Kinds Of Fractal Dimensions Research Articles

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

AbstractIn 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.

Fractal Analysis Of Colors And Shapes For Natural And Urbanscapes URBANSCAPES

Abstract. Fractal analysis has been applied in many fields since it was proposed by Mandelbrot in 1967. Fractal dimension is a basic parameter of fractal analysis. According to the difference of fractal dimensions for images, natural landscapes and urbanscapes could be differentiated, which is of great significance. In this paper, two methods were used for two types of landscape images to discuss the difference between natural landscapes and urbanscapes. Traditionally, a box-counting method was adopted to evaluate the shape of grayscale images. On the other way, for the spatial distributions of RGB values in images, the fractal Brownian motion (fBm) model was employed to calculate the fractal dimensions of colour images for two types of landscape images. From the results, the fractal dimensions of natural landscape images were lower than that of urbanscapes for both grayscale images and colour images with two types of methods. Moreover, the spatial distributions of RGB values in images were clearly related with the fractal dimensions. The results indicated that there was obvious difference (about 0.09) between the fractal dimensions for two kinds of landscapes. It was worthy to mention that when the correlation coefficient is 0 in the semivariogram, the fractal dimension is 2, which means that when the RGB values are completely random for their locations in the colour image, the fractal dimension becomes 3. Two kinds of fractal dimensions could evaluate the shape and the color distributions of landscapes and discriminate the natural landscapes from urbanscapes clearly.

The Blanket Fractal Dimension Based on the Directed Pattern Plate

Blanket fractal dimension is a kind of fractal dimension,which is usually used in image processing , and doesn't have a direction. This paper will propose to introduce the direction of pixel into the calculation of the Blanket fractal dimension, and use it to detect the edge of image. This algorithm will calculate the new blanket fractal dimension of the image at first, and then detect the edge with the method of the edge segmentation algorithm based on the traditional Blanket fractal dimension. After this, it will eliminate a part of pixels based on the IFS and Collage Theorem. Experiments will show that this algorithm is able to overcome the issue of double-border in the edge segmentation algorithm base on traditional Blanket fractal dimension, and extract the precision edges with the different directions. DOI: http://dx.doi.org/10.11591/telkomnika.v11i9.3258

Fractal Researches on the Spatial Structure of the Urban System in Sheyang Economic Zone

This paper used three kinds of fractal dimensions to characterize spatial structure of the urban system in Shenyang Economic Zone, namely aggregation dimension, spatial correlation dimension, and box-counting dimension. It was found that the spatial structure of the Shenyang Economic Zone urban system has obvious fractal properties. In analysing the results of our research the following conclusions emerged: The space distribution of the Shenyang Economic Zone urban system is centripetal gathered around the central city of Shenyang, the accessibility of the transport network between cities is good, and the spatial shape is more compact than other urban systems of Liaoning Province. The authors believe that the structure of the Shenyang Economic Zone urban system is more effective than the structure of Liaoning Province urban system, and put forward rationalization proposals to optimize the urban system of the Shenyang Economic Zone.

On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on R n . We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.

Characterization of irregular seeds on gibbsites precipitated from caustic aluminate solutions

The irregular surface of seeds on which gibbsites are precipitated from caustic aluminate solutions, was investigated according to the fractal theory. Two kinds of fractal dimensions were used to characterize these irregularity. Box-dimension and spectral dimension are based on the SEM images of seeds and diffusive dynamic equation of the precipitation respectively. Both these two dimensions are affected by the reaction temperature, evolved with different reaction conditions and can reflect the influence of irregularity of seeds on the precipitation rate. Box dimension is fit for the characterization of the irregular morphology of seeds, while spectral dimension can explain the fractal dynamic behavior.

Statistics of the galaxy distribution

The universe appears from recent observational results to be a highly structured but also highly disordered medium. This accounts for the difficulties with a conventional statistical approach. Since the statistics of disordered media is an increasingly well-studied field in physics, it is tempting to try to adapt its methods for the study of the universe (the use of correlation functions also resulted from the adaptation of techniques from a very different field to astrophysics). This is already the case for the fractal analysis, which, mainly developed in microscopic statistics, is increasingly used in astrophysics. I suggest a new approach, also derived from the study of disordered media, both from the study of percolation clusters and from the dynamics of so-called “cluster aggregation” gelification models. This approach is briefly presented. Its main interest lies in two points. First, it suggests an analysis able to characterize features of unconventional statistics (those that seem to be present in the galaxy distribution and which conventional indicators are unable to take into account). It appears alsoa priori very convenient for a synthetic approach, since it can be related to the other indicators used up to now: the link with the void probability function is very straightforward. The connexion with fractals can be said to be contained in the method, since the objects defined during this analysis are themselves fractal: different kinds of fractal dimensions are very easy to extract from the analysis. The link with the percolation studies is also very natural since the method is adapted from the study of percolation clusters. It is also expected that the information concerning the topology is contained in this approach; this seems natural since the method is very sensitive to the topology of the distribution and posses some common characteristics with the topology analysis already developed by Gottet al. (1986). The quantitative relations remain however to be calculated. Additionally, this approach concerns the variation of clustering properties of galaxy groups and clusters with their richness. Although such studies have been made for various cases (like comparison of the correlation functions between galaxies and clusters, or between clusters of different richness classes), the analysis presented here deals with it in a more systematic and synthetic way.

４種類の電算機シミュレーションで作成した均一球形粒子集合体のフラクタル次元

The three dimensional randomly packed systems of uniform-sized spheres were simulated by four kinds of algorithms. From a multi-fractal analysis, two different kinds of fractal dimensions in the packed system were obtained by the scaling method. The results show that the difference in particle packed structure, which cannot be clearly discriminated from the relationship between the average coordination number and the void fraction in a packed system, became discriminatable by using fractal dimensions either on long or short length scales. Therefore, fractal dimensions are useful to quantitatively express the packing structure in the particle packed systems.