Methodological principles for fractal analysis of trabecular bone

A standardized methodology for the fractal analysis of histological sections of trabecular bone has been established.
A modified box counting method has been developed for use on a PC-based image analyser. The effect of image analyser settings, magnification, image orientation and threshold levels was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to calculate objectively more than one fractal dimension from the modified Richardson plot.
The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (λ < 25 μm) and the upper limit is 4250 μm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0.
It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than one fractal dimension, describing spatial structural entities. Fractal analysis is a model-independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and complements conventional histomorphometric and stereological techniques.     

非金属夹杂物分形特征研究

材料的宏观力学性能与材料的微观组织密切相关，以显微组织中离散的点状、线状分布的典型夹杂物为研究对象，用计盒维数测量了标准图谱中不同等级的非金属夹杂物图形的分形维数，考察了夹杂物等级与分维之间的关系。结果表明，非金属夹杂物可用分形维数定量描述；随着夹杂物等级的上升，分形维数变大。     

Evaluation of the wavelet transform method for machined surface topography 2: fractal characteristic analysis

In this paper, a new method based on wavelet transform is proposed as a means for studying the fractal characteristics of rough surfaces. Through estimation of normal mathematical curves with known fractal dimensions, generated by the Weierstrass-Mandelbrot function, Majumdar-Bhushan function, Fractal Brownian motion (including three methods: the Midpoint FBm, the Additions FBm, the Interpolated FBm) and Interposed method (Kiesswetter curve), it is validated that the wavelet transform method can accurately calculate the fractal dimension. These fractal functions have been used to simulate some surface profiles. The results indicate that the Wavelet transform method is the most precise in its calculation of the fractal dimensions of the curves. It obtains more accurate results than seven other methods, named the Box counting method, the Yardstick method, the Co-variation method, the Structure function method, the Variation method, the Power Spectrum method and the Rescaled range analysis method. Precisely calculating the fractal dimensions of the curves is the first step in characterising machined surface topography. In addition, this paper aims to further develop the evaluation procedure for the fractal characteristics of machined surface topography.     

磨合表面形貌变化的分形表征

用结构函数法计算磨损表面轮廓的分形维数和尺度系数。研究表明 :分形维数或尺度系数不能实现表面的唯一性表征。因此 ,把分形维数和尺度系数相结合 ,提出一个新的分形参数———特征粗糙度 ,给出了其定义和计算表达式 ,并在推进式试验机上进行摩擦磨损试验 ,对试件表面某一位置在不同磨合阶段的形貌进行精确复位测量 ,用特征粗糙度表征形貌的变化。表征结果表明 :特征粗糙度对反映磨合表面形貌的变化不但表现出很好的灵敏性 ,而且比分形维数更具有规律性。     

分形维数计算的流水线优化方法研究

分形维数计算具有计算复杂度高、计算时间长等特点,严重影响计算的实时性。针对此问题,在充分分析分形维数计算内在特性的基础上,利用分形维数具有流水线计算的特点,提出了一种计算分形维数的流水线体系结构,可有效提高分形维数计算的实时性。由于嵌入式并行处理硬件平台资源有限,对分形维数计算实时性进行优化的同时还需要考虑资源消耗的优化。通过对不同级数流水线运行时间和资源消耗的分析,建立基于运行时间与资源消耗的优化目标函数,从而得到运行时间与资源消耗最优的流水线结构。并与已有的计算分形维数的并行算法进行对比分析,实验结果表明,本文提出的优化方法在提高计算实时性的同时有效降低了资源消耗,实现了运行时间与资源消耗的优化。     

PIM粉末颗粒的分形特征及其分形维数

分析了粉末注射成形几种常用粉末颗粒形状、投影边界、表面的分形特征。介绍了一种适用于粉末颗粒的分维测量方法。根据民镜图片，用“数盒子”法测算了羰基铁和羰基镍粉投影边界图形的分形维数，它们分别在1.068-1.080、1.225-1.235之间，说明羰基镍粉末颗粒的形状特征有可能用Koch曲线分形性质来进行描述和分析，分形理论的引入可为研究粉末注射成形提供更准确的定量描述原料特征的方法，为粉末注射成形过程的控制提供了更精确的工艺参数。     

Influence of scanning rate on quality of AFM image: Study of surface statistical metrics

The purpose of this work is to study the dependence of AFM‐data reliability on scanning rate. The three‐dimensional (3D) surface topography of the samples with different micro‐motifs is investigated. The analysis of surface metrics for estimation of artifacts from inappropriate scanning rate is presented. Fractal analysis was done by cube counting method and evaluation of statistical metrics was carrying out on the basis of AFM‐data. Combination of quantitate parameters is also presented in graphs for every measurement. The results indicate that the sensitivity to scanning rate growths with fractal dimension of the sample. This approach allows describing the distortion of the images against scanning rate and could be applied for dependences on the other measurement parameters. The article explains the relevance and comparison of fractal and statistical surface parameters for characterization of data distortion caused by inappropriate choice of scanning rate.     

