Fractal analysis of leaf shapes

1986 ◽  
Vol 16 (1) ◽  
pp. 124-127 ◽  
Author(s):  
J. Vlcek ◽  
E. Cheung
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Sugar Maple ◽  
Shape Change ◽  
Leaf Shape ◽  
Fractal Dimensions ◽  
Imaging Method ◽  
Analysis Program ◽  
White Oak ◽  
Starting Position

An application of fractal mathematics to the analysis of leaf shapes is presented. Six leaves randomly selected from nine tree species were used in the study. A video imaging method together with microcomputer-based image processing was used to generate leaf outlines. A fractal analysis program was written to calculate the fractal dimensions of the leaves. Recalling a leaf outline from a diskette and specifying both the starting position on it (e.g., the beginning of the petiole) and six step lengths (explained later), the program then generates the fractal dimension according to the theory described. The results show that the fractal dimension is sensitive to leaf shape variations within a species. For example, two types of ginkgo leaves (one with and one without a notch in the middle of the leaf outline) showed distinctly different fractal values. Similar sensitivity to shape change was observed among the leaves of white oak, red oak, and sugar maple where such variables as width to length ratio and the degree of jaggedness of the leaf caused a departure of the fractal value from the average.

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 Fractal analysis of urban catchments and their representation in semi-distributed models: imperviousness and sewer system

2016 ◽  
Author(s):  
Auguste Gires ◽  
Ioulia Tchiguirinskaia ◽  
Daniel Schertzer ◽  
Susana Ochoa Rodriguez ◽  
Patrick Willems ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Fractal Dimensions ◽  
Hydrological Models ◽  
Scale Invariant ◽  
Sewer System ◽  
Distributed Models ◽  
Urban Catchments ◽  
Distributed Hydrological Models ◽  
Dimension Characteristic

Abstract. Fractal analysis relies on scale invariance and the concept of fractal dimension enables to characterise and quantify the space filled by a geometrical set exhibiting complex and tortuous patterns. Fractal tools have been widely used in hydrology but seldom in the specific context of urban hydrology. In this paper fractal tools are used to analyse surface and sewer data from 10 urban or peri-urban catchments located in 5 European countries. The aim was to characterise urban catchment properties accounting for the complexity and inhomogeneity typical of urban water systems. Sewer system density and imperviousness (roads or buildings), represented in rasterized maps of 2 m × 2 m pixels, were analysed to quantify their fractal dimension, characteristic of scaling invariance. The results showed that both sewer density and imperviousness exhibit scale invariant features and can be characterized with the help of fractal dimensions ranging from 1.6 to 2, depending on the catchment. In a given area consistent results were found for the two geometrical features, yielding a robust and innovative way of quantifying the level of urbanization. The representation of imperviousness in operational semi-distributed hydrological models for these catchments was also investigated by computing fractal dimensions of the geometrical sets made up of the sub-catchments with coefficients of imperviousness greater than a range of thresholds. It enabled to quantify how well spatial structures of imperviousness were represented in the urban hydrological models.

Image Reconstruction and Analysis of Three-Dimensional Fracture Surfaces Based on the Stereo Matching Method

2004 ◽  
Vol 261-263 ◽  
pp. 1593-1598
Author(s):  
M. Tanaka ◽  
Y. Kimura ◽  
A. Kayama ◽  
L. Chouanine ◽  
Reiko Kato ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Image Reconstruction ◽  
Fracture Surface ◽  
Fractal Analysis ◽  
Stereo Matching ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Matching Method ◽  
Fracture Surfaces ◽  
Box Counting Method

A computer program of the fractal analysis by the box-counting method was developed for the estimation of the fractal dimension of the three-dimensional fracture surface reconstructed by the stereo matching method. The image reconstruction and fractal analysis were then made on the fracture surfaces of materials created by different mechanisms. There was a correlation between the fractal dimension of the three-dimensional fracture surface and the fractal dimensions evaluated by other methods on ceramics and metals. The effects of microstructures on the fractal dimension were also experimentally discussed.

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 Fractal analysis of leaves: are all leaves self-similar along the cane?

