color texture analysis

Florindo¹,

Bruno²

2011

Chaos

View full text Add to dashboard Cite

A wavelet transform algorithm for peak detection and application to powder x-ray diffraction data Rev. Sci. Instrum. 82, 015105 (2011) Support vector machine-based feature extractor for L/H transitions in JET Rev. Sci. Instrum. 81, 10E123 (2010) Determining the absorption tolerance of single chromophore photodiodes for machine vision Appl. Phys. Lett. 96, 253303 (2010) Determining the absorption tolerance of single chromophore photodiodes for machine vision APL: Org. Electron. Photonics 3, 123 (2010) The present work proposes the development of a novel method to provide descriptors for colored texture images. The method consists of two steps. First, we apply a linear transform in the color space of the image aiming at highlighting spatial structuring relations among the color of pixels. Second, we apply a multiscale approach to the calculus of fractal dimension based on Fourier transform. From this multiscale operation, we extract the descriptors that are used to discriminate the texture represented in digital images. The accuracy of the method is verified in the classification of two color texture datasets, by comparing the performance of the proposed technique to other classical and state-of-the-art methods for color texture analysis. The results showed an advantage of almost 3% of the proposed technique over the second best approach. Fractal objects constitute a particular category of nonlinear dynamic system. Essentially, fractals are geometric objects which do not obey the classical Euclidian rules. Besides, they are characterized by the self-similarity, that is, we observe some geometrical and/or statistical patterns which repeat themselves along different observation scales. In a true fractal, the self-similarity is observed at infinite observation scales. The inherent composition rules of fractals yield to some interesting characteristics. For instance, fractals present non-proportionality between cause and effect. Still they present infinite complexity (complexity in the sense of observed details) and they show a high dependence level from initial conditions. Such characteristics allow for fractals are also to be considered as a chaotic system. This physical interpretation collaborated for authors, such as Mandelbrot, 1 to suggest the use of fractals to model objects found in the nature. More recently, Manoel et al. 2 developed a method for using fractal theory in the extraction of features from natural objects represented in a digital image. This is an important problem in computational vision. Bruno et al. 3 applied the technique to a shape recognition problem. Here, we propose the development and study of a novel fractal-based method for the analysis of color texture. We verify the efficiency of the proposed approach in the solution of a color texture classification problem. Additional information on Chaos

show abstract

Multifractals

Cited by 234 publications

References 0 publications

Fractal analysis of the dark matter and gas distributions in the Mare-Nostrum universe

Fractal analysis of the dark matter and gas distributions in the Mare-Nostrum universe

Generalized dimensions of measures on almost self-affine sets

Fractal descriptors in the Fourier domain applied to color texture analysis

Contact Info

Product

Resources

About