Improved Triangle Box-Counting Method for Fractal Dimension Estimation

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.

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Study on the Fractal Dimension and Growth Time of the Electrical Treeing Degradation at Different Temperature and Moisture

Advances in Materials Science and Engineering ◽

10.1155/2018/6019269 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10

Author(s):

Youping Fan ◽

Dai Zhang ◽

Jingjiao Li

Keyword(s):

Fractal Dimension ◽

Tree Growth ◽

Fractal Dimensions ◽

Moisture Level ◽

Growth Time ◽

Counting Method ◽

Box Counting ◽

Electrical Trees ◽

Box Counting Method ◽

Electrical Treeing

The paper aims to understand how the fractal dimension and growth time of electrical trees change with temperature and moisture. The fractal dimension of final electrical trees was estimated using 2-D box-counting method. Four groups of electrical trees were grown at variable moisture and temperature. The relation between growth time and fractal dimension of electrical trees were summarized. The results indicate the final electrical trees can have similar fractal dimensions via similar tree growth time at different combinations of moisture level and temperature conditions.

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A BOX-COUNTING METHOD TO CHARACTERIZE DEGREES OF FOLIAGE CLUMPING USING AIRBORNE AND SIMULATED LIDAR DATA

ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences ◽

10.5194/isprsarchives-xl-7-w3-1325-2015 ◽

2015 ◽

Vol XL-7/W3 ◽

pp. 1325-1331

Author(s):

M. van Leeuwen ◽

J. A. N. van Aardt ◽

T. Kampe ◽

K. Krause

Keyword(s):

Fractal Dimension ◽

Leaf Area ◽

Optical Methods ◽

Lidar Data ◽

Counting Method ◽

Area Index ◽

Cloud Data ◽

Box Counting ◽

Box Counting Method ◽

Foliage Clumping

Monitoring forest productivity and health is key to sustainable ecosystem management and informed decision making. A key parameter used in monitoring forest resources is the leaf area index (LAI), which is defined as the one-sided leaf area per unit ground area and is used to describe the canopy radiation regime, among other forest biophysical dynamics. Traditional optics-based methods to estimate LAI rely on the measurement of canopy transmission and foliage clumping. Extending optical methods to LiDAR data has been challenging and studies have reported effective LAI assessments, with no further quantification of foliage clumping. This study investigates the use of the box-counting method to assess the fractal dimension of point cloud data for contrasting forest types and along a gradient of foliage dispersal. We demonstrate the box-counting method on simulated ‘range-to-hit’, as well as acquired airborne discrete LiDAR data. Coherent results obtained from the different test cases hint at the potential of the box-counting fractal dimension to characterize foliage clumping and bode well for the use of clumping assessments in support of airborne, wall-to-wall estimates of LAI.

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Fractal Characterization of Earthquake occurrences in Nigeria

NIGERIAN ANNALS OF PURE AND APPLIED SCIENCES ◽

10.46912/napas.50 ◽

2019 ◽

Vol 1 ◽

pp. 281-287

Author(s):

N N Abdulsalam ◽

O Ologe

Keyword(s):

Fractal Dimension ◽

Active Zone ◽

Hazard Analysis ◽

B Value ◽

Google Earth ◽

Counting Method ◽

Microsoft Word ◽

Box Counting ◽

Box Counting Method

Fractal characterization of Earthquake occurrences in Nigeria was carried out in order to know the b-value of tremor occurrences in the country. This will help in hazard analysis and research in the geological and geophysical structures of Nigeria. The method used in determining the b-value is the box counting method, but for simplicity, we used circle. The areas that are tremor prone were posted on a digitized Nigeria map using Google earth and Surfer 7.0 software. The computation with the box counting method was performed with picked radius of the circle from 50km - 350km and the average number of points that falls within each circle were recorded. The graph of log r (the logarithms of radius of circle or scale) against log <N> (logarithms of average number of points) was plotted using grapher and excels Microsoft word and the slope of the graph was determined. The determined slope gave the fractal dimension and the b-value was thus calculated. In this work, a b-value of 0.6 was obtained indicating that Nigeria falls within seismically less active zone.

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Binarization methods and investigation of their influence on the fractal dimension of functional coatings

Modern Problems of Metalurgy ◽

10.34185/1991-7848.2020.01.04 ◽

2020 ◽

pp. 30-42

Author(s):

Anna Zhurba ◽

Michail Gasik

Keyword(s):

Fractal Dimension ◽

Fractal Analysis ◽

Quantitative Characteristic ◽

Image Binarization ◽

Counting Method ◽

Functional Coatings ◽

Binary Images ◽

Box Counting ◽

Box Counting Method ◽

Binarization Threshold

An essential element of fractal analysis of functional coatings is the fractal dimension, which is an important quantitative characteristic. Typically, coating images are represented as colored or halftone, and most fractal dimension algorithms are for binary images. Therefore, an important step in fractal analysis is binarization, which is a threshold separation operation and the result of which is a binary image.The purpose of the study is to study and program the methods of image binarization and to study the influence of these methods on the value of fractal dimension of functional coatings.As a result of the binarization threshold, the image is split into two regions, one containing all pixels with values below a certain threshold and the other containing all pixels with values above that threshold. Of great importance is the determination of the binarization threshold.The study analyzed a number of functional coating images, determined the fractal dimension of the image by the Box Counting method at different binarization thresholds and when applying different binarization methods (binarization with lower and upper threshold, with double restriction, and the average method for determining the optimal binarization threshold) images. The Box Counting method is used to depict any structure on a plane. This method allows us to determine the fractal dimension of not strictly self-similar objects. Each image binarization method is used for different types of images and for solving different problems.As a result, the methods of image binarization were developed and implemented, the fractal dimension of binary images was calculated, and the influence of these methods on the value of fractal dimension of functional coatings was investigated.The surfaces of composite steel structure, metallic porous materials, and natural cave structures are analyzed.

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