Embedding a Signature in an Image Based on Fractal Compress Transformations

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Fractal Surface Synthesis Based on Two Dimensional Discrete Fourier Transform

The discrete Fourier transform（DFT） is used for fractional Brownian motion（FBM） surface synthesis in tribology（i.e., contact, sliding, and sealing, etc）. However, the relationship between fractal parameters（fractal dimension and scale factor） and traditional parameters, the influence of fractal parameters on surface appearance, have not been deeply discussed yet. These lead to some kind of difficulty to ensure the synthesized surfaces with ideal fractal characteristic, required traditional parameters and geometric appearance. A quantitative relationship between fractal parameters and the root mean square deviation of surface （Sq） is derived based on the energy conservation property between the space and frequency domain of DFT. Under the stability assumption, the power spectrum of a FBM surface is composed of concentric circles strictly, a series of FBM surfaces with prescribed Sq could be synthesized with given fractal dimension, scale factor, and sampling numbers, but the ten-point height（Sz）, the skewness（Ssk） and the kurtosis（Sku） are still in random, where the probability distributions of Sz and Ssk are approximately normal distribution. Furthermore, by iterative searching, a surface with desired Abbott-Firestone curve could be obtained among those surfaces. An intuitive explanation for the influence of fractal dimension and scale factor on surface appearance is obtained by discussing the effects on the ratio of energy between high and low frequency components. Based on the relationship between Sq and surface energy, a filtering method of surface with controllable Sq is proposed. The proposed research ensures the synthesized surfaces possess ideal FBM properties with prescribed Sq, offers a method for selecting desired Abbott-Firestone curve of synthesized fractal surfaces, and makes it possible to control the Sq of surfaces after filtering.

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.

Realization of Fractal Affine Transformation

This paper gives the definition of fractal affine transformation and presents a specific method for its realization and its corresponding mathematical equations which are essential in fractal image construction.

Application of Fractal Theory in Simulation of Ferromagnetic Elements' Hysteresis Loop in Transformer

The simulation of the transformer transient is one of the indispensable qualifications for improving the performance of transformer protection, the key technique of the transformer's transient simulation is the treatment of ferromagnetic elements' loop. Thus the shapes of the primary hysteresis loop and each internal secondary hysteresis loop in the identical magnetism conducting are analyzed, and then it is proposed that there are some fractal characteristics in the relation between them. The fractal phenomenon of the ferromagnetic elements' hysteresis loop in the transformer's transient simulation is first brought forward, the mutuality between the ferromagnetic elements' primary hysteresis loop and its secondary hysteresis loops is revealed in mechanism by using the fractal theory. According to the iterated function system of fractal theory, the secondary hysteresis loops can be generated by the iterative calculation of the primary loop. The simulation results show the validity of this idea.

Classification of forearm action surface EMG signals based on fractal dimension

Surface electromyogram （EMG） signals were identified by fractal dimension.Two patterns of surface EMG signals were acquired from 30 healthy volunteers＇ right forearm flexor respectively in the process of forearm supination （FS） and forearm pronation （FP）.After the raw action surface EMG （ASEMG） signal was decomposed into several sub-signals with wavelet packet transform （WPT）,five fractal dimensions were respectively calculated from the raw signal and four sub-signals by the method based on fuzzy self-similarity.The results show that calculated from the sub-signal in the band 0 to 125 Hz,the fractal dimensions of FS ASEMG signals and FP ASEMG signals distributed in two different regions,and its error rate based on Bayes decision was no more than 2.26%.Therefore,the fractal dimension is an appropriate feature by which an FS ASEMG signal is distinguished from an FP ASEMG signal.

高碳钢中片状马氏体的分形(Fractal)特征

本文研究了T12钢不同温度加热淬火后得到的片状马氏体组织,认为在一个奥氏体晶粒内部,片状马氏体呈分形结构。实验表明在不同的相区,随淬火加热温度增加,分形维数D_f增加。在T=820℃附近D_f有一较大的变化,是由于A+FesC两相区向单相A区的转变所引起,同时也讨论了分形维数D_f与淬火组织硬度之间的关系。

FPGA implementation of fractal patterns classifier for multiple cardiac arrhythmias detection

This paper proposes the fractal patterns classifier for multiple cardiac arrhythmias on field-programmable gate array (FPGA) device. Fractal dimension transformation (FDT) is employed to adjoin the fractal features of QRS-complex, including the supraventricular ectopic beat, bundle branch ectopic beat, and ventricular ectopic beat. FDT with fractal dimension (FD) is addressed for constructing various symptomatic patterns, which can produce family functions and enhance features, making clear differences between normal and unhealthy subjects. The probabilistic neural network (PNN) is proposed for recognizing multiple cardiac arrhythmias. Numerical experiments verify the efficiency and higher accuracy with the software simulation in order to formulate the mathematical model logical circuits. FDT results in data self-similarity for the same arrhythmia category, the number of dataset requirement and PNN architecture can be reduced. Its simplified model can be easily embedded in the FPGA chip. The prototype classifier is tested using the MIT-BIH arrhythmia database, and the tests reveal its practicality for monitoring ECG signals.

