Fractal Dimension Spectrum Research Articles

A Study on the Computer Simulation for the Fractal Growth of Semiconductor Thin Films

This paper is based on theoretical methods to study the computer simulation and analysis of the growth of semiconductor thin films. First, according to the traditional theory of thin-film growth, the relationship between the growth morphology and the physical parameters are discussed. Then, fractal theory has been applied to improve the diffusion-limited aggregation (DLA) model. And the simulations of the two-dimensional and three-dimensional thin-film growth are proposed. A computer program of the simulation of the thin-film growth is developed with help of MATLAB. Finally, the results of the simulation of the thin-film growth have been analyzed by the fractal dimension and multifractal spectra. The results of this paper can be applied to the dynamic simulation of nanometer thin-film growth, and an effective simulation tool is to provide the semiconductor process.

Real-Time Mental Arithmetic Task Recognition From EEG Signals

Electroencephalography (EEG)-based monitoring the state of the user's brain functioning and giving her/him the visual/audio/tactile feedback is called neurofeedback technique, and it could allow the user to train the corresponding brain functions. It could provide an alternative way of treatment for some psychological disorders such as attention deficit hyperactivity disorder (ADHD), where concentration function deficit exists, autism spectrum disorder (ASD), or dyscalculia where the difficulty in learning and comprehending the arithmetic exists. In this paper, a novel method for multifractal analysis of EEG signals named generalized Higuchi fractal dimension spectrum (GHFDS) was proposed and applied in mental arithmetic task recognition from EEG signals. Other features such as power spectrum density (PSD), autoregressive model (AR), and statistical features were analyzed as well. The usage of the proposed fractal dimension spectrum of EEG signal in combination with other features improved the mental arithmetic task recognition accuracy in both multi-channel and one-channel subject-dependent algorithms up to 97.87% and 84.15% correspondingly. Based on the channel ranking, four channels were chosen which gave the accuracy up to 97.11%. Reliable real-time neurofeedback system could be implemented based on the algorithms proposed in this paper.

Fractal Application in Lithological Classification and Alteration Extraction

In order to promote the application of multi-fractal theory to remote sensing exploration,this paper takes the West Junggar Region as the study area to build fractal dimension spectra by analyzing the characteristics of the region and the remote sensing image.Combined with spectra,the authors completed the lithologic classification and pyrite alteration extraction of the region.And then mineral targets of the region was identified with the support of the existing statistical data of mines and expert's experience and knowledge.Practice has proved that the appropriate prospecting model and criteria established on remote sensing images of fractal theory and spectral characteristics can satisfactorily supply decision support to regional mineral exploration.

Modeling the relationship between Higuchi’s fractal dimension and Fourier spectra of physiological signals

The exact mathematical relationship between FFT spectrum and fractal dimension (FD) of an experimentally recorded signal is not known. In this work, we tried to calculate signal FD directly from its Fourier amplitudes. First, dependence of Higuchi's FD of mathematical sinusoids on their individual frequencies was modeled with a two-parameter exponential function. Next, FD of a finite sum of sinusoids was found to be a weighted average of their FDs, weighting factors being their Fourier amplitudes raised to a fractal degree. Exponent dependence on frequency was modeled with exponential, power and logarithmic functions. A set of 280 EEG signals and Weierstrass functions were analyzed. Cross-validation was done within EEG signals and between them and Weierstrass functions. Exponential dependence of fractal exponents on frequency was found to be the most accurate. In this work, signal FD was for the first time expressed as a fractal weighted average of FD values of its Fourier components, also allowing researchers to perform direct estimation of signal fractal dimension from its FFT spectrum.

Characterization of Surface Roughness Effects on Laminar Flow in Microchannels by Using Fractal Cantor Structures

The fractal characterization of surface topography by using Cantor set structures is introduced to quantify the microchannel surface. Based on this fractal characterization of surface, a model of laminar flow in rough microchannels is developed and numerically analyzed in this paper. The effects of Reynolds number, relative roughness, and fractal dimension on laminar flow are all discussed. The results indicate that the presence of roughness leads to the form of detachment, and the eddy generation is observed at the shadow of the roughness elements. The pressure drop in the rough microchannels along the flow direction is larger than that in the smooth channel. It is no longer in a linear fashion and the fluctuation in pressure drop along the stream due to the vortex near the wall is found. Differing from the smooth channel, the Poiseuille number for laminar flow in rough microchannels is no longer only dependent on the cross-sectional shape of the channel, but also strongly influenced by the Reynolds number, roughness height, and fractal dimension (spectrum) of the surface.

