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.

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.

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 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.

scholarly journals Rock Mechanical Property Influenced by Inhomogeneity

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yunliang Tan ◽  
Dongmei Huang ◽  
Ze Zhang
Keyword(s):  
Mechanical Property ◽  
Fractal Dimension ◽  
Rock Type ◽  
Fractal Dimensions ◽  
Rock Material ◽  
The Self ◽  
Self Similarity ◽  
Red Sandstone ◽  
Variation Patterns ◽  
Rock Microstructure

In order to identify the microstructure inhomogeneity influence on rock mechanical property, SEM scanning test and fractal dimension estimation were adopted. The investigations showed that the self-similarity of rock microstructure markedly changes with the scanned microscale. Different rocks behave in different fractal dimension variation patterns with the scanned magnification, so it is conditional to adopt fractal dimension to describe rock material. Grey diabase and black diabase have high suitability; red sandstone has low suitability. The suitability of fractal-dimension-describing method for rocks depends on both investigating scale and rock type. The homogeneities of grey diabase, black diabase, grey sandstone, and red sandstone are 7.8, 5.7, 4.4, and 3.4, separately; their average fractal dimensions of microstructure are 2.06, 2.03, 1.72, and 1.40 correspondingly, so the homogeneity is well consistent with fractal dimension. For rock material, the stronger brittleness is, the less profile fractal dimension is. In a sense, brittleness is an image of rock inhomogeneity in macroscale, while profile fractal dimension is an image of rock inhomogeneity in microscale. To combine the test of brittleness with the estimation of fractal dimension with condition will be an effective approach for understanding rock failure mechanism, patterns, and behaviours.

Preliminary Evidence for a Theory of the Fractal City

1996 ◽  
Vol 28 (10) ◽  
pp. 1745-1762 ◽  
Author(s):  
M Batty ◽  
Y Xie
Keyword(s):  
Fractal Dimension ◽  
Urban Form ◽  
Scaling Laws ◽  
Fractal Dimensions ◽  
Power Laws ◽  
Preliminary Evidence ◽  
Residential Development ◽  
Self Similarity ◽  
Estimation Techniques ◽  
Data Source

In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States, with an innovative data source from the TIGER/Line files. The results support the theory of the fractal city outlined in books by Batty and Longley and Frankhauser, but with the clear conclusion that different scale and estimation techniques generate different types of fractal dimension.

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 Classification of texture using random box counting and binarization methods

2021 ◽  
Vol 10 (1) ◽  
pp. 533-540
Author(s):  
Wijdan Jaber AL-kubaisy ◽  
Maha Mahmood
Keyword(s):  
Image Processing ◽  
Fractal Dimension ◽  
Blood Cells ◽  
Multifractal Analysis ◽  
Vision System ◽  
White Blood Cells ◽  
Real Life ◽  
Self Similarity ◽  
Box Counting

The heterogeneous texture classifications with the complexity of structures provide variety of possibilities in image processing, as an example of the multifractal analysis features. The task of texture analysis is a highly significant field of study in the field of machine vision. Most of the real-life surfaces exhibit textures and an efficiently modelled vision system must have the ability to deal with this variety of surfaces. A considerable number of surfaces maintain a self-similarity quality as well as statistical roughness at different scales. Fractals could provide a great deal of advantages; also, they are popular in the process of modelling these properties in the tasks related to the field of image processing. With two distinct methods, this paper presents classification of texture using random box counting and binarization methods calculate the estimation measures of the fractal dimension BCM. There methods are the banalization and random selecting boxes. The classification of the white blood cells is presented in this paper based on the texture if it is normal or abnormal with the use of a number of various methods.

scholarly journals The linkage between box-counting and geomorphic fractal dimensions in the fractal structure of river networks: the junction angle

2020 ◽  
Vol 51 (6) ◽  
pp. 1397-1408
Author(s):  
Xianmeng Meng ◽  
Pengju Zhang ◽  
Jing Li ◽  
Chuanming Ma ◽  
Dengfeng Liu
Keyword(s):  
Fractal Dimension ◽  
Fractal Structure ◽  
Fractal Dimensions ◽  
River Networks ◽  
Stream Length ◽  
Box Counting ◽  
Linear Relationships ◽  
Box Counting Dimension ◽  
Junction Angle

Abstract In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.

