Design and analysis of fractal detector for high resolution radars

Recently, fractal geometry has been used as a tool for improving the detection of targets in radar systems. The fractal dimension is utilized as a feature to distinguish between target and clutter in fractal detectors. In this paper, a general model is proposed for the target and clutter signals in high resolution radar (HRR). The fractal dimensions of the clutter and the target plus clutter are evaluated. Performing statistical tests on the distribution of the fractal dimension, it is proved that a gaussian distribution can approximately model the distribution of the fractal dimension for HRR signals. The fractal detector is designed based on the gaussian distribution of the fractal dimension and its performance is compared with a semi-optimum detector. It is demonstrated that the fractal detector has great capabilities in the rejection of colored clutter. Moreover, we show that the fractal detector is CFAR, i.e., the probability of false alarm remains approximately constant in different interference powers.     

Temporal variation of velocity components in a turbulent open channel flow: Identification of fractal dimensions

Fractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst’s rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold’s numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold’s number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold’s numbers.     

Fractal analysis of time series in oil and gas production

The peculiarities of fractal characteristics’ calculations for time series are described in this article. An algorithm for calculation of fractal dimension is suggested. It has been proved that the suggested method possesses high accuracy and the rapidity of convergence on the limited number of measurements compared to the methods of covering.The criteria of early diagnosis for changes in the condition of hydrodynamic processes, which do not vary by fractal dimension, have been recommended.The presented method is applicable for practical engineering calculations with self-affine, chaotic data, usually with relatively limited number of measurements. It is quite a simple method for calculation of fractal dimension, algorithm can be easily realized and it should be useful for engineers.The applicability of the proposed algorithm for fractal dimension calculation and early diagnosis criteria of qualitative changes in the behaviour of various dynamic systems has been tested both on simulated as well as practical examples of oil and gas production.     

Linear correlation between fractal dimension of surface EMG signal from Rectus Femoris and height of vertical jump

Fractal dimension was demonstrated to be able to characterize the complexity of biological signals. The EMG time series are well known to have a complex behavior and some other studies already tried to characterize these signals by their fractal dimension.This paper is aimed at studying the correlation between the fractal dimension of surface EMG signal recorded over Rectus Femoris muscles during a vertical jump and the height reached in that jump.Healthy subjects performed vertical jumps at different heights. Surface EMG from Rectus Femoris was recorded and the height of each jump was measured by an optoelectronic motion capture system.Fractal dimension of sEMG was computed and the correlation between fractal dimension and eight of the jump was studied.Linear regression analysis showed a very high correlation coefficient between the fractal dimension and the height of the jump for all the subjects.The results of this study show that the fractal dimension is able to characterize the EMG signal and it can be related to the performance of the jump. Fractal dimension is therefore an useful tool for EMG interpretation.     

岩石裂隙网络的分形特征

分形特征与分形维数广泛应用于岩石裂隙网络的量化,及与工程参数的关系模型建立.然而,严格的分形维数的极限定义形式难以直接应用,工程应用中多用近似分形维数值代替,近似的结果在建立量化关系模型时会产生蝴蝶效应,在量化及预测过程中产生巨大偏差.本文回顾了分形研究一系列的发展过程,并基于最新的分形定义提出了一种新的分形维数计算方法.通过对于十个岩石裂隙网络分形维数的计算,证明该方法能够准确有效的计算出图形的复杂度,避免了以往计算分形维数所产生的问题.     

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.     

Affective property of image and fractal dimension

Affective information processing is an advanced research direction in the AI world. Affective Information of image was taken as the objective of research in this paper. The influence of color vision properties’ histograms of image on human emotions was analyzed. Then based on fractal theory, the fractal aspect of different kinds of images was analyzed in keeping with the space domain. After that, psychological testing method of semantic difference was applied to verify the uniformity of the objective and subjective evaluations. At last, a conclusion was drawn that image having different affective properties could be classified by their fractal dimensions.     

Universality of fractal dimension on time-independent Hamiltonian systems

This paper summarizes a numerical study of the dependence of the fractal dimension on the energy of certain open Hamiltonian systems, which present different kind of symmetries. Owing to the presence of chaos in these systems, it is not possible to make predictions on the way and the time of escape of the orbits starting inside the potential well. This fact causes the appearance of fractal boundaries in the initial-condition phase space. In order to compute its dimension, we use a simple method based on the perturbed orbits’ behavior. The results show that the fractal dimension function depends on the structure of the potential well, contrary to other properties, such us the probability of escape, which has already been postulated as universal in earlier papers (see for instance [C. Siopis, H.E. Kandrup, G. Contopoulos, R. Dvorak, Universal properties of escape in dynamical systems, Celest. Mech. Dyn. Astr. 65 (57-68) (1997)]), from the study of Hamiltonians with different number of possible exits.     