Classification of Prostate Histopathology Images Based on Multifractal Analysis

Histopathology is a microscopic anatomical study of body tissues and widely used as a cancer diagnosing method. Generally, pathologists examine the structural deviation of cellular and sub-cellular components to diagnose the malignancy of body tissues. These judgments may often subjective to pathologists' skills and personal experiences. However, computational diagnosis tools may circumvent these limitations and improve the reliability of the diagnosis decisions. This paper proposes a prostate image classification method by extracting textural behavior using multifractal analysis. Fractal geometry is used to describe the complexity of self-similar structures as a non-integer exponent called fractal dimension. Natural complex structures (or images) are not self-similar, thus a single exponent (the fractal dimension) may not be adequate to describe the complexity of such structures. Multifractal analysis technique has been introduced to describe the complexity as a spectrum of fractal dimensions. Based on multifractal computation of digital imaging, we obtain two textural feature descriptors; i) local irregularity: α and ii) global regularity: f(α). We exploit these multifractal feature descriptors with a texton dictionary based classification model to discriminate cancer/non-cancer tissues of histopathology images of H&E stained prostate biopsy specimens. Moreover, we examine other three feature descriptors; Gabor filter bank, LM filter bank and Haralick features to benchmark the performance of the proposed method. Experiment results indicated that the performance of the proposed multifractal feature descriptor outperforms the other feature descriptors by achieving over 94% of correct classification accuracy.

Fractures distribution modeling using fractal and multi-fractal–neural network analysis in Tabnak hydrocarbon field, Fars, Iran

Modeling of fractures distribution in naturally fractured hydrocarbon reservoirs is a complex process that contains large amount of uncertainty. High gas production from the Dehram group of Tabnak hydrocarbon reservoir in Fars province indicates the presence of natural fractures. This study presents a novel methodology that integrates various features of geological, statistical and artificial intelligence techniques in a nested loop to characterize field fractures and then to model them. Their characterization is, to some extent, technique dependent. Secondary properties such as fracture density, fractal dimension and fractal spectrum are there defined for better description of fractures’ spatial distribution. Irregular geometry laws showed that the connectivity of fractured media depends on power-law exponent and some fracture characteristics. A neural network is incorporated in the proposed methodology to determine these relationships, by processing field data available from the six outcrops with similar lithology, image logs and core analyses. The value of 1/82 was calculated for fractal dimension of surface fractures. The fracture density in the range of 0.2 to 1.4, fractal dimension and fractal spectrum (D(q = 0)) of fractures in the range of 1.0 to 1.6 and 1.8 to 3.0 respectively were calculated for the 10-m intervals within the well and extended with fractal–neural network algorithm. Determination of fractal dimension and fractal spectrum from image logs and dual application of neural network and fractal geometry in fracture modeling are innovative in this study. Finally fracture distribution models estimated are of great interest with gas production rates of the Tabnak hydrocarbon reservoir and have more reality results compared with DFN algorithm models using Open Flow software.

Domain Wall Dynamics Near the Phase Transition in PZN-9%PT

To investigate the paraelectric–ferroelectric phase transition in relaxor Pb(Zn1/3Nb2/3)O3-9% PbTiO3, correlations and scaling relations were measured from the domain structure of single crystals near the Curie temperature using a multifractal analysis. The dynamics of the domain walls were observed in (100) platelets using polarized light microscopy. The complexity of the domain wall distribution was evaluated by means of its fractal dimension and singularity spectrum. The changes in the energy state of the samples are assessed by measuring changes of the spatial distribution of the walls as a function of temperature; this can be used to estimate the specific heat.

A fast MATLAB program to estimate the multifractal spectrum of multidimensional data: Application to fractures

A MATLAB® program based on the Hou algorithm for estimation of fractal dimension and multifractal spectrum of fractures is presented. The program performance was tested with many synthetical fractals and field data. Interpolation and sampling effects on the fractal dimension and multifractal spectrum estimation were also studied. Some common problems related to the fractal dimension and multifractal spectrum are also discussed.

Multifractal Spectra of INGaN/GaN Self-Assembled Quantum Dots Films

The surface shape and microstructure of semiconductor thin films, especially nanometer thin films, have important influence to construct physical characteristics, such as electricity, magnetic, and optics nature to the thin films. In this work, we use the multifractal spectra to study the surface morphology of InGaN/GaN self-assembled quantum dot films after the annealed process. Samples used in this study were grown on the (0001)-oriented sapphire (Al2O3) substrates in a vertical low-pressure metal-organic chemical vapor deposition reactor with a high-speed rotation disk. The fractal dimension and multifractal spectra can be used to describe the influence of different annealed conditions on surface characterization. Fractal analysis reveals that both the average surface roughness and root-mean-square roughness of nanostructure surfaces are decreased after the thermal annealing process. It can be seen that a smoother surface was obtained under an annealing temperature at 800°C, and it implies that the surface roughness of this case is minimum in all tests. The results of this paper also described a mathematical modeling method for the observation of the fractal and multifractal characteristics in a semiconductor nanostructure films.