Fault Diagnosis Method Based on Fractal Theory and Its Application in Wind Power Systems

The non-linear dynamic theory brought a new method for recognizing and predicting complex non-linear dynamic behaviors. The non-linear behavior of vibration signals can be described by using fractal dimension quantitatively. In this paper, a fractal dimension calculation method for discrete signals in the fractal theory was applied to extract the fractal dimension feature vectors and classified various fault types. Based on the wavelet packet transform, the energy feature vectors were extracted after the vibration signal was decomposed and reconstructed. Then, a wavelet neural network was used to recognize the mechanical faults. Finally, the fault diagnosis for a wind power system was taken as an example to show the method's feasibility.

GENERALIZED FLOW ANALYSIS OF NON-NEWTONIAN VISCO-ELASTICFLUID FLOW THROUGH FRACTAL RESERVOIR

In this paper, fractal geometry theory is used to combine with the seepage flow mechanics to establish the relaxation models of non_Newtonian visco_elastic fluid flow in fractal reservoirs. A method to scale the fractal properties of a fractal reservoir by the double parameters (d f ,d s ) and to describe the generalized flow characteristics of visco_elastic fluid by four parameters (d f ,d s ,λ v,λ p) are presented. Exact solutions and asymptotic solutions have been obtained by using the Laplace_Weber and Laplace_orthogonal transforms with both infinite and finite reservoirs. The pressure transient behavior of non_Newtonian visco_elastic fluid flow through a fractal reservoir are studied by using the numerical Laplace transform inversion and asymptotic solutions. The law of pressure change for various fractal parameters is obtained.

MULTISCALE IMAGE SEGMENTATION USING FRACTAL AND NEURAL NETWORK

Clustering algorithms in feature space are important methods in image segmentation. The choice of the effective feature parameters and the construction of the clustering method are key problems encountered with clustering algorithms. In this paper, the multifractal dimensions are chosen as the segmentation feature parameters which are extracted from original image and wavelet-transformed image. SOM (Self-Organizing Map) network is applied to cluster the segmentation feature parameters. The experiment shows that the performance of the presented algorithm is very good.

Equivalence Transformations among Ising Models

Using topology, fractal analysis and investigation of lattice formation process we find two types of equivalence transformations among Ising models: topological equivalence transformation and formation equivalence transformation. With the help of the transformations and the known data of the critical points of simple cubic (sc) lattice and planar square (sq) lattice we get directly the critical points for face-centered cubic (fcc) lattice, body-centered cubic (bcc) lattice and diamond (d) lattice. The transformation itself results no error in the calculation. Other than Monte Carlo method and series expansion approach the equivalence transformations help us simplify much more greatly the calculation of the critical points for the three-dimensional models and understand much more deeply the structural connection among Ising models.

Performance analysis of a rate adaptive multi-band ultra-wideband system based on quadrature fractal modulation scheme

A rate adaptive multi-band ultra-wideband (UWB) system based on the quadrature fractal modulation (QFM) scheme was proposed.Exploring the use of homogeneous signals as modulating waveforms in UWB system,the signal within each 528MHz sub-band was divided into 8 different frequency bandwidths using wavelets transform and these data sequences to be transmitted were embedded into homogeneous waveforms.It is found that the use of homogeneous signals in such UWB system is quite feasible,leadings to a novel multi-rate diversity strategy.Within each 528MHz sub-band,the UWB-QFM system can provide much higher data rates than that of the UWB orthogonal frequency division multiplexing (OFDM) system.Simulation results also show that the bit error rate (BER) performance of the UWB-QFM system achieves a greatly improvement over existing UWB-OFDM system.Due to the fractal properties of the homogeneous signals,these data sequences to be transmitted can be recovered using arbitrarily short receiver signal.

FACE RECOGNITION BASED ON WAVELET-CURVELET-FRACTAL TECHNIQUE

In this paper,a novel face recognition method,named as wavelet-curvelet-fractal technique,is proposed. Based on the similarities embedded in the images,we propose to utilize the wave-let-curvelet-fractal technique to extract facial features. Thus we have the wavelet’s details in diagonal,vertical,and horizontal directions,and the eight curvelet details at different angles. Then we adopt the Euclidean minimum distance classifier to recognize different faces. Extensive comparison tests on dif-ferent data sets are carried out,and higher recognition rate is obtained by the proposed technique.

