A Modified Box-Counting Method to Estimate the Fractal Dimensions

2011 ◽  
Vol 58-60 ◽  
pp. 1756-1761 ◽  
Author(s):  
Jie Xu ◽  
Giusepe Lacidogna
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
The Self ◽  
Whole Body ◽  
Counting Method ◽  
Border Effect ◽  
Self Similarity ◽  
Box Counting ◽  
Self Similar ◽  
Box Counting Method

A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.

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.

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

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.

FRACTAL DIMENSION AND LACUNARITY OF TRACTOGRAPHY IMAGES OF THE HUMAN BRAIN

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 181-189 ◽  
Author(s):  
P. KATSALOULIS ◽  
D. A. VERGANELAKIS ◽  
A. PROVATA
Keyword(s):  
Fractal Dimension ◽  
Human Brain ◽  
Fractal Dimensions ◽  
Resonance Imaging ◽  
Self Similarity ◽  
Healthy Human ◽  
Box Counting ◽  
Self Similar ◽  
Box Method ◽  
The Brain

Tractography images produced by Magnetic Resonance Imaging scans have been used to calculate the topology of the neuron tracts in the human brain. This technique gives neuroanatomical details, limited by the system resolution properties. In the observed scales the images demonstrated the statistical self-similar structure of the neuron axons and its fractal dimensions were estimated using the classic Box Counting technique. To assess the degree of clustering in the neural tracts network, lacunarity was calculated using the Gliding Box method. The two-dimensional tractography images were taken from four subjects using various angles and different parts in the brain. The results demonstrated that the average estimated fractal dimension of tractography images is approximately Df = 1.60 with standard deviation 0.11 for healthy human-brain tissues, and it presents statistical self-similarity features similar to many other biological root-like structures.

scholarly journals FRACTAL DIMENSION AND SELF-SIMILARITY IN ASPARAGUS PLUMOSUS

Fractals ◽  
2002 ◽  
Vol 10 (04) ◽  
pp. 429-434 ◽  
Author(s):  
J. R. CASTREJÓN PITA ◽  
A. SARMIENTO GALÁN ◽  
R. CASTREJÓN GARCÍA
Keyword(s):  
Fractal Dimension ◽  
Counting Method ◽  
Self Similarity ◽  
Box Counting ◽  
Box Counting Method

We measure the fractal dimension of an African plant that is widely cultivated as an ornamental – the Asparagus plumosus. This plant presents self-similarity, remarkable in at least two different scalings. In the following, we present the results obtained by analyzing this plant via the box counting method for three different scalings. We show in a quantitative way that this species is a fractal.

AGGREGATION OF CRYSTALLINE COPPER SULPHATE SALT IN A GELATIN MEDIUM

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750038 ◽  
Author(s):  
SUBARNAREKHA BHATTACHARYYA ◽  
BISWAJIT ROY ◽  
SUJATA TARAFDAR
Keyword(s):  
Fractal Dimension ◽  
Salt Solution ◽  
Growth Pattern ◽  
Copper Sulphate ◽  
The Self ◽  
Counting Method ◽  
Morphological Pattern ◽  
Box Counting ◽  
Pattern Forming ◽  
Box Counting Method

A drying droplet changes its morphological pattern depending upon complex pattern forming system. To control the distribution of solute particles in a droplet during drying is an important aspect in many scientific and industrial purposes. In this work, with the help of optical microscopy, we study characteristic patterns generated in dried drops of colloidal copper sulphate (CuSO[Formula: see text][Formula: see text]5H2O) solution on surface of glass. At lower concentration of salt solution the growth pattern follows a monofractal structure whereas at higher concentration, the self-assembled pattern gradually gets disappeared. Calculating the fractal dimension (FD) of the generated patterns by box counting method with help of imageJ, it is observed that the patterns resemble DLA structure through a specific range of concentration of the salt solution.

