In the research of fractal cities, the fractal dimension is very important. It is used to describe the fractal character of the city. The authors have designed two approaches to calculate the fractal dimension by the ...In the research of fractal cities, the fractal dimension is very important. It is used to describe the fractal character of the city. The authors have designed two approaches to calculate the fractal dimension by the box-counting method through an example of Beijing, which are called the vector method and the grid method, respectively. The former calculates the fractal dimension through an intersecting analysis in ArcView; and the latter is carried out by programming in Matlab. They are compared from three aspects： the calculating process, the limits in use, and the results. As a result, the conclusion is made that there are merits and faults on both methods, and they should be chosen to use properly in practical situation.展开更多
Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media withi...Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.展开更多
Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This ...Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This paper proposes a successive shift box-counting method,in which the studied object is divided into small sub-objects that are composed of a series of gridsaccording to its characteristic scaling. The terrain fractal dimensions in the grids are calculatedwith the successive shift box-counting method and the scattered points with values of fractaldimensions are obtained. The present research shows that the planar variation of fractal dimensionsis well consistent with fault traces and geological boundaries.展开更多
This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-cou...This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T. Meanwhile, the box dimension of the velocity histories Dye is negatively correlated with the characteristic period T and improved characteristic period Tgi.展开更多
A set in Rd is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..
In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.
AIM：To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina.METHODS：Fifty volunteers were enrolled in ...AIM：To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina.METHODS：Fifty volunteers were enrolled in this study in the Ophthalmological Clinic of Cluj-Napoca,Romania,between January 2012 and January 2014. A set of 100 segmented and skeletonised human retinal images,corresponding to normal states of the retina were studied. An automatic unsupervised method for retinal vessel segmentation was applied before multifractal analysis. The multifractal analysis of digital retinal images was made with computer algorithms,applying the standard boxcounting method. Statistical analyses were performed using the Graph Pad In Stat software.RESULTS：The architecture of normal human retinal microvascular network was able to be described using the multifractal geometry. The average of generalized dimensions（D_q）for q=0,1,2,the width of the multifractal spectrum（Δα=α_（max）-α_（min））and the spectrum arms＇ heights difference（│Δf│）of the normal images were expressed as mean±standard deviation（SD）：for segmented versions,D_0=1.7014±0.0057; D_1=1.6507±0.0058; D_2=1.5772±0.0059; Δα=0.92441±0.0085; │Δf│= 0.1453±0.0051; for skeletonised versions,D_0=1.6303±0.0051; D_1=1.6012±0.0059; D_2=1.5531± 0.0058; Δα=0.65032±0.0162; │Δf│= 0.0238±0.0161. The average of generalized dimensions（D_q）for q=0,1,2,the width of the multifractal spectrum（Δα）and the spectrum arms＇ heights difference（│Δf│）of the segmented versions was slightly greater than the skeletonised versions.CONCLUSION：The multifractal analysis of fundus photographs may be used as a quantitative parameter for the evaluation of the complex three-dimensional structure of the retinal microvasculature as a potential marker for early detection of topological changes associated with retinal diseases.展开更多
Fractal geometry can effectively quantify naturally nonlinear or much complicated phenomena, and fractal dimension is the quantitative parameter. In this paper, we applied fractal geometry to analyze spatial fault dis...Fractal geometry can effectively quantify naturally nonlinear or much complicated phenomena, and fractal dimension is the quantitative parameter. In this paper, we applied fractal geometry to analyze spatial fault distribution of the southern South China Sea. Calculation results indicate that the spatial fault distribution of this area have remarkable fractal features in a non-scale interval of 25 - 250 km, with fractal dimensions of 1.660 1, 1.387 5, and 1.269 3 for all faults, NE-trending faults and NW-trending faults, respectively. These dimensions （capacity dimensions）, obtained by box-counting method, represent well the characteristics of spatial fault distribution. Displayed by isoline-maps of fractal dimensions, faults in the study area arrange symmetrically along the spreading axis of the Southwest Sub-basin, and density of the NE-trending faults supplements to that of the NW-trending faults, revealing a property of mutual restriction between these two groups of faults. At last, by correlating the fractal features of spatial fault distribution with the evolution of the South China Sea and the distribution of oil or gas basins, we tried to reveal their relationships.展开更多
The spatial distribution patterns of species are always scale-dependent and spatially self-similar in ecological systems. In this work, vegetation distribution data collected from the vegetation map of the Xigaze regi...The spatial distribution patterns of species are always scale-dependent and spatially self-similar in ecological systems. In this work, vegetation distribution data collected from the vegetation map of the Xigaze region was analyzed using a box-counting method. The power law of the box-counting dimension （DB） across a range of scales （5-160 km） confirms the fractal patterns for most vegetation formations, while the fluctuations of the scale-specific DB among the different abundance groups indicate limitations of fractal coherence. The fractal method is shown to be a useful tool for measuring the distribution patterns of vegetation formations across scales, which provides important information for both species and habitat conservation, especially in landscape management.展开更多
A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper,we prove that the random self-conformal set is regular almost surely. Also we determine the dimensions for a ...A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper,we prove that the random self-conformal set is regular almost surely. Also we determine the dimensions for a class of random self-conformal sets.展开更多