Quantification of structural changes in acute inflammation by fractal dimension,angular second moment and correlation

The aim of the study was to examine alteration and possible application of fractal dimension, angular second moment, and correlation for quantification of structural changes in acutely inflamed tissue. Acute inflammation was induced by injection of turpentine oil into the right and left hind limb muscles of mice, whereas control animals received intramuscular saline injection. After 12 h, animals were anesthetised and treated muscles collected. The tissue was stained by hematoxylin and eosin, digital micrographs produced, enabling determination of fractal dimension of the cells, angular second moment and correlation of studied tissue. Histopathological analysis showed presence of inflammatory infiltrate and tissue damage in inflammatory group, whereas tissue structure in control group was preserved, devoid of inflammatory infiltrate. Fractal dimension of the cells, angular second moment and correlation of treated tissue in inflammatory group decreased in comparison to the control group. In this study, we were first to observe and report that fractal dimension of the cells, angular second moment, and correlation were reduced in acutely inflamed tissue, indicating loss of overall complexity of the cells in the tissue, the tissue uniformity and structure regularity. Fractal dimension, angular second moment and correlation could be useful methods for quantification of structural changes in acute inflammation.     

机械加工表面形貌分形特征的计算方法

提出应用小波变换计算表面形貌分形特征参数，基于Weierstrass-Mandelbrot函数（W-M函数）和Majumdar-Bhushan函数（M－B函数）这2种常用于表征和模拟机械加工表面轮廓曲线的标准分形函数，验证了小波变换计算分形维数具有很高的精度。与其它计算表面形貌分形维数的方法进行了比较，结果表明小波变换方法的稳定性和准确性好。应用小波变换计算了不锈钢和铜2种材料的机械加工表面的分形维数。     

Intelligent fault diagnosis of rolling element bearing based on SVMs and fractal dimension

The development of non-linear dynamic theory brought a new method for recognising and predicting the complex non-linear dynamic behaviour. Fractal dimension can quantitatively describe the non-linear behaviour of vibration signal. In the present paper, the capacity dimension, information dimension and correlation dimension are applied to classify various fault types and evaluate various fault conditions of rolling element bearing, and the classification performance of each fractal dimension and their combinations are evaluated by using SVMs. Experiments on 10 fault data sets showed that the classification performance of the single fractal dimension is quite poor on most data sets, and for a given data set, each fractal dimension exhibited different classification ability, this indicates that various fractal dimensions contain various fault information. Experiments on different combinations of the fractal dimensions demonstrated that the combination of all these three fractal dimensions gets the highest score, but the classification performance is still poor on some data sets. In order to improve the classification performance of the SVM further, 11 time-domain statistical features are introduced to train the SVM together with three fractal dimensions, and the classification performance of the SVM is improved significantly. At the same time, experimental results showed that the classification performance of the SVM trained with 11 time-domain statistical features in tandem with three fractal dimensions outperforms that of the SVM trained only with 11 time-domain statistical features or with three fractal dimensions.     

Application of regularization dimension to gear damage assessment

Fractal dimension provides a measure of the complexity of a dynamic system, and contains the health information of a machine. The basics of regularization dimension and the effects of Gaussian kernel parameters on the regularization of a signal are introduced. Regularization dimension has advantages over other fractal dimensions because the scale-independent range can be selected according to the signal frequency components of interest. Experimental gearbox vibration signals are analyzed by means of spectral analysis firstly, and then according to the spectral structure, the scale-independent range is selected for computing the regularization dimension, which increases monotonically with increasing gear damage degree. Comparison with correlation dimension and kurtosis shows the advantages of regularization dimension in assessing the localized gear damage.     

Fe3Al基复合材料摩擦表面的分形特性研究

利用结构函数法对Fe3A l基复合材料干滑动摩擦表面的分形特性进行了研究,并计算其分形维数。结果表明Fe3A l基复合材料摩擦表面具有明显的分形特征,分形维数和摩擦因数以及磨损率之间有着密切相关性,随着分形维数的增大,摩擦因数和磨损率都逐渐减小,磨损率尤为明显,说明材料摩擦表面形貌特征对其摩擦学性能具有显著影响。     

小波变换方法评价曲线的分形特征

应用小波变换对Kiesswetter曲线和3种方法生成的分数维布朗运动（FBm）进行了分析,验证了该方法计算分形维数具有较高的精度。在宽广的分形维数范围内,与其他7种计算方法比较表明,小波变换方法的精确性和一致性都最好。小波变换为进一步分辨确定性信号、分形特征的信号或完全随机性的信号提供了一种有效工具,为评价粗糙表面形貌的分形特征提供了前提条件。     