A Unified Approach To Fractal Dimensions

Many scientific papers treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This paper presents a new description of three essential classes of fractal dimensions based on: (1) morphology, (2) entropy, and (3) transforms, all unified through the generalized-entropy-based Rényi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this paper is unique in terms of (1) its progressive development of the fractal dimension concept, (2) similarity in the definitions and expressions, (3) analysis of the relation between the dimensions, and (4) their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioral processes (such as dishabituation), irreversible and birth-death growth phenomena (e.g., diffusion-limited aggregates (DLAs), dielectric discharges, and cellular automata), as well as dynamical non-stationary transient processes (such as speech and transients in radio transmitters), multi-fractal optimization of image compression using learned vector quantization with Kohonen’s self-organizing feature maps (SOFMs), and multi-fractal-based signal denoising.

SCALING BEHAVIOR OF DIFFUSION LIMITED AGGREGATION IN PERCOLATION CLUSTER

This paper presents a computer model of diffusion limited aggregation (DLA) in percolation cluster. Simulation of the aggregation clusters in percolation cluster with varying occupancy probability is performed, and their fractal dimension and multifractal spectrum are obtained. The simulation results show that the percolation cluster has stronger effects on the aggregation clusters' pattern structure when occupancy probability is smaller. The dimension Df of aggregation clusters increases together with the increase of occupancy probability. Furthermore, the multifractal spectra f(α) curve becomes higher and the range of singularity α wilder. The bigger the occupancy probability is, the more irregular and non-uniform the aggregation clusters becomes.

Obtaining the size distribution of fault gouges with polydisperse bearings

We generalize a recent study of random space-filling bearings to a more realistic situation, where the spacing offset varies randomly during the space-filling procedure, and show that it reproduces well the size distributions observed in recent studies of real fault gouges. In particular, we show that the fractal dimensions of random polydisperse bearings sweep predominantly the low range of values in the spectrum of fractal dimensions observed along real faults, which strengthen the evidence that polydisperse bearings may explain the occurrence of seismic gaps in nature. In addition, the influence of different distributions on the offset is studied and we find that a uniform distribution is the best choice for reproducing the size distribution of fault gouges.

A Relative Fractal Dimension Spectrum for a Perceptual Complexity Measure

This article presents a derivation of a new relative fractal dimension spectrum, DRq, to measure the dissimilarity between two finite probability distributions originating from various signals. This measure is an extension of the Kullback-Leibler (KL) distance and the Rényi fractal dimension spectrum, Dq. Like the KL distance, DRq determines the dissimilarity between two probability distibutions X and Y of the same size, but does it at different scales, while the scalar KL distance is a single-scale measure. Like the Rényi fractal dimension spectrum, the DRq is also a bounded vectorial measure obtained at different scales and for different moment orders, q. However, unlike the Dq, all the elements of the new DRq become zero when X and Y are the same. Experimental results show that this objective measure is consistent with the subjective mean-opinion-score (MOS) when evaluating the perceptual quality of images reconstructed after their compression. Thus, it could also be used in other areas of cognitive informatics.

A Unified Approach To Fractal Dimensions

Many scientific papers treat the diversity of fractal dimensions as mere variations on either the same theme or a single definition. There is a need for a unified approach to fractal dimensions for there are fundamental differences between their definitions. This paper presents a new description of three essential classes of fractal dimensions based on: (1) morphology, (2) entropy, and (3) transforms, all unified through the generalized-entropy-based Rényi fractal dimension spectrum. It discusses practical algorithms for computing 15 different fractal dimensions representing the classes. Although the individual dimensions have already been described in the literature, the unified approach presented in this paper is unique in terms of (1) its progressive development of the fractal dimension concept, (2) similarity in the definitions and expressions, (3) analysis of the relation between the dimensions, and (4) their taxonomy. As a result, a number of new observations have been made, and new applications discovered. Of particular interest are behavioral processes (such as dishabituation), irreversible and birth-death growth phenomena (e.g., diffusion-limited aggregates (DLAs), dielectric discharges, and cellular automata), as well as dynamical non-stationary transient processes (such as speech and transients in radio transmitters), multi-fractal optimization of image compression using learned vector quantization with Kohonen’s self-organizing feature maps (SOFMs), and multi-fractal-based signal denoising.