Two-dimensional wave equations with fractal boundaries

This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The first case is the equation with classical derivative. The formal solution is obtained. And a definition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, differentiable nowhere in its domain. Next, for specific given initial position and 3 different initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given.     

基于分形理论的金融系统波动行为分数维理论计算

运用分形理论中分数维的定义和方法,对金融系统的波动行为进行了描述和研究,并且对金融系统中的时间序列数据介绍了两种分数维理论计算方法.     

Counterexamples in theory of fractal dimension for fractal structures

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.     

Fractal dimension of zeolite surfaces by calculation

A theoretical method for the estimation of the fractal dimensions of the pore surfaces of zeolites is proposed. The method is an analogy to the commonly employed box-counting method and uses imaginary meshes of various sizes (s) to trace the pore surfaces determined by the frameworks of crystalline zeolites. The surfaces formed by the geometrical shapes of the secondary building units of zeolites are taken into account for the calculations performed. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using equations proposed in this study. As a result, the fractal dimension values of the zeolites 13X, 5A and silicalite are generally observed to vary in significant amounts with the range of mesh size used, especially for the relatively larger mesh sizes that are close to the sizes of real adsorbates. For these relatively larger mesh sizes, the fractal dimension of silicalite falls below 1.60 while the fractal dimension values of zeolite 13X and 5A tend to rise above 2. The fractal dimension values obtained by the proposed method seem to be consistent with those determined by using experimental adsorption data in their relative magnitudes while the absolute magnitudes may differ due to the different size ranges employed. The results of this study show that fractal dimension values much different from 2 (both higher and lower than 2) may be obtained for crystalline adsorbents, such as zeolites, in ranges of size that are close to those of real adsorbates.     

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

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.     

基于分形维数的液体爆炸分散中界面破碎形态研究

通过对爆炸抛撒图象的处理,得到液体界面的曲线.采用盒维数的计算方式,计算界面曲线的分形维数.通过对各时刻液体界面分形维数的变化研究,分析爆炸抛撒近场阶段的变化过程,同时观察到蘑菇状尖顶的出现与破碎,以及空化区域的形成和消失现象。     

Fractal structure and fractal dimension determination at nanometer scale

Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of frsctal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in real metallic fractures is correlated to the intrinsic characteristics of the materials in addition to the effect of boundaries. The relationship of fractal dimensions with the mechanical properties of materials at macrometer scale also exists at nanometer scale. Project supported by the National Natural Science Foundation of China (Grant Nos. 59771050 and 59872004) and the Foundation Fund of Ministry of Metallurgical Industry.     

One-dimensional drug release from finite Menger sponges: In silico simulation

The purpose of this work was to evaluate the consequences of the spatial distribution of components in pharmaceutical matrices type Menger sponge on the drug release kinetic from this kind of platforms by means of Monte Carlo computer simulation. First, six kinds of Menger sponges (porous fractal structures) with the same fractal dimension, d_f=2.727, but with different random walk dimension, d_w[2.149,3.183], were constructed as models of drug release device. Later, Monte Carlo simulation was used to describe drug release from these structures as a diffusion-controlled process. The obtained results show that drug release from Menger sponges is characterized by an anomalous behavior: there are important effects of the microstructure anisotropy, and porous structures with the same fractal dimension but with different topology produce different release profiles. Moreover, the drug release kinetic from heteromorphic structures depends on the axis used to transport the material to the external medium. Finally, it was shown that the number of releasing sites on the matrix surface has a significant impact on drug release behavior and it can be described quantitatively by the Weibull function.     

线弹性动力学的某些一般定理及广义与广义分区变分原理*

从四维空间思想出发,在四种时端条件下,系统地推导得出了弹性动力学有关的一般定理,如:可能功作用量原理,虚位移原理,虚应力一动量原理,互易定理及由此导出的位移互等定理与始末时刻条件关系定理等;得出了线弹性动力学的位能作用量变分原理,余能作用量变分原理,动力问题的胡-鹫原理,H-R原理及本构关系变分原理.Hamilton原理,Toupin原理及有关文献如[5]、[17]～[24]的工作均可作为文中一般结果的特例.对应于有限元分析.在空间分区,时间分区及时空均分区情况.给出了动力学问题的分区位能作用量原理.分区余能作用量原理,分区混合能作用量原理及相应的分区广义变变分原理.导出了分区原理的一般形式.若去掉时间维及有关量,文中有关结果可转化为静力问题中有关的相应结果.     

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.