Abstract 17136: Pulmonary Arterial and Venous Fractal Dimension as a Marker of Pulmonary Arterial Hypertension

Circulation ◽  
2020 ◽  
Vol 142 (Suppl_3) ◽  
Author(s):  
Andrew Tsao ◽  
Pietro Nardelli ◽  
Eileen Harder ◽  
Gonzalo Vegas Sanchez-Ferrero ◽  
James C Ross ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Mean Pulmonary Arterial Pressure ◽  
Box Counting ◽  
Wilcoxon Rank Sum Test ◽  
History Of ◽  
Pulmonary Arterial ◽  
Box Counting Method ◽  
Definition Of ◽  
Vascular Trees

Introduction: PAH is characterized by a loss of pulmonary vascular complexity. In this study, total, arterial, and venous vasculatures of patients with PAH and with ePAH were analyzed using fractal analysis and compared against controls Methods: Data from 1514 consecutive right heart catheterizations from 4/27/2011 to 10/2/2018 representing subjects referred to our dyspnea center were searched for availability of imaging. 388 CT angiography (CTA) scans were identified (used given retrospective availability of thin slice reconstructions). Three initial cohorts (no overlap) were identified from individuals in this set. Control patients had normal resting and exercise hemodynamics and no history of cardiopulmonary disease. The second group met the current definition of PAH (resting mean pulmonary arterial pressure >20mmHg, pulmonary vascular resistance >3 Wood Units, pulmonary capillary wedge pressure <15mmHg). The third group (ePAH) had normal resting hemodynamics but age adjusted evidence of PAH with exercise. Pulmonary vascular trees were reconstructed; total, arterial, and venous trees were separated; and fractal dimensions were measured using a 3D box counting method for each tree. Comparisons were made using the Wilcoxon Rank Sum test (R 3.5). Results: Venous fractal dimensions of controls (2.10±0.07) were higher than those of PAH (2.03±0.08; p=3e-6) and of ePAH (2.04±0.13; p=0.008). Total fractal dimension also yielded higher values for controls (2.30±0.05) compared against PAH (2.28±0.07; p=0.009) and ePAH (2.26±0.10; p=0.04). No significant differences were found between arterial fractal dimensions of controls (2.17±0.04) against those of PAH (2.16±0.07; p=0.15) and of ePAH (2.15±0.10; p=0.14). Conclusions: Fractal dimension allows for non-invasive characterization of pulmonary vascular complexity. Using this method, patients with PAH or ePAH were found to have lower total and venous vascular complexities than controls without PAH or ePAH.

Concentration Influence on Diffusion Limited Cluster Aggregation

1994 ◽  
Vol 367 ◽  
Author(s):  
ST.C. Pencea ◽  
M. Dumitrascu
Keyword(s):  
Brownian Motion ◽  
Fractal Dimensions ◽  
Simple Random Walk ◽  
Two Dimensional ◽  
Diffusion Limited Aggregation ◽  
Dimensional Lattice ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Diffusion Limited ◽  
Diffusion Limited Cluster Aggregation

AbstractDiffusion-limited cluster aggregation has been simulated on a square two dimensional lattice. In order to simulate the brownian motion, we used both the algorithm proposed initially by Kolb et all. and a new algorithm intermediary between a simple random walk and the ballistic model.The simulation was performed for many values of the concentration, from 1 to 50%. By using a box-counting algorithm one has calculated the fractal dimensions of the obtained clusters. Its increasing vs. concentration has been pointed out. The results were compared with those of the classical diffusion-limited aggregation (DLA).

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