In this paper, the box-counting dimension is used to derive an explicit formula for the dimension of a fractal constructed using several contractions or by combining fractals. This dimension agrees with the Hausdorff ...In this paper, the box-counting dimension is used to derive an explicit formula for the dimension of a fractal constructed using several contractions or by combining fractals. This dimension agrees with the Hausdorff dimension in the particular case when the scales factors considered are all the same. A more general sufficient condition for the box-counting dimension and the Hausdorff dimension to be the same is given. It is also shown that the dimension of the fractal obtained by combining two fractals is the weighted average of the dimensions of the two fractals.展开更多
The goal of this study is to establish relationships between the hot compression deformation behaviors and the fractal dimension of primary phase morphology of TA15 titanium alloy using the analytical methods of metal...The goal of this study is to establish relationships between the hot compression deformation behaviors and the fractal dimension of primary phase morphology of TA15 titanium alloy using the analytical methods of metallurgical microscope and transmission electron microscope coupled with box-counting dimension method. The hot compression deformation behaviors vary with decreasing fractal dimension owing to the change of microstructure caused by different parameters of the hot compressive deformation.The results indicate that TA15 alloy shows dynamic recrystallization characteristics at deformation temperature lower than 850℃while fractal dimension exhibits a moderate decreasing trend with the temperature increasing,and shows dynamic recovery characteristics at deformation temperature higher than 850℃while fractal dimension reduces rapidly with the temperature increasing.The fractal dimension displays non-linear relationship with fraction of primary phase and with aspect ratio of primary phase.展开更多
Inclusions with sizes less than 1 μm in molten steel are difficult to float up to the molten steel and slag interface owing to their slow terminal velocity. Thus, increasing the size of inclusion is essential for acc...Inclusions with sizes less than 1 μm in molten steel are difficult to float up to the molten steel and slag interface owing to their slow terminal velocity. Thus, increasing the size of inclusion is essential for accelerating the removal of inclusions. Polystyrene particles simulating inclusions in molten steel were quantified by direct observation of the particle collision behavior in a turbulent flow in a water model. The box-counting fractal dimension of particles was calculated by processing the binary images of aggregated particles. The fractal dimension of the outer contours of the single plastic particles was smaller than that of the aggregated particles. The fractal dimension was varied from 1.14 to 1.35. When two or more monomer particles collide, the aggregates are separated more easily, as the temperature increases from 40 to 80 ℃. The aggregated particles were loose and easy to separate in the high-temperature aqueous solution. The effect of temperature on the surface tension of liquid and the interracial tension of solid and liquid is obvious. The particles are wetting in the water solution at a temperature more than 60 ℃. The relationship between the velocity of the particles and the fractal dimension of the solid particles with the equivalent diameter was discussed.展开更多
<strong>Introduction:</strong> Current knowledge postulated glia as active participants in various metabolic processes within nervous tissue. The most numerous glial cells were astrocytes, and qualitative ...<strong>Introduction:</strong> Current knowledge postulated glia as active participants in various metabolic processes within nervous tissue. The most numerous glial cells were astrocytes, and qualitative analysis divided them into two types based on their anatomical locations: fibrous and protoplasmic. The main goal of this research was to examine the morphological difference between types, analyzing four features of the image. The secondary objective of this research was to explore their morphology through maturation and aging. <strong>Materials and Methods: </strong>The material originated from bilateral sections of the human principal olivary nucleus, without disorders in the central nervous system. The brains were taken from 30 human cadavers (35 - 90 years) and cut into samples corresponding to dimensions of the principal olivary nucleus. A light microscope digitized the histological preparations. The selection of 294 images was analyzed by monofractal parameters derived from the box-counting. These parameters quantified four image properties (size, shape, complexity and homogeneity) of the glial body or whole glial cell. <strong>Results: </strong>The first results showed that images of two types of astrocytes were significantly different (p < 0.05 and higher) in all properties of whole cells. The second results examined the differences between three age groups in both types of astrocytes. The differences between groups were more evident for protoplasmic than fibrous (nine vs. three parameters). <strong>Conclusions: </strong>The main limitation of this study lies in the fact that the quantification was performed only by fractal analysis techniques. Nevertheless, a detailed monofractal analysis of astrocytes was performed for the first time. Thus, although this study can be seen as an improvement of the previous qualitative results, future research will provide the complete procedure of the image analysis.展开更多
文摘In the research of fractal cities, the fractal dimension is very important. It is used to describe the fractal character of the city. The authors have designed two approaches to calculate the fractal dimension by the box-counting method through an example of Beijing, which are called the vector method and the grid method, respectively. The former calculates the fractal dimension through an intersecting analysis in ArcView; and the latter is carried out by programming in Matlab. They are compared from three aspects： the calculating process, the limits in use, and the results. As a result, the conclusion is made that there are merits and faults on both methods, and they should be chosen to use properly in practical situation.