AFFINE TRANSFORMATION IN RANDOM ITERATED FUNCTION SYSTEMS

Random iterated function systems (IFSs) is discussed, which is one of the methods for fractal drawing. A certain figure can be reconstructed by a random IFS. One approach is presented to determine a new random IFS, that the figure reconstructed by the new random IFS is the image of the origin figure reconstructed by old IFS under a given affine transformation. Two particular examples are used to show this approach.

On Wavelet Transform General Modulus Maxima Metric for Singularity Classification in Mammograms

Continuous wavelet transform is employed to detect singularities in 2-D signals by tracking modulus maxima along maxima lines and particularly applied to microcalcification detection in mammograms. The microcalcifications are modeled as smoothed positive impulse functions. Other target property detection can be performed by adjusting its mathematical model. In this application, the general modulus maximum and its scale of each singular point are detected and statistically analyzed locally in its neighborhood. The diagnosed microcalcification cluster results are compared with health tissue results, showing that general modulus maxima can serve as a suspicious spot detection tool with the detection performance no significantly sensitive to the breast tissue background properties. Performed fractal analysis of selected singularities supports the statistical findings. It is important to select the suitable computation parameters-thresholds of magnitude, argument and frequency range-in accordance to mathematical description of the target property as well as spatial and numerical resolution of the analyzed signal. The tests are performed on a set of images with empirically selected parameters for 200 μm/pixel spatial and 8 bits/pixel numerical resolution, appropriate for detection of the suspicious spots in a mammogram. The results show that the magnitude of a singularity general maximum can play a significant role in the detection of microcalcification, while zooming into a cluster in image finer spatial resolution both magnitude of general maximum and the spatial distribution of the selected set of singularities may lead to the breast abnormality characterization.

HERMITE EXPANSIONS AND FRACTALMEASURES

Let mu be a locally uniformly alpha-dimensional measure on R-n and P-t(fd mu) be the Able Poisson means of Hermite expansions for f is an element of L-p(d mu), it is studied that the asymptotis properties of P-t(fd mu) as t --> 1_. Analogue of Wiener's theorem is obtained. Author also establishs the boundedness of the alpha-dimensional maximal conjugate Poisson integral operators from L-p(d mu) to the Lebesgne p-power integrable function spaces L-p(dx), and this derives directly the boundedness of Riesz transforms.

A Study on Some Aspects of Wavelet Fractal Compression

Based on explanation of wavelet fractal compression method, the significance of introducing wavelet decomposition into conventional fractal compression method is deeply investigated from the point of theoretical and practical view. The result of study can be regarded as valuable guidelines for taking advantages of wavelet transform to develop more effective image compression algorithm.

基于主元-分形理论的电子式电压互感器计量误差自主辨识方法研究

电子式电压互感器(EVT)由大量的电子单元组成,易受变电站复杂运行环境的影响,计量误差的长期稳定性较差。常规检测方法是定期与标准互感器比对校准,属于一种设备状态的抽样检测方法,无法评估EVT的计量数据对节点电能累积作用的影响。针对这一问题,文章基于电力系统节点三相对称的运行特性,通过对三相EVT的计量数据进行主元分形,在残差子分量中建立平方预测方差SPE统计量实现了EVT计量误差的自检测;通过计算EVT运行过程中SPE统计量的分形维数,实现了EVT计量误差劣化形式的自主辨识。研究成果有利于推动EVT的状态检测由定期检测发展为按需检测。

基于对数比变换和分形模型的地球化学异常识别与提取——以云南个旧锡多金属矿集区为例

覆盖区弱缓异常识别与靶区圈定,是矿产勘查的难点。云南个旧锡多金属矿集区50%以上面积被晚三叠统个旧组灰岩覆盖,属于典型老矿山,有必要开展新一轮找矿工作。本文以个旧锡多金属矿集区为研究对象,利用1∶20万水系沉积物数据,分别进行对数比变换、因子分析、S-A分析和靶区圈定。结果表明:加性对数比变换增强了因子分析的地质解释性,因子F1和F2的元素组合分别代表了酸性岩浆热液高-中-低温叠加的Sn矿化和基性岩浆体系的Cu矿化特征;S-A分析得到的异常区与个旧东西矿床高度契合,证实了S-A分形方法能有效消除“闭合效应”,提高覆盖区的异常识别效率;在未充分探明的异常区,圈定了4个潜在靶区,为后续的锡矿资源勘查提供了科学依据。