Soil Mass, Surface, and Spectral Fractal Dimensions Estimated from Thin Section Photographs
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Abstract Aspects of fractal geometry have been used to give quantitative measurements of soil structure. Fractal dimensions measured were the mass fractal dimension ( D m ), surface fractal dimension ( D s ), and the spectral dimension ( d ). We investigated the fractal component of a computer program, STRUCTURA, which measures the fractal dimension of soil from images of soil thin sections. Six thin sections, each showing different structural characteristics, were analyzed in order to obtain a range of fractal dimensions. The dimensions, in particular D m and d , were shown to discriminate the different structures. The values of D m and d ranged from 1.682 to 1.852 and 1.236 to 1.668, respectively. A further objective was to use these results, together with fractal theory, to show the potential fractal geometry has in predicting physical processes such as diffusion within the soil. To assist with the interpretation of fractal dimensions, the dimensions of different soil samples with the same porosity were compared.Keywords:
Fractal derivative
Fractal landscape
Shape is one of the most important visual attributes to characterize objects, playing a important role in pattern recognition. There are various approaches to extract relevant information of a shape. An approach widely used in shape analysis is the complexity, and Fractal Dimension and Multi-Scale Fractal Dimension are both well-known methodologies to estimate it. This papers presents a comparative study between Fractal Dimension and Multi-Scale Fractal Dimension in a shape analysis context. Through experimental comparison using a shape database previously classified, both methods are compared. Different parameters configuration of each method are considered and a discussion about the results of each method is also presented.
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Abstract The application of fractal theory provides a new approach for the study of soil science. The calculation of fractal dimension is the basis. The purpose of the study is to find a suitable method for calculating the fractal dimension of the soil particle size and to analyse how the fractal dimension reflects soil properties. We mostly use comparative analysis to study. Comparing the mass fractal dimension method with volume fractal dimension method, we analyze their advantages and disadvantages, and the fractal dimension performance of the soil in different land types.
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The fractal shapes in the nature, their characteristics, also, opinions about fractal dimension are given in this article. The phenomena of geophysical origin of fractal geometric shapes and potential useful tools for describing complex shapes are illustrated. Despite, the fractals are widely used in geographical areas; the opinions which cause inconsistent results from different fractal computational algorithms have been expressed.Fractal dimension was firstly introduced as the coefficient which describes geometrically complex shapes, the details are considered more important than a completely drawen picture. The theoretical fractal dimension for sets which describes simple geometric shapes is equal to the usual Euclidean or topological dimension. The theoretical fractal dimension for sets which describe points, is equal to 0, the fractal dimension for sets which describe straight line with only length, is equal to 1, the fractal dimension for sets which describe surface, is equal to 2, and the fractal dimension for sets which describe volume, is equal to 3.
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Effective dimension
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Shape is one of the most important visual attributes to characterize objects, playing a important role in pattern recognition. There are various approaches to extract relevant information of a shape. An approach widely used in shape analysis is the complexity, and Fractal Dimension and Multi-Scale Fractal Dimension are both well-known methodologies to estimate it. This papers presents a comparative study between Fractal Dimension and Multi-Scale Fractal Dimension in a shape analysis context. Through experimental comparison using a shape database previously classified, both methods are compared. Different parameters configuration of each method are considered and a discussion about the results of each method is also presented.
Fractal landscape
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A mathematical model was set up to calculate the surface fractal of single particle.The surface fractal of single particles samples was measured and calculated based on a high precision digital optics micro-system.A simple analytical method was generated to calculate the surface fractal of single particle.In addition,we put forward the conception of mid-position fractal dimension to characterize the average surface fractal of particle group,and proved the reasonability of fulfiling the conception mid-position fractal dimension to characterize the average surface fractal of particle group.
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A concept of sub-fractal structure - a fractal structure growing in a space with a fractal dimension (base-fractal) - was put forward to describe the shape of a single partial discharge in an electrical tree with a base-fractal dimension. A computer simulation model was proposed to study the relation between the sub-fractal and the base-fractal dimensions. The tree growth rate was suggested to be related to the difference between these two fractal dimensions.
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Base (topology)
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Since fractal functions are widely applied in dynamic systems and physics such as fractal growth and fractal antennas, this paper concerns fundamental problems of fractal continuous functions like cardinality of collection of fractal functions, box dimension of summation of fractal functions, and fractal linear space. After verifying that the cardinality of fractal continuous functions is the second category by Baire theory, we investigate the box dimension of sum of fractal continuous functions so as to discuss fractal linear space under fractal dimension. It is proved that the collection of 1-dimensional fractal continuous functions is a fractal linear space under usual addition and scale multiplication of functions. Particularly, it is revealed that the fractal function with the largest box dimension in the summation represents a fractal dimensional character whenever the other box dimension of functions exist or not. Simply speaking, the fractal function with the largest box dimension can absorb the other fractal features of functions in the summation.
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Fractal landscape
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Fractal landscape
Fractal derivative
Correlation dimension
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