基金supported by the National Natural Science Foundation of China (Nos.52374078 and 52074043)the Fundamental Research Funds for the Central Universities (No.2023CDJKYJH021)。
文摘Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.
文摘Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This paper proposes a successive shift box-counting method,in which the studied object is divided into small sub-objects that are composed of a series of gridsaccording to its characteristic scaling. The terrain fractal dimensions in the grids are calculatedwith the successive shift box-counting method and the scattered points with values of fractaldimensions are obtained. The present research shows that the planar variation of fractal dimensionsis well consistent with fault traces and geological boundaries.
基金National Natural Science Foundation of China under Grant Nos.50978047 and 11332004National Basic Research Program of China under Grant No.2010CB832703
文摘This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T. Meanwhile, the box dimension of the velocity histories Dye is negatively correlated with the characteristic period T and improved characteristic period Tgi.
文摘A set in Rd is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..
文摘In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.
基金the Program"Partnerships in priority domains"with the support of the National Education Ministry,the Executive Agency for Higher Education,Research,Development and Innovation Funding (UEFISCDI),Romania (Project code:PN-II-PT-PCCA-2013-4-1232)
文摘AIM：To apply the multifractal analysis method as a quantitative approach to a comprehensive description of the microvascular network architecture of the normal human retina.METHODS：Fifty volunteers were enrolled in this study in the Ophthalmological Clinic of Cluj-Napoca,Romania,between January 2012 and January 2014. A set of 100 segmented and skeletonised human retinal images,corresponding to normal states of the retina were studied. An automatic unsupervised method for retinal vessel segmentation was applied before multifractal analysis. The multifractal analysis of digital retinal images was made with computer algorithms,applying the standard boxcounting method. Statistical analyses were performed using the Graph Pad In Stat software.RESULTS：The architecture of normal human retinal microvascular network was able to be described using the multifractal geometry. The average of generalized dimensions（D_q）for q=0,1,2,the width of the multifractal spectrum（Δα=α_（max）-α_（min））and the spectrum arms＇ heights difference（│Δf│）of the normal images were expressed as mean±standard deviation（SD）：for segmented versions,D_0=1.7014±0.0057; D_1=1.6507±0.0058; D_2=1.5772±0.0059; Δα=0.92441±0.0085; │Δf│= 0.1453±0.0051; for skeletonised versions,D_0=1.6303±0.0051; D_1=1.6012±0.0059; D_2=1.5531± 0.0058; Δα=0.65032±0.0162; │Δf│= 0.0238±0.0161. The average of generalized dimensions（D_q）for q=0,1,2,the width of the multifractal spectrum（Δα）and the spectrum arms＇ heights difference（│Δf│）of the segmented versions was slightly greater than the skeletonised versions.CONCLUSION：The multifractal analysis of fundus photographs may be used as a quantitative parameter for the evaluation of the complex three-dimensional structure of the retinal microvasculature as a potential marker for early detection of topological changes associated with retinal diseases.
基金This study is supported by the National Natural Science Foundation of China under contract No. 40476026the Guangdong Natural Science Foundation under contract No. 04001309
文摘Fractal geometry can effectively quantify naturally nonlinear or much complicated phenomena, and fractal dimension is the quantitative parameter. In this paper, we applied fractal geometry to analyze spatial fault distribution of the southern South China Sea. Calculation results indicate that the spatial fault distribution of this area have remarkable fractal features in a non-scale interval of 25 - 250 km, with fractal dimensions of 1.660 1, 1.387 5, and 1.269 3 for all faults, NE-trending faults and NW-trending faults, respectively. These dimensions （capacity dimensions）, obtained by box-counting method, represent well the characteristics of spatial fault distribution. Displayed by isoline-maps of fractal dimensions, faults in the study area arrange symmetrically along the spreading axis of the Southwest Sub-basin, and density of the NE-trending faults supplements to that of the NW-trending faults, revealing a property of mutual restriction between these two groups of faults. At last, by correlating the fractal features of spatial fault distribution with the evolution of the South China Sea and the distribution of oil or gas basins, we tried to reveal their relationships.