Quantitative discrimination of magma fragmentation and pyroclastic transport processes using the fractal spectrum technique

The morphology of volcanic particles can yield insight into magma fragmentation, transport processes, and style of eruption. However, the complexity and variability of volcanic particle shapes make quantitative characterization difficult. The technique applied in this study is based on fractal geometry, which has been successfully used to characterize a wide variety of particles and shapes. Here, fractal data is produced by dilation of the 2-D particle boundary to produce a full spectrum of fractal dimensions over a range of scales for each particle. Multiple fractal dimensions, which can be described as a fractal spectrum curve, are calculated by taking the first derivative of data points on a standard Richardson plot. Use of multiple fractal dimensions results in more effective discrimination than expressions of shape based on one or two fractal dimensions. Quantitative comparisons are carried out using multivariate statistical techniques such as cluster and principal components analysis. Applications to samples from well-documented eruptions (e.g. Mt. St. Helens 1980, Tambora 1815, Surtsey 1963–64) indicate that the fractal spectrum technique provides a useful means of characterizing volcanic particles and can be helpful for identifying the products of specific fragmentation processes (volatile exsolution, phreatomagmatic, quench granulation) and modes of volcanic transport/deposition (tephra fall, pyroclastic flow, blast/surge).

Scaling Behaviour of Diffusion Limited Aggregation with LinearSeed

We present a computer model of diffusion limited aggregation with linearseed. The clusters with varying linear seed lengths are simulated, andtheir pattern structure, fractal dimension and multifractal spectrum areobtained. The simulation results show that the linear seed length haslittle effect on the pattern structure of the aggregation clusters ifits length is comparatively shorter. With its increasing, the linearseed length has stronger effects on the pattern structure, while thedimension Df decreases. When the linear seed length is larger, thecorresponding pattern structure is cross alike. The larger the linearseed length is, the more obvious the cross-like structure with moreparticles clustering at the two ends of the linear seed and along thevertical direction to the centre of the linear seed. Furthermore, themultifractal spectra curve becomes lower and the range of singularitynarrower. The longer the length of a linear seed is, the less irregularand nonuniform the pattern becomes.

Characterizing chaos through Lyapunov metrics

Science, engineering, medicine, biology, and many other areas deal with signals acquired in the form of time series from different dynamical systems for the purpose of analysis, diagnosis, and control of the systems. The signals are often mixed with noise. Separating the noise from the signal may be very difficult if both the signal and the noise are broadband. The problem becomes inherently difficult when the signal is chaotic because its power spectrum is indistinguishable from broadband noise. This paper describes how to measure and analyze chaos using Lyapunov metrics. The principle of characterizing strange attractors by the divergence and folding of trajectories is studied. A practical approach to evaluating the largest local and global Lyapunov exponents by rescaling and renormalization leads to calculating the m Lyapunov exponents for m-dimensional strange attractors either modeled explicitly (analytically) or reconstructed from experimental time-series data. Several practical algorithms for calculating Lyapunov exponents are summarized. Extensions of the Lyapunov exponent approach to studying chaos are also described briefly as they are capable of dealing with the multiscale nature of chaotic signals. The extensions include the Lyapunov fractal dimension, the Kolmogorov--Sinai and Re/spl acute/nyi entropies, as well as the Re/spl acute/nyi fractal dimension spectrum and the Mandelbrot fractal singularity spectrum.

A random multifractal tilling

We develop a multifractal random tilling that fills the square. The multifractal is formed by an arrangement of rectangular blocks of different sizes, areas and number of neighbors. The overall feature of the tilling is an heterogeneous and anisotropic random self-affine object. The multifractal is constructed by an algorithm that makes successive sections of the square. At each n-step there is a random choice of a parameter ρi related to the section ratio. For the case of random choice between ρ1 and ρ2 we find analytically the full spectrum of fractal dimensions.

Multifractality as a Measure of Spatial Distribution of Geochemical Patterns

To quantify the spatial distribution of geochemical elements, the multifractality indices for Zn, Cu, Pt, Pd, Cr, Ni, Co, Pb, and As in lake-sediment samples in the Shining Tree area in the Abitibi area of Ontario are determined. The characterization of multifractal distribution patterns is based on the box-counting moment method and involves three functions: a mass exponent function τ(q); Coarse Holder Exponent α (q); and fractal dimension spectrum f(α (q)). Properties of these functions at different values of q, characterize the spatial distribution of the variable under study. It is shown that the degree of multifractality defined by τ”(1) can be used as a measure of irregularity of geochemical spatial dispersion patterns. The variations of Zn and Cu in the study area are characterized by relatively low degree of multifractality, whereas those for Pt, Pd, Cr, Ni, and Co; and particularly for As and Pb are characterized by higher multifractality indices.In the case of Zn and Cu, singularity spectra are close to a monofractal compared to the ones for As an Pb. The determination of multifractality indices allows us, in a quantitative way, to study the pattern of metal dispersions and link them to different physical processes, such as metal adsorption by organic material or glaciogenic processes.