Lacunarity and multifractal analysis of the large DLA mass distribution

Rodríguez‐Romo, Suemi; Sosa-Herrera, Antonio

doi:10.1016/j.physa.2013.03.044

Cited by 7 publications

(12 citation statements)

References 47 publications

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“…This observation is incorporated into our model for by means of the formalism of multifractal sets as applied to the mass-distribution of fractal clusters 5,35 . In particular, through the relation between the Rényi entropies, S q , and the generalized dimension, D q 36,40,66 . First, the Rényi entropies are defined as, , where q are the momenta, n is the number of partitions (boxes) of the set, and is the probability of finding an element of the cluster at a spatial observation scale, .…”

Section: Resultsmentioning

confidence: 99%

“…In this analysis it was considered that the mass-distribution of the DLA cluster is self-similar ( for all q ) 5,37–40 . However, even in the event of weak multifractality for and 35,36,40 , these results would still apply to the box-counting or Hausdorff dimension, , and the correlation dimension, , which are the most common dimensions reported for DLA; and even to the information-dimension, given the fact that for sufficiently large clusters, 36,40 .…”

Section: Discussionmentioning

confidence: 99%

“…Although there are diverse works that have rigorously proven the consistency of the two-dimensional DLA within a self-similar picture, with a fractal dimension very close to 11,13,15,20 , there are other results based on multifractal analysis where this is not conclusive. In these studies, the DLA cluster is found to be a weak mass-multifractal, that in terms of the generalized dimension, D q and momenta q , goes from to 35,36 ; while in others, it is a monofractal with for all q 37–40 . Nevertheless, among the most important theoretical results for DLA one has that: first, its fractal dimensions are an exclusive function of the embedding Euclidean space, 34 ; secondly, they must satisfy Kesten’s inequality 17 , and Ball’s inequality 27 , , where the equality holds at (see Fig.…”

Section: Introductionmentioning

confidence: 96%

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A universal dimensionality function for the fractal dimensions of Laplacian growth

Nicolás-Carlock

Carrillo-Estrada

2019

Sci Rep

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Laplacian growth, associated to the diffusion-limited aggregation (DLA) model or the more general dielectric-breakdown model (DBM), is a fundamental out-of-equilibrium process that generates structures with characteristic fractal/non-fractal morphologies. However, despite diverse numerical and theoretical attempts, a data-consistent description of the fractal dimensions of the mass-distributions of these structures has been missing. Here, an analytical model of the fractal dimensions of the DBM and DLA is provided by means of a recently introduced dimensionality equation for the scaling of clusters undergoing a continuous morphological transition. Particularly, this equation relies on an effective information-function dependent on the Euclidean dimension of the embedding-space and the control parameter of the system. Numerical and theoretical approaches are used in order to determine this information-function for both DLA and DBM. In the latter, a connection to the Rényi entropies and generalized dimensions of the cluster is made, showing that DLA could be considered as the point of maximum information-entropy production along the DBM transition. The results are in good agreement with previous theoretical and numerical estimates for two- and three-dimensional DBM, and high-dimensional DLA. Notably, the DBM dimensions conform to a universal description independently of the initial cluster-configuration and the embedding-space.

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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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A universal dimensionality function for the fractal dimensions of Laplacian growth

Nicolás-Carlock

Carrillo-Estrada

2019

Sci Rep

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“…This proves (45). To complete the proof of (42), it suffices now to combine (45) with (41) to see that…”

Section: Asymptotics Of the First Methodsmentioning

confidence: 99%

“…Algorithms for determining lacunarity and related texture parameters have been proposed and analyzed, e.g. in [3,47,4,5] and used successfully in very different fields such as pattern recognition [4], signal processing [32], DNA classification [9], the analysis of aggregation clusters in statistical mechanics [41] and of breast tumors in medicine [43], to mention just a few recent studies. Despite these many positive examples, one can not hope that a single texture parameter like lacunarity will always be able to successfully distinguish or classify fractal structures of a given class.…”

Section: Introductionmentioning

confidence: 99%

Estimation of fractal dimension and fractal curvatures from digital images

Spodarev

Straka

Winter

2015

Chaos, Solitons & Fractals

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Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.

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