A Fractal Analysis of Human Cranial Sutures

2003 ◽  
Vol 40 (4) ◽  
pp. 409-415 ◽  
Author(s):  
Jack C. Yu ◽  
Ronald L. Wright ◽  
Matthew A. Williamson ◽  
James P. Braselton ◽  
Martha L. Abell
Keyword(s):  
Fractal Analysis ◽  
Cranial Sutures ◽  
Box Dimension ◽  
Counting Method ◽  
Scale Invariant ◽  
Box Counting ◽  
Invariant Features ◽  
Iteration Functions ◽  
Self Similar ◽  
Box Counting Method

Objectives Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). Design One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. Results The log-log plots of the number of boxes containing the sutural pattern, Nr, and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, Db, was 1.29289 ± 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. Conclusions Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such Db. This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log Nr = 1.29 log 1/r, where Nr is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.

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

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.

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%.

scholarly journals Fractal dimension of external linear contour of human cerebellum (magnetic resonance imaging study)

2021 ◽  
Vol 27 (2) ◽  
pp. 16-22
Author(s):  
N.I. Maryenko ◽  
O.Y. Stepanenko
Keyword(s):  
Fractal Dimension ◽  
Magnetic Resonance ◽  
Fractal Analysis ◽  
Spatial Configuration ◽  
Magnetic Resonance Images ◽  
Counting Method ◽  
Average Value ◽  
Box Counting ◽  
Human Cerebellum ◽  
Box Counting Method

Fractal analysis is a method of mathematical analysis, which provides quantitative assessment of the spatial configuration complexity of the anatomical structures and may be used as a morphometric method. The purpose of the study was to determine the values of the fractal dimension of the outer linear contour of human cerebellum by studying the magnetic resonance images of the brain using the authors’ modification of the caliper method and compare to the values determined using the box counting method. Brain magnetic resonance images of 30 relatively healthy persons aged 18-30 years (15 men and 15 women) were used in the study. T2-weighted digital magnetic resonance images were studied. The midsagittal MR sections of the cerebellar vermis were investigated. The caliper method in the author’s modification was used for fractal analysis. The average value of the fractal dimension of the linear contour of the cerebellum, determined using the caliper method, was 1.513±0.008 (1.432÷1.600). The average value of the fractal dimension of the linear contour of the cerebellum, determined using the box counting method, was 1.530±0.010 (1.427÷1.647). The average value of the fractal dimension of the cerebellar tissue as a whole, determined using the box counting method, was 1.760±0.006 (1.674÷1.837). The values of the fractal dimension of the outer linear contour of the cerebellum, determined using the caliper method and the box counting method were not statistically significantly different. Therefore, both methods can be used for fractal analysis of the linear contour of the cerebellum. Fractal analysis of the outer linear contour of the cerebellum allows to quantify the complexity of the spatial configuration of the outer surface of the cerebellum, which is difficult to estimate using traditional morphometric methods. The data obtained from this study and the methodology of the caliper method of fractal analysis in the author’s modification can be used for morphometric investigations of the human cerebellum in morphological studies, as well as in assessment of cerebellar MR images for diagnostic purposes.

Effect of Perma-Zyme on Properties of Tabia Used to Protect Ancient Earthen Ruins

2011 ◽  
Vol 243-249 ◽  
pp. 327-334
Author(s):  
Hong Tao Peng ◽  
Qi Zhang ◽  
Nai Sheng Li ◽  
De Fa Wang
Keyword(s):  
Color Difference ◽  
Box Dimension ◽  
Unconfined Compression ◽  
Counting Method ◽  
Sem Images ◽  
Box Counting ◽  
The Difference ◽  
Box Counting Method ◽  
The One ◽  
Sem Microstructure

Perma-Zyme is an enzymatic stabilizer. To test the feasibility of adding Perma-Zyme to tabia used to protect ancient earthen ruins, the 28d unconfined compression strength, permeability and color-difference were tested to determine the difference between the tabia and tabia treated with Perma-Zyme. The experimental results show that the 28d unconfined compression strengths of specimens treated with Perma-Zyme are higher than those without Perma-Zyme, and the color-difference(DE*) between the tabia and tabia mixed Perma-zyme is larger than 3(suggested color tolerance Chinese standardized GB/T 15608-2006), and the permeability coefficient of tabia mixed Perma-Zyme is smaller than that without Perma-Zyme, i.e., has better impermeability. Box dimension values of SEM images(with different scale bars) of samples were computed by box-counting method. Box dimension values of SEM images of the same sample are different on different scale bars. The analysis shows that SEM microstructure of the tabia sample treated with Perma-Zyme is finer and denser than the one without Perma-Zyme.