2013 ◽  
Vol 32 (1) ◽  
Author(s):  
Anna Gazda
Keyword(s):  
Fractal Dimension ◽  
Fractal Analysis ◽  
Leaf Shape ◽  
Scale Dependency ◽  
Size Change ◽  
Self Similar ◽  
The Mean ◽  
Shape Complexity ◽  
Shape And Size

AbstractThe fractal dimension can be used to quantify the shape of a natural curve. Curves with similar degrees of irregularity will tend to have the same fractal dimension. The fractal exponent describes the complexity of a shape and characterizes the scale-dependency of the pattern. This article presents an application of the fractal dimension in the analysis of leaves shape. In this paper I attempt to ask question if leaves of blackberry characterized by fractal dimension differ significantly in relation to the leaf ’s position along the cane. The fractal dimension of 49 leaves of blackberry from 8 primocanes, and 53 leaves from 19 lateral canes, from 9 individuals was estimated. The mean of D of a leaf is 1.12. There are no significant differences between D for leaves from two different cane types. Previous studies were focused on measurements of fractal dimension of leaves randomly chosen from one or a few individuals so there was necessity to measure fractal dimension all leaves growing along the same shoot, because usually leaf shape and size change more or less along a shoot. This research confirmed that fractal dimension is much more related to the shape complexity than to the size of leaves.

scholarly journals Bi-Fractal Characterization of the Pore Network of Tight Sandstone

2021 ◽  
Vol 9 ◽  
Author(s):  
Zezhang Song ◽  
Junyi Zhao ◽  
Yuanyin Zhang ◽  
Dailin Yang ◽  
Yunlong Wang ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Pore Structure ◽  
Thin Section ◽  
Fractal Analysis ◽  
Pore Space ◽  
Fractal Dimensions ◽  
Pore Network ◽  
Physical Analysis ◽  
Tight Sandstone ◽  
Gas Bearing

Fluid seepage performance and accumulation in tight sandstone is a critical research topic for in-depth exploration and development, closely related to the heterogeneity of the pore network. The fractal characterization is one of the most compelling and direct ways for quantitative investigation of heterogeneity. However, only one kind of fractal is used in most studies, and the differences and relations between different fractal dimensions are rarely discussed. This paper chose one of the most representative tight sandstone formations in China, the second member of the Xujiahe Formation, as the research object. First, based on physical analysis and XRD analysis, we carried out a qualitative investigation on pore structure utilizing thin-section and scanning electron microscopy. Then, detailed pore structure parameters were obtained using high-pressure mercury intrusion (HPMI). Lastly, we combined two-dimensional fractal analysis on thin-section images and three-dimensional fractal analysis on HPMI data to characterize the pore network heterogeneity quantitatively. The Xu2 tight sandstone is mainly medium- to fine-grained lithic feldspathic sandstone or feldspathic lithic sandstone with low porosity and permeability. Also, the Xujiahe tight sandstone is mainly composed of quartz, feldspar, and clay. The pore types of Xu2 tight sandstones are primarily intergranular pores, micro-fractures, and intra- and intergranular dissolution pores. Moreover, most of the micro-fractures in gas-bearing formation are open-ended, while most are filled by clay minerals in the dry formation. The r50 (median pore radius) is the most sensitive parameter to seepage capability (permeability) and gas-bearing status. The 2D fractal dimension (Ds) of gas-bearing samples is significantly larger than that of dry samples, while the 3D fractal dimension (D1, D2) of gas-bearing samples is lower than that of dry samples. There is a strong negative correlation between D2 and gas-bearing status, permeability, quartz content, and r50, but a positive correlation between Ds and these parameters. D2 represents the heterogeneity of pore space, while the Ds indicates the development of the pore network. Tectonic movements that generate micro-fractures and clay cementation that blocks the seepage channels are the two main controlling factors on fractal dimensions. Combining 2D and 3D fractal analysis could give a more in-depth investigation of pore structure.

scholarly journals Fractal analysis of the ULF geomagnetic data obtained at Izu Peninsula, Japan in relation to the nearby earthquake swarm of June–August 2000

2003 ◽  
Vol 3 (3/4) ◽  
pp. 229-236 ◽  
Author(s):  
K. Gotoh ◽  
M. Hayakawa ◽  
N. Smirnova
Keyword(s):  
Time Series ◽  
Fractal Dimension ◽  
Active Region ◽  
Fractal Analysis ◽  
Flicker Noise ◽  
Earthquake Swarm ◽  
Fractal Dimensions ◽  
Izu Peninsula ◽  
Geomagnetic Data ◽  
Fractal Characteristics

