A simple method for estimating the fractal dimension from digital images: The compression dimension

The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem.     

On the local fractional wave equation in fractal strings

The key aim of the present study is to attain nondifferentiable solutions of extended wave equation by making use of a local fractional derivative describing fractal strings by applying local fractional homotopy perturbation Laplace transform scheme. The convergence and uniqueness of the obtained solution by using suggested scheme is also examined. To determine the computational efficiency of offered scheme, some numerical examples are discussed. The results extracted with the aid of this technique verify that the suggested algorithm is suitable to execute, and numerical computational work is very interesting.     

WAVELET TRANSFORM IN FRACTAL IMAGE ENCODING

This paper presents some results of the relation between wavelet transform and fractal transform. The wavelet transform of the attractor of fractal transform posseses translational and scale invariance. So we speed the fractal image encoding by testing the invariance of the wavelet transform appropriate for image encoding. The classfication scheme of range blocks by wavelet transform is given in this paper.     

星积分形曲面及其维数

通过分形曲线定义了一类分形曲面(被称为星积分形曲面),讨论了这类分形曲面的分形维数,得出了分形曲线的维数与它们所构造出的分形曲面维数之间的关系。     

The Samarskii problem for the fractal diffusion equation

We prove the existence and uniqueness of the solution of the Samarskii problem in a modified setting for a loaded differential fractal diffusion equation and propose a constructive scheme for the equivalent reduction of this nonlocal boundary-value problem to the corresponding local problem.     

Fractal compression coding based on wavelet transform with diamond search

In this paper, a fractal image compression coding scheme based on wavelet transform with diamond search is proposed. The goal is to offer fast positioning. According to search pattern and search path of diamond search, the proposed scheme just needs to search in the domain blocks in the fixed place around the range blocks. Our proposed method has benefits in reducing the search time and enhancing the coding speed compared with other image compression techniques.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

弱阻尼KdV方程的惯性分形集*

本文对非自共轭情形下弱阻尼非线性KdV方程,证明了惯性分形集存在且进行了分形维度的上界估计.     

The Minkowski dimension of the bivariate fractal interpolation surfaces

We present a new construction of continuous bivariate fractal interpolation surface for every set of data. Furthermore, we generalize this construction to higher dimensions. Exact values for the Minkowski dimension of the bivariate fractal interpolation surfaces are obtained.     

Fractal Approximation

In the present article every complex square integrable function defined in a real bounded interval is approached by means of a complex fractal function. The approximation depends on a partition of the interval and a vectorial parameter of the iterated function system providing the fractal attractor. The original may be discontinuous or undefined in a set of zero measure. The fractal elements can modify the features of the originals, for instance their character of smooth or non-smooth. The properties of the operator mapping every function into its fractal analogue are studied in the context of the uniform and least square norms. In particular, the transformation provides a decomposition of the set of square integrable maps. An orthogonal system of fractal functions is constructed explicitly for this space. Sufficient conditions for the uniform convergence of the fractal series expansion corresponding to this basis are also deduced. The fractal approximation of real functions is obtained as a particular case.     

Closed contour fractal dimension estimation by the Fourier transform

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its fractal dimension using the Fourier transform. The Fourier power spectrum is obtained and an exponential relation is verified between the power and the frequency. From the parameter (exponent) of the relation, is obtained the fractal dimension. The method is compared to other classical fractal dimension estimation methods in the literature, e.g., Bouligand–Minkowski, box-counting and classical Fourier. The comparison is achieved by the calculus of the fractal dimension of fractal contours whose dimensions are well-known analytically. The results showed the high precision and robustness of the proposed technique.     

Structure function and spectral density of fractal profiles

Spectral density and structure function for fractal profile are analyzed. It is found that the fractal dimension obtained from spectral density is not exactly the same as that obtained from structure function. The fractal dimension of structure function is larger than that of spectral density for small fractal dimension, and is smaller than that of spectral density for larger fractal dimension. The fractal dimension of structure function strongly depends on the spectral density at low and high wave numbers. The spectral density at low wave number affects the structure function at long distance, especially for small fractal dimension. The spectral density at high wave number affects the structure function at short distance, especially for large fractal dimension. This problem is more serious for bifractal profiles. Therefore, in order to obtain a correct fractal dimension, both spectral density and structure function should be checked.     

二元分形插值的拟合误差估计

A given bivariate continuous function is fitted by using a bivariate fractal interpolation function, and the error of fitting is studied in this paper. The results of error estimates are obtained in two metric cases. This provides a theoretical basis for the algorithms of fractal surface reconstruction.     

曲线造型的本征离散法

本文提出了一种基于弧长和转角这两个本质几何参数的曲线离散造型方法．该方法计算简单，几何意义明显，适用于逼近（拟合）和插值（补角），还可作出一类分形图形．其特点是一切细分操作都在转角关于弧长的对应关系图上进行．     

Separatrix Map Analysis for Fractal Scatterings in Weak Interactions of Solitary Waves

Previous studies have shown that fractal scatterings in weak interactions of solitary waves in the generalized nonlinear Schrödinger (NLS) equations are described by a universal second-order separatrix map. In this paper, this separatrix map is analyzed in detail, and hence a complete characterization of fractal scatterings in these weak interactions is obtained. In particular, scaling laws of these fractals are derived analytically for different initial conditions, and these laws are confirmed by direct numerical simulations. In addition, an analytical criterion for the occurrence of fractal scatterings is given explicitly.     

Fractal dimension and classification of music

The fractal aspect of different kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great number of different musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal dimension helps discriminate different categories of music. Then, we used an original statistical technique based on the Bootstrap assumption to find a time window in which fractal dimension reaches a high power of music discrimination. The best discrimination is obtained between 1/44100 and 16/44100 Hertz. We admit that to distinguish some different aspects of music well, the high information quantity is obtained in the high frequency domain. By calculating fractal dimension with the ANAM method, it was statistically proven that fractal dimension could distinguish different kinds of music very well: musics could be classified by their fractal dimensions.     

Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen.Our pointwise tube formulas are expressed as a sum of the residues of the “tubular zeta function” of the fractal spray in Rd. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers 0,1,…,d. The resulting “fractal tube formulas” are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula.     

一个Lie代数的子代数及其相关的两类Loop代数

本文构造了Lie代数A2的一个子代数A2,通过选取恰当的基元阶数得到相应的一个loop代数A2,由此设计一个等谱问题,利用屠格式得到了一个新的Liouville可积的Hamilton方程族.作为其约化情形,得到了一个非线性有理分式型演化方程.再由一个矩阵变换,得到了换位运算与A2等价的Lie代数A1的一个子代数A1,将A1再扩展成一个新的高维loop代数G,利用G获得了所得方程族的一类扩展可积系统.     

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set,and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a C1-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global C2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1)(2007), pp. 41-53].     

一类分形曲面的精细计盒维数

研究一类分形曲面的精细计盒维数,得到了星积分形曲面与其生成元的精细计盒维数的关系.