Fractal analysis of dendrite morphology using modified box-counting method

2014 ◽  
Vol 84 ◽  
pp. 64-67 ◽  
Author(s):  
Dušan Ristanović ◽  
Bratislav D. Stefanović ◽  
Nela Puškaš
Keyword(s):  
Fractal Analysis ◽  
Counting Method ◽  
Dendrite Morphology ◽  
Box Counting ◽  
Box Counting Method

scholarly journals ADAPTED BOX-COUNTING METHOD FOR ASSESSMENT OF THE FRACTAL DIMENSION OF FINANCIAL TIME SERIES

2020 ◽  
pp. 185-218
Author(s):  
Robert Garafutdinov ◽  
◽  
Sofya Akhunyanova ◽  
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Fractal Analysis ◽  
Computer Image ◽  
Financial Time Series ◽  
Counting Method ◽  
Financial Time ◽  
Box Counting ◽  
Modeling And Prediction ◽  
Box Counting Method

This paper continues research within the framework of the scientific direction in econophysics at the Department of Information Systems and Mathematical Methods in Economics of the faculty of Economics of PSU. Modeling and prediction of financial time series is quite a perspective area of research, because it allows participants of financial processes to reduce risks and make effective decisions. For example, we could research financial processes with the help of fractal analysis. In the article there is studied and worked out in detail one of the methods of fractal analysis of financial time series – the box-counting method for assessment of the fractal dimension. This method is often used in studies conducted by domestic authors, but the authors do not delve into the characteristics and problems of using the box-counting method for analysis of time series, that means that the answers to the interested questions have not yet been given. The main problem is that, as a rule, the analyzed object in the tasks of applying the box-counting method to time series is a computer image of the plot of series. In the article there is proposed the procedure of adaptation of the box-counting method for assessment of the fractal dimension of time series, the procedure does not require the formation of a computer image of the plot. In the article there is considered following difficulties developed from this adaptation: 1) high sensitivity of the resulting estimation of the dimension to the input parameters of the method (the ratio of the sides of the covered by cells plane with the plot; the used range of lengths of the side of the cell; the number of partitions of the plane into cells); 2) the non-obviousness of choosing the optimal values ​​of these parameters. In the article there are analyzed approaches to the selection of these parameters that were proposed by other authors, and there are determined the most suitable approaches for the adapted box-counting method. Also there are developed unique methods for determining the ratio of the sides of the plane with the plot. In the paper there is written the computer program that implements the developed method, and this program is tested on the generated data. The study obtained the following results. The fact of sensitivity of the adapted box-counting method to input parameters is confirmed, that indicates the high importance of the correct choice of these parameters. According to the study, there is found out inability of the proposed methods of automatic determination the ratio of the sides of the plane in relation to artificial time series. There are obtained the most precise (in a statistical sense) estimates of fractal dimension, those found by means of the adapted box-counting method, with the fixed ratio of the sides 1:1. According to comparing the adapted box-counting method and R/S analysis, there are obtained the most precise estimates by the second method (R/S analysis). Finally in the paper there are formulated the possible directions for further research: 1) comparison of the accuracy of various methods for assessment of the fractal dimension on series of different lengths; 2) comparison of the methods of fractal analysis and p-adic analysis for modeling and prediction of financial time series; 3) determination of the conditions of applicability of various methods; 4) approbation of the developed methods for determining of the ratio of the sides of the plane with the plot on real economic data.

Interpretation of sieve analysis data using the box-counting method for gravelly cobbles

2001 ◽  
Vol 38 (6) ◽  
pp. 1201-1212 ◽  
Author(s):  
Zon-Yee Yang ◽  
Jian-Liang Juo
Keyword(s):  
Particle Size ◽  
Fractal Dimension ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Analysis Data ◽  
Box Dimension ◽  
Sieve Analysis ◽  
Counting Method ◽  
Box Counting ◽  
Box Counting Method

In fractal theory, the fractal dimension (D) is a measure of the complexity of particle distribution in nature. It can provide a description of how much space a particle set fills. The box-counting method uses squared grids of various sizes to cover the particles to obtain a box dimension. This sequential counting concept is analogous to the sieve analysis test using stacked sieves. In this paper the box-counting method is applied to describe the particle-size distribution of gravelly cobbles. Three approaches to obtain the fractal dimension are presented. In the first approach the data obtained from a classic laboratory sieve analysis are rearranged into a double-logarithmic plot, according to a fractal model, to obtain the fractal dimension of the particle collection. In addition, an equivalent number of covered grids on each sieve during the sieve analysis are counted to produce the box dimension. According to the box-counting method concept, a photo-sieving technique used in scanning electron microscope microstructure analysis is adopted for use on gravelly cobbles in the field. The box-counting method concept is capable of explaining the sieve analysis data to clarify the information on the particle-size distribution. Using photo-sieving to produce the fractal dimension from field photographs can provide another approach for understanding the particle-size distribution. However, the representative cross-profile should be chosen carefully. The composition of the particle-size distribution for gravelly cobbles with higher D values is more complicated than those at sites with smaller D values.Key words: sieve analysis, box-counting method, fractal dimension, particle-size distribution, gravelly cobbles.

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