基金Supported by the Japan Society for the Promotion of Science (No. L-02711)
文摘The spatial distribution patterns of species are always scale-dependent and spatially self-similar in ecological systems. In this work, vegetation distribution data collected from the vegetation map of the Xigaze region was analyzed using a box-counting method. The power law of the box-counting dimension （DB） across a range of scales （5-160 km） confirms the fractal patterns for most vegetation formations, while the fluctuations of the scale-specific DB among the different abundance groups indicate limitations of fractal coherence. The fractal method is shown to be a useful tool for measuring the distribution patterns of vegetation formations across scales, which provides important information for both species and habitat conservation, especially in landscape management.
文摘A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper,we prove that the random self-conformal set is regular almost surely. Also we determine the dimensions for a class of random self-conformal sets.
文摘In this paper, the box-counting dimension is used to derive an explicit formula for the dimension of a fractal constructed using several contractions or by combining fractals. This dimension agrees with the Hausdorff dimension in the particular case when the scales factors considered are all the same. A more general sufficient condition for the box-counting dimension and the Hausdorff dimension to be the same is given. It is also shown that the dimension of the fractal obtained by combining two fractals is the weighted average of the dimensions of the two fractals.
文摘The goal of this study is to establish relationships between the hot compression deformation behaviors and the fractal dimension of primary phase morphology of TA15 titanium alloy using the analytical methods of metallurgical microscope and transmission electron microscope coupled with box-counting dimension method. The hot compression deformation behaviors vary with decreasing fractal dimension owing to the change of microstructure caused by different parameters of the hot compressive deformation.The results indicate that TA15 alloy shows dynamic recrystallization characteristics at deformation temperature lower than 850℃while fractal dimension exhibits a moderate decreasing trend with the temperature increasing,and shows dynamic recovery characteristics at deformation temperature higher than 850℃while fractal dimension reduces rapidly with the temperature increasing.The fractal dimension displays non-linear relationship with fraction of primary phase and with aspect ratio of primary phase.
文摘Inclusions with sizes less than 1 μm in molten steel are difficult to float up to the molten steel and slag interface owing to their slow terminal velocity. Thus, increasing the size of inclusion is essential for accelerating the removal of inclusions. Polystyrene particles simulating inclusions in molten steel were quantified by direct observation of the particle collision behavior in a turbulent flow in a water model. The box-counting fractal dimension of particles was calculated by processing the binary images of aggregated particles. The fractal dimension of the outer contours of the single plastic particles was smaller than that of the aggregated particles. The fractal dimension was varied from 1.14 to 1.35. When two or more monomer particles collide, the aggregates are separated more easily, as the temperature increases from 40 to 80 ℃. The aggregated particles were loose and easy to separate in the high-temperature aqueous solution. The effect of temperature on the surface tension of liquid and the interracial tension of solid and liquid is obvious. The particles are wetting in the water solution at a temperature more than 60 ℃. The relationship between the velocity of the particles and the fractal dimension of the solid particles with the equivalent diameter was discussed.
文摘<strong>Introduction:</strong> Current knowledge postulated glia as active participants in various metabolic processes within nervous tissue. The most numerous glial cells were astrocytes, and qualitative analysis divided them into two types based on their anatomical locations: fibrous and protoplasmic. The main goal of this research was to examine the morphological difference between types, analyzing four features of the image. The secondary objective of this research was to explore their morphology through maturation and aging. <strong>Materials and Methods: </strong>The material originated from bilateral sections of the human principal olivary nucleus, without disorders in the central nervous system. The brains were taken from 30 human cadavers (35 - 90 years) and cut into samples corresponding to dimensions of the principal olivary nucleus. A light microscope digitized the histological preparations. The selection of 294 images was analyzed by monofractal parameters derived from the box-counting. These parameters quantified four image properties (size, shape, complexity and homogeneity) of the glial body or whole glial cell. <strong>Results: </strong>The first results showed that images of two types of astrocytes were significantly different (p < 0.05 and higher) in all properties of whole cells. The second results examined the differences between three age groups in both types of astrocytes. The differences between groups were more evident for protoplasmic than fibrous (nine vs. three parameters). <strong>Conclusions: </strong>The main limitation of this study lies in the fact that the quantification was performed only by fractal analysis techniques. Nevertheless, a detailed monofractal analysis of astrocytes was performed for the first time. Thus, although this study can be seen as an improvement of the previous qualitative results, future research will provide the complete procedure of the image analysis.