GOODNESS OF FIT ON A MODIFIED RICHARDSON PLOT BY RESIDUAL ANALYSIS

Fractals ◽  
2000 ◽  
Vol 08 (03) ◽  
pp. 261-265
Author(s):  
IAN H. PARKINSON ◽  
NIC L. FAZZALARI
Keyword(s):  
Fractal Dimension ◽  
Standard Deviation ◽  
Trabecular Bone ◽  
Serial Correlation ◽  
Goodness Of Fit ◽  
Fractal Dimensions ◽  
Accurate Estimation ◽  
Counting Method ◽  
Box Counting ◽  
Box Counting Method

Modified Richardson plots obtained by a box counting method on outlines of trabecular bone were tested for linearity. The degree of deviation from true linearity was quantified. The results showed that although there was evidence of nonlinearity or serial correlation in the Richardson plots, the magnitude of deviation from true linearity was less than 0.3% for the residuals and less than 4% for the standard deviation of the residuals. This study shows that the modified box counting method for estimating overall fractal dimension or sectional fractal dimensions of trabecular bone is efficacious. The low magnitude of deviation from linearity confirms that over a defined range of scale the Richardson plot provides an accurate estimation of the fractal dimension of trabecular bone.

On the Fractal Characters of Seawater Temperature in the Kuroshio

2011 ◽  
Vol 403-408 ◽  
pp. 2931-2935
Author(s):  
Yan Xia Zhou ◽  
Mei Han ◽  
Liang Long Da
Keyword(s):  
Fractal Dimension ◽  
Seawater Temperature ◽  
Sound Field ◽  
Fractal Dimensions ◽  
Three Dimension ◽  
Counting Method ◽  
Acoustic Environment ◽  
Box Counting ◽  
Box Counting Method ◽  
The Kuroshio

The Kurshio can affect sonar detection notably. The software named Gis ArcView is used to identify the Kuroshio area by analyzing underwater acoustics data. And seawater temperature isolines are drawn based on the temperature data which are distributed at some intervals in terms of latitude and longitude. The box counting method is applied to calculating the fractal dimension of the Kuroshio seawater temperature isolines in different seasons at the depth of 60m and 150m.The results indicate that the fractal dimension at 150m deep is bigger than that at 60m, but the former fluctuates less than the latter.It is conformed to the facts about the Kuroshio. Acoustic wave is the best medium through which acoustic information can be well transmitted. In the Kurshio area where underwater acoustic environment is rather complex, many singular areas are formed in the three-dimension sound field, in which sonar detection and tracking will be affected to a great extent. However, with the development of remote sensing technologies, it is possible to observe and forecast such mid-scale phenomena as the Kurshio, ocean fronts and interwaves. Therefore, how to effectively extract and analyze the main characters of these phenomena is quite important for Navies. Some people still get used to judging Kurshio’s existence and even intensity as well by simple criteria for a long time. For example, when the exchange of seawater temperature is exceeds 0.1°C/n mile, it will be considered that an ocean front exists. This criterion in fact is so ambiguous not only in this regard, but also in describing the Kurshio’s intensity[1]. In vies of this, the author operates the software, GIS ArcView, to extract and analyze the Kurshio, and take the box counting method[2] to calculate out the fractal dimensions in the area between 20º~30º N and 120 º~130ºE respectively at the depths of 60m and 150m.

scholarly journals PENDETEKSIAN CITRA DAUN TANAMAN MENGGUNAKAN METODE BOX COUNTING

2020 ◽  
Vol 20 (1) ◽  
pp. 35
Author(s):  
Novita Anggraini Juwitarty ◽  
Kosala Dwidja Purnomo ◽  
Kiswara Agung Santoso
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Plant Leaves ◽  
Counting Method ◽  
Image Detection ◽  
Box Counting ◽  
Irregular Shapes ◽  
Different Types ◽  
Box Counting Method ◽  
Leaf Part

Different types of plants make identification difficult. Therefore, we need a system that can identify the similarity of the leaves based on a reference leaf. Extraction can be done by taking one part of the plant and the most easily obtained part is the leaf part. Natural objects such as leaves have irregular shapes and are difficult to measure, but this can be overcome by using fractal dimensions. In this research, image detection of plant leaves will be carried out using the box counting method. The box counting method is a method of calculating fractal dimensions by dividing images into small boxes in various sizes. Image detection using fractal dimension values, we know which leaves the match with the reference. In this study,10 species of leave were tested, with each species 10 samples of plant leaves. Image testing of plant leaves uses a variety of r box size, namely 1/2 ,1/4 , 1/8 , 1/16 ,1/32 , 1/64 , 128which obtain an average match accuracy of 44%. Keywords: Box Counting, Fractal dimension

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

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