scholarly journals Fractal analysis of images in medicine and morphology: basic principles and methodologies

Morphologia ◽  
2021 ◽  
Vol 15 (3) ◽  
pp. 196-206
Author(s):  
N.I. Maryenko ◽  
O.Yu. Stepanenko
Keyword(s):  
Fractal Analysis ◽  
Spatial Configuration ◽  
Linear Segment ◽  
Image Size ◽  
Fractal Measure ◽  
Box Counting ◽  
Morphological Studies ◽  
Fractal Measures ◽  
Minimum Number ◽  
Box Counting Method

Background. Fractal analysis is an informative and objective method of mathematical analysis that can complement existing methods of morphometry and provides a comprehensive quantitative assessment of the spatial configuration of irregular anatomical structures. Objective: a comparative analysis of fractal analysis methods used for morphometry in biomedical research. Methods. A comprehensive analysis of morphological studies, based on fractal analysis. Results. Different types of medical images with different preprocessing algorithms can be used for fractal analysis. The parameter determined by fractal analysis is the fractal dimension, which is a measure of the complexity of the spatial configuration and the degree of filling of space with a certain geometric object. The most known methods of fractal analysis are the following: box counting, caliper, pixel dilation, "mass-radius", cumulative intersection, grid intercept. The box counting method and its modifications is the most commonly used method due to the simplicity and versatility. Different methods of fractal analysis have a similar principle: fractal measures (different geometric figures) of a certain size completely cover the structure in the image, size of fractal measure is iteratively changed, and the minimum number of fractal measures covering the structure is calculated. Methods of fractal analysis differ in the type of fractal measure, which can be a linear segment, a square of a fractal grid, a cube, a circle, a sphere etc. Conclusion. The choice of the method of fractal analysis and image preprocessing method depends on the studied structure, features of its spatial configuration, the type of image used for the analysis, and the aim of the study.

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.

scholarly journals FRACTAL ANALYSIS OF TRABECULAR BONE: A STANDARDISED METHODOLOGY

2011 ◽  
Vol 19 (1) ◽  
pp. 45 ◽  
Author(s):  
Ian Parkinson ◽  
Nick Fazzalari
Keyword(s):  
Fractal Dimension ◽  
Trabecular Bone ◽  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Box Counting ◽  
Cell Tissue ◽  
Image Analyser ◽  
Model Independent ◽  
Box Counting Method ◽  
Image Orientation

A standardised methodology for the fractal analysis of histological sections of trabecular bone has been established. A modified box counting method has been developed for use on a PC based image analyser (Quantimet 500MC, Leica Cambridge). The effect of image analyser settings, magnification, image orientation and threshold levels, was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to objectively calculate more than one fractal dimension from the modified Richardson plot. The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (λ<25 μm) and the upper limit is 4250 μm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0. It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than 1 fractal dimension, describing spatial structural entities. Fractal analysis is a model independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and compliments conventional histomorphometric and stereological techniques.

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 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 ANALYSIS OF DENDRITIC ARBORISATION PATTERNS OF STALKED AND ISLET NEURONS IN SUBSTANTIA GELATINOSA OF DIFFERENT SPECIES

Fractals ◽  
2007 ◽  
Vol 15 (01) ◽  
pp. 1-7 ◽  
Author(s):  
NEBOJŠA T. MILOŠEVIĆ ◽  
DUŠAN RISTANOVIĆ ◽  
JOVAN B. STANKOVIĆ ◽  
RADMILA GUDOVIĆ
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Islet Cells ◽  
Fractal Dimensions ◽  
Substantia Gelatinosa ◽  
Neuronal Populations ◽  
Box Counting ◽  
Morphological Descriptor ◽  
The Mean ◽  
Box Counting Method

Through analysis of the morphology of dendritic arborisation of neurons from the substantia gelatinosa of dorsal horns from four different species, we have established that two types of cells (stalked and islet) are always present. The aim of the study was to perform the intra- and/or inter-species comparison of these two neuronal populations by fractal analysis, as well as to clarify the importance of the fractal dimension as an objective and usable morphological parameter. Fractal analysis was carried out adopting the box-counting method. We have shown that the mean fractal dimensions for the stalked cells are significantly different between species. The same is true for the mean fractal dimensions of the islet cells. Still, no significant differences were found for the fractal dimensions of the stalked and islet cells within a particular species. The human species has shown as the only exception where fractal dimensions of these two types of cells differ significantly. This study shows once more that the fractal dimension is a useful and sensitive morphological descriptor of neuronal structures and differences between them.

scholarly journals Use of fractal analysis in dental images: a systematic review

2020 ◽  
Vol 49 (2) ◽  
pp. 20180457 ◽  
Author(s):  
Camila NAO Kato ◽  
Sâmila G Barra ◽  
Núbia PK Tavares ◽  
Tânia MP Amaral ◽  
Cláudia B Brasileiro ◽  
...  
Keyword(s):  
Fractal Analysis ◽  
English Language ◽  
Bone Structure ◽  
Cone Beam Ct ◽  
Cone Beam ◽  
Inclusion Criteria ◽  
Mineral Density ◽  
Box Counting ◽  
Cephalometric Radiograph ◽  
Box Counting Method

Objectives: This study reviewed the use of fractal analysis (FA) in dental images. Methods: A search was performed using PubMed, MEDLINE, LILACS, Web of Science and SCOPUS databases. The inclusion criteria were human studies in the English language, with no date restriction. Results: 78 articles were found in which FA was applied to panoramic radiographs (34), periapical radiographs (21), bitewing radiographs (4), cephalometric radiograph (1), cone beam CT (15), micro-CT (3), sialography (2), and ultrasound (2). Low bone mineral density (21) and systemic or local diseases (22) around the bone of dental implants were the main subjects of the study of FA. Various sizes and sites of the regions of interest were used to evaluate the bone structure. Different ways were used to treat the image and to calculate FA. FA of 43 articles showed significant differences in the comparison of groups, mainly between healthy and sick patients. Conclusions: FA in Dentistry has been widely applied to the study of images. Panoramic and periapical radiographs were those most frequently used. The Image J software and the box-counting method were extensively adopted in the studies reviewed herein. Further studies are encouraged to improve clarification of the parameters that directly influence FA.

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.

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