Abstract. In our recent papers we applied fractal methods to extract the earthquake precursory signatures from scaling characteristics of the ULF geomagnetic data, obtained in a seismic active region of Guam Island during the large earthquake of 8 August 1993. We found specific dynamics of their fractal characteristics (spectral exponents and fractal dimensions) before the earthquake: appearance of the flicker-noise signatures and increase of the time series fractal dimension. Here we analyze ULF geomagnetic data obtained in a seismic active region of Izu Peninsula, Japan during a swarm of the strong nearby earthquakes of June–August 2000 and compare the results obtained in both regions. We apply the same methodology of data processing using the FFT procedure, Higuchi method and Burlaga-Klein approach to calculate the spectral exponents and fractal dimensions of the ULF time series. We found the common features and specific peculiarities in the behavior of fractal characteristics of the ULF time series before Izu and Guam earthquakes. As a common feature, we obtained the same increase of the ULF time series fractal dimension before the earthquakes, and as specific peculiarity – this increase appears to be sharp for Izu earthquake in comparison with gradual increase of the ULF time series fractal dimension for Guam earthquake. The results obtained in both regions are discussed on the basis of the SOC (self-organized criticality) concept taking into account the differences in the depths of the earthquake focuses. On the basis of the peculiarities revealed, we advance methodology for extraction of the earthquake precursory signatures. As an adjacent step, we suggest the combined analysis of the ULF time series in the parametric space polarization ratio – fractal dimension. We reason also upon the advantage of the multifractal approach with respect to the mono-fractal analysis for study of the earthquake preparation dynamics.

scholarly journals Fractal analysis of the flour

2020 ◽  
Vol 24 (2) ◽  
pp. 67-71
Author(s):  
Ľubomír Kubík ◽  
Monika Božiková ◽  
Peter Hlaváč ◽  
Viera Kažimírová
Keyword(s):  
Fractal Dimension ◽  
Correlation Analysis ◽  
Fractal Analysis ◽  
Particle Distribution ◽  
Fractal Dimensions ◽  
Small Particles ◽  
Digital Microscope ◽  
Powder Samples ◽  
Powder Particles

The aim of the paper was the evaluation of the microscopic, powder samples of flour by utilizing the fractal analysis. The powder particles were compared and submitted to fractal analysis. Three types of flour were studied, smooth flour, semi-flour and thick flour. The five samples of each sort of flour were tested by fractal analysis. The samples were digitized by the digital microscope Motic DM 1802-A with software Motic Image Plus ver. 2.0. Each image was processed by the thresholding operation and the fractal analysis was realized by the software Harfa ver. 5.1.0 and the samples were compared by the correlation analysis. The obtained fractal dimensions described the segmentation and distribution of flour powder and the fractions of the flour. The fractal dimension of the smooth flour was DWBW = 1.29266, of the semi-flour DWBW = 1.70734 and of the thick flour DWBW = 1.57978. The smooth flour was composed of microscopic powder particles of wheat. Small particles of about 10 mm were mainly found in the smooth flour. However, sporadic particles greater than 47.6 mm were also observed. The size of the smooth flour particles was from 2.38 mm to 47.6 mm. The semi-flour contained mainly particles the size of up to 71.2 mm. Practically half the particles obtained from semi-flour were the size of up to 71.2 mm. The thick flour was created mainly by particles to the size of 73.78 mm. Greater particles, the size from 130.9 mm to 314.2 mm, were obtained in a small number. On the base of the particle distribution, the semi-flour and the thick flour were very similar, but on the base of fractal analysis they were different and we can distinguish them.

Abstract MP59: Retinal Vascular Metrics Predict Pial Collateral Status in Patients With Acute Ischemic Stroke

Stroke ◽  
2021 ◽  
Vol 52 (Suppl_1) ◽  
Author(s):  
Adnan Khan ◽  
Saadat Kamran ◽  
Patrick D Boever ◽  
Nele Gerrits ◽  
Maher Saqqur ◽  
...  
Keyword(s):  
Ischemic Stroke ◽  
Acute Stroke ◽  
Acute Ischemic Stroke ◽  
Fractal Analysis ◽  
Management Strategies ◽  
Experimental Studies ◽  
Fractal Dimensions ◽  
Retinal Vessel ◽  
Imaging Method ◽  
Stroke Patients

