Network Box Counting Dimension

2020 ◽  
pp. 131-144
Author(s):  
Eric Rosenberg
Keyword(s):  
Box Counting ◽  
Box Counting Dimension

The dimension of self-affine fractals II

1992 ◽  
Vol 111 (1) ◽  
pp. 169-179 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Exact Expression ◽  
Singular Value ◽  
Compact Set ◽  
Value Functions ◽  
Affine Transformations ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Almost All

AbstractA family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

2021 ◽  
Author(s):  
Nicholas Dudu ◽  
Arturo Rodriguez ◽  
Gael Moran ◽  
Jose Terrazas ◽  
Richard Adansi ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Atmospheric Turbulence ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Counting Method ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Box Counting Method

Abstract Atmospheric turbulence studies indicate the presence of self-similar scaling structures over a range of scales from the inertial outer scale to the dissipative inner scale. A measure of this self-similar structure has been obtained by computing the fractal dimension of images visualizing the turbulence using the widely used box-counting method. If applied blindly, the box-counting method can lead to misleading results in which the edges of the scaling range, corresponding to the upper and lower length scales referred to above are incorporated in an incorrect way. Furthermore, certain structures arising in turbulent flows that are not self-similar can deliver spurious contributions to the box-counting dimension. An appropriately trained Convolutional Neural Network can take account of both the above features in an appropriate way, using as inputs more detailed information than just the number of boxes covering the putative fractal set. To give a particular example, how the shape of clusters of covering boxes covering the object changes with box size could be analyzed. We will create a data set of decaying isotropic turbulence scenarios for atmospheric turbulence using Large-Eddy Simulations (LES) and analyze characteristic structures arising from these. These could include contours of velocity magnitude, as well as of levels of a passive scalar introduced into the simulated flows. We will then identify features of the structures that can be used to train the networks to obtain the most appropriate fractal dimension describing the scaling range, even when this range is of limited extent, down to a minimum of one order of magnitude.

Edge extraction of optical subaperture based on differential box-counting dimension method with window merge replication

2016 ◽  
Vol 55 (9) ◽  
pp. 093107 ◽  
Author(s):  
Yunqi Wang ◽  
Mei Hui ◽  
Ming Liu ◽  
Liquan Dong ◽  
Xiaohua Liu ◽  
...  
Keyword(s):  
Edge Extraction ◽  
Box Counting ◽  
Box Counting Dimension

scholarly journals The Meso-Analysis of the Rock-Burst Debris of Rock Similar Material Based on SEM

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Manqing Lin ◽  
Lan Zhang ◽  
Xiqi Liu ◽  
Yuanyou Xia ◽  
Jiaqi He ◽  
...  
Keyword(s):  
Rock Burst ◽  
Fractal Dimensions ◽  
Similar Material ◽  
Failure Characteristics ◽  
True Triaxial ◽  
Sem Image ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Combined Test ◽  
Experimental Process

In order to explore the specimen failure characteristics during rock-burst under different gradient stress conditions, in this paper, a novel experimental technique was proposed; a common series of tests under two gradient stress paths were conducted on rock similar material specimens using the true-triaxial gradient and hydraulic-pneumatic combined test apparatus. And plaster was used as the rock similar material. In the experimental process, several rock-burst debris with area sizes of 100 mm2 were collected, and the fractal dimensions of typical detrital section crystal contours were analyzed and calculated using a scanning electron microscopy (SEM) method. The results showed that the specimens’ failure characteristics which had been induced by the two gradient stress processes were various. Also, the mesoscopic morphology of the rock-burst detrital section had effectively reflected its macroscopic failure characteristics. It was found that the fractal dimensions of the crystal contours of the specimen’s detrital section had fractal characteristics, and the box-counting dimension based on the SEM image could quantitatively describe the rock-burst failure characteristics. Furthermore, under the same magnification, the fractal dimensions of the crystal contours of the splitting failures were found to be relatively smaller than those of the shearing failures.

scholarly journals Influence of Soil Parameters on Detecting Voids behind a Tunnel Lining Using an Impact Echo Method

2019 ◽  
Vol 9 (24) ◽  
pp. 5403
Author(s):  
Meng Ma ◽  
Rongning Cao ◽  
Chao Niu ◽  
Hougui Zhang ◽  
Weining Liu
Keyword(s):  
Element Model ◽  
Tunnel Lining ◽  
Parameter Analysis ◽  
Soil Parameters ◽  
Impact Echo ◽  
Time Frequency ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Frequency Domains ◽  
The Impact

Tunnel lining cavities are a common problem that may affect the bearing capacity of the tunnel-supporting structure, as well as the tunnel service life. The impact echo (IE) method can be used to detect voids behind tunnel linings. For a long tunnel, the surrounding rocks/soils are inhomogeneous and anisotropic, with parameters that vary with tunnel mileages. It is interesting to analyse whether alterations of the soil parameters affect the non-destructive test results. A laboratory experiment was performed in this study, in which voids behind a concrete plate, representing the tunnel lining, were designed to model the ineffective contact between the soil and the tunnel. The IE method was employed to inspect the existence of the void using different signal analysis approaches in the time, frequency and time–frequency domains. Furthermore, the fractal box-counting dimension was calculated for the purpose of quantitative evaluation. Different soil parameters and void sizes were considered, and finally, a finite element model was built and parameter analysis was accomplished using the software ABAQUS. The results demonstrated that: (1) A comprehensive analysis of vibration signals in the time, frequency and time–frequency domains was useful for identifying voids, while the box-counting dimension was useful for evaluating voids quantitatively. (2) Soils with large density and Young’s modulus differences had a certain influence on void detection, while those with large water content and Poisson’s ratio differences had little influence. (3) The box-counting dimension value was stable within the area where the void existed behind the tunnel; when the detection point was beyond twice that of the void dimension, it was difficult to locate the void.

ON THE MENDÈS FRANCE FRACTAL DIMENSION OF STRANGE ATTRACTORS: THE HÉNON CASE

2002 ◽  
Vol 12 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
M. PIACQUADIO ◽  
R. HANSEN ◽  
F. PONTA
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Strange Attractors ◽  
Planar Curves ◽  
Box Counting ◽  
Box Counting Dimension

We consider the Hénon attractor ℋ as a curve limit of continuous planar curves H(n), as n → ∞. We describe a set of tools for studying the Hausdorff dimension dim H of a certain family of such curves, and we adapt these tools to the particular case of the Hénon attractor, estimating its Hausdorff dimension dim H (ℋ) to be about 1.258, a number smaller than the usual estimates for the box-counting dimension of the attractor. We interpret this discrepancy.

Canonical number systems, counting automata and fractals

2002 ◽  
Vol 133 (1) ◽  
pp. 163-182 ◽  
Author(s):  
KLAUS SCHEICHER ◽  
JÖRG M. THUSWALDNER
Keyword(s):  
Direct Methods ◽  
Number Fields ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Box Counting ◽  
Number Systems ◽  
Rings Of Integers ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Affine Set

In this paper we study properties of the fundamental domain [Fscr ]β of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since [Fscr ]β defines a tiling of the n-dimensional vector space, we ask, which tiles of this tiling ‘touch’ [Fscr ]β. It turns out that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above-mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of [Fscr ]β. Since this boundary is a so-called graph-directed self-affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

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