基于小波变换的分形曲线维数计算方法的研究

研究了基于小波变换的分形曲线维数计算方法，具有算法简单和容易实现的优点；通过构造典型分形曲线并加以应用研究，提出并总结了小波分解尺度对维数计算精度的影响规律。根据影响规律，采用小波变换计算分形曲线维数，首先应该估计曲线的采样长度，根据曲线特征选择特定的小波函数，确定最佳的小波分解尺度，这样既提高了计算精度，又缩短了计算时间；其次，当分形曲线有限长度较短时，应该采用信号周期延拓的方法可以减少计算误差。     

铝合金加速腐蚀形貌分形特征提取

基于数字图像技术,将分形理论应用于腐蚀形貌图像的特征提取,结合LY12CZ铝合金在EXCO(exfoliationcorrosion)溶液中的加速腐蚀试验,用分形维数表征腐蚀形貌的演化过程,计算得到加速腐蚀不同时间的分维值。结果表明,腐蚀初期蚀坑数目增加明显,分形维数迅速增大。随后,腐蚀速率加快,蚀坑逐渐连成一片,分形维数增加减缓,并趋于平稳。     

分形在齿轮磨损监测中的应用

将分形理论用于齿轮磨损的监测,从工程应用角度介绍了振动信号盒维数的计算方法。通过对齿轮振动信号分形维数的计算,揭示了分形维数与信号复杂程度之间的内在联系。结果表明:随齿轮磨损的增加,齿轮振动信号盒维数呈下降趋势,运用振动信号的分形维数特征可有效实现齿轮磨损监测。     

A fractal-dimension-based signal-processing technique and its use for nondestructive testing

A fractal-dimension-based signal-processing technique has been extensively applied to various fields, but the use of the method to characterize discrete time-domain ultrasonic signals reflecting defects and any other structural-material inhomogeneities has not been fully investigated. The fractal features of the ultrasonic echoes with fractal dimensions and their implementation in nondestructive testing are investigated. In order to obtain a faithful representation of the fractal dimensions, two improved fractal dimension algorithms are presented: the box-counting method and the R/S (range/standard deviation) method. Their capabilities are evaluated with two kinds of fractal signals: the FBM (fractal Brownian motion) and WM (Weierstrass-Mandelbrot) signals. A new method to guarantee the feasibility of the calculated fractal dimensions is proposed on the basis of the analysis of the results simulated above. Then, the fractal dimensions of ultrasonic signals measured from a pipeline sample and from carbon-steel and aluminum specimens are calculated and statistically analyzed to find the fractal properties of the ultrasonic signals. The experimental results show that ultrasonic signals have the property of scale invariance that the fractal set possessed. The fractal dimension is indicative of the complexity and degree of irregularity of the waveform of an ultrasonic signal. The fractal dimensions of ultrasonic signals from various defects and microstructures are found to possess solid distribution intervals, which can be used to identify the presence of defects and the features of materials. The potential of the technique for testing defects and assessing the microstructure of materials via the use of ultrasonic echoes is revealed. The text was submitted by the authors in English.     

Fractal analysis and stereological evaluation of microstructures

Fractal properties and the concept of fractal dimension has been studied. Emphasis is given to the applicability in structure analysis. Comparison between different measurement procedures, analyses of mathematically defined lines and surfaces as well as measurements on real surfaces have been performed. The stereological consequences have been considered. A restrictive use of the fractal analysis results as an indicator of size, shape and self-similarity is recommended. If results obtained by quantitative microscopy at different magnification-resolution levels are to be compared, fractal analysis may be of advantage. The actual choice of resolution should yet be determined from the physical relevance of the geometrical details.     

基于分形理论的玻璃切削力和表面粗糙度研究

基于分形理论研究了玻璃切削时不同切削参数对切削力及其分形维数以及表面粗糙度的影响。试验结果表明:切削力分形维数与玻璃切削过程的四个阶段密切相关,如玻璃切削过程以某一阶段(如大块破碎)为主,则切削力分形维数越小,表面粗糙度越大;如切削过程包含多个阶段,则切削力分形维数越大,表面粗糙度越大。     

Fractal roundness modelling of a measured profile of a cylindrical object

Recently, a fractal analysis has been used as a useful tool in the characterization of the geometry. Especially in fractal analysis, a fractal interpolation method has been widely used to estimate and analyse experimental geometry. Due to the chaotic nature of the dynamic roundness profile in round shape machining, a more desirable method must be used for the analysis of experimental data which is natural to sequential data. A fractal method which is analysed in this paper focused its objects on the scope of the fractal interpolation and fractal dimension. Also for the calculation of fractal dimension, two methods for computing the fractal dimension have been introduced and discussed. These two methods can calculate the fractal dimensions of dynamic roundness profile according to the number of data points in which the fixed data are generally lower than 120 data points. This fractal analysis shows that it is possible to analysis fractal characteristics of roundness profile that have some different geometric properties. The fractal parameters were calculated and analysed using the measured profile of workpiece after turning.