Background: The extent of the pial collateral circulation may determine outcomes in an acute ischemic stroke. Experimental studies suggest that retinal vessel metrics and geometric patterning may predict the pial collateral status. We have undertaken a translational study to quantify and relate retinal vascular metrics to the grade of pial collaterals in patients with acute ischemic stroke. Method: 35 patients admitted with acute stroke underwent computed tomography angiography (OCT) and were graded as having good (n=20)(47.55 ± 10.65 years) or poor (n=15) (48.93 ± 10.91 years) pial collaterals and compared to healthy controls (n=21)(44.26 ± 10.15 years). Retinal images were generated using OCT and central retinal artery equivalent (CRAE), central retinal vein equivalent (CRVE), artery-to-vein ratio (AVR), segmented fractal analysis and lacunarity, tortuosity index and fractal dimensions (capacity D 0 , information D 1 and correlation D 2 , curve asymmetry, singularity length and f-alfa-max using MONA software) were quantified. Results: Age ( p =0.709), BMI ( p =0.451), total cholesterol ( p =0.845), triglycerides ( p =0.679), LDL ( p =0.953), HDL ( p =0.361) and HbA 1c ( p =0.210) were comparable but the national institute of health stroke scale ( p =0.031) and modified Rankin Scale ( p =0.048) were higher in patients with poor compared to good collaterals. CRAE ( p =0.114), CRVE ( p =0.946), AVR ( p =0.114), lacunarity ( p =0.442), tortuosity index ( p =0.681), fractal analysis ( p =0.656), curve asymmetry ( p =0.619) and singularity length ( p =0.944) did not differ between patients with poor compared to good collaterals. However, fractal capacity D 0 (1.673 ± 0.029 vs 1.654 ± 0.025, p =0.042), fractal information D 1 (1.610 ± 0.027 vs 1.591 ± 0.024, p =0.036), fractal correlation D 2 (1.581± 0.028 vs 1.564 ± 0.024, p =0.060), and f alfa max (1.674 ± 0.027 vs 1.654 ± 0.025, p =0.030) were higher in patients with poor compared to good collaterals. Conclusion: This study shows differences in retinal vessel fractal dimensions between acute stroke patients with poor compared to good pial collaterals. This represents a non-invasive imaging method to define the pial collateral status and develop personalized intervention management strategies in acute ischemic stroke patients.

The Percolation Phenomenon and Fractal Dimension of Natural Silicates

2015 ◽  
Vol 365 ◽  
pp. 128-135
Author(s):  
Maria A. Vasilyeva ◽  
Yuri A. Gusev ◽  
Valery G. Shtyrlin ◽  
Yury N. Osin
Keyword(s):  
Fractal Dimension ◽  
Electron Micrographs ◽  
Fractal Analysis ◽  
Dielectric Spectroscopy ◽  
Fractal Dimensions ◽  
Adsorbed Water ◽  
Ambient Air ◽  
Silicate Structure ◽  
Percolation Phenomenon

Many physical effects, such asdcconductivity and percolation, depend on the morphology of the silicate structure and its relationship to adsorbed water. These effects play an important role in numerous technological applications, in geology, oil-extracting industry, and other practical fields. In this study, all the samples: natural montmorillonite, kaolinite, and сlinoptilolite with different exchangeable cations in their structures, – were stored in ambient air humidity. The investigation was carried by using two separate techniques, namely Dielectric Spectroscopy and a fractal analysis of electron micrographs. The aims of this work were to analyze the complex relaxation behavior of the relaxation process in temperature range –70°C ÷ +70°C and to determine the fractal dimensions of silicates from the dielectric response at percolation. Dielectric measurements in the frequency range of 1 Hz ÷ 1 MHz were performed using a BDS 80 Dielectric Spectrometer based on an Alpha Impedance Analyzer (Novocontrol). The micrographs were analyzed using a special Matlab based program. The analysis of aspects of the dielectric relaxation spectra related to percolation was used for the determination of the numerical characteristics of geometric heterogeneity of natural silicates. The percolation temperatures of the studied samples were determined. The percolation phenomenon in the silicates is related to the transfer of the electric excitation within the developed network of open pores due to the migration of protons and ions along the surface of connected pores on the outer surfaces of the granules. The analysis of these processes allows one to extract the fractal dimensions associated with the migration of charge carriers within the porous medium. Fractal dimensions of the silicates calculated in two ways: from dielectric spectroscopy study and from fractal analysis of the micrographs, – are in good agreement with each other. It was demonstrated that conventional method of the spatial fractal dimension determination using fractal analysis of electron micrographs leads to overestimation in the case of spatial fractal bounded by a surface fractal. The dielectric spectroscopy method is free from such overestimation.

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