Fractal Analysis Method Research Articles

Using a Fractal Analysis and Polarization Method for Phase Identification in Three-Component Seismograms

This study presents the automatic P-wave and S-wave arrivals picking algorithm which is essentially based on the fractal dimension and polarized method. With an estimate of the spectral exponent γ in a 1/f process, an interval that indicates the preferred intersection containing both noise and the P-wave is well-detected by corresponding to the minimum absolute spectral exponent γ value along the trace. Based on the different properties of background noise and deterministic signal, the fractal dimension technique can detect the position of the P-wave. The place where the fractal dimension value changes suddenly within the intersection interval indicates the location of arrival of the P-wave. Testing that adds various levels of noise to the seismic signal shows the method can prove able to tolerate noise to a signal-to-noise (S/N) ratio 1.5. Based on the P-wave arrival, the polarized P-wave could be detected by a genetic algorithm (GA) with the strength of polarization and phase difference between the vertical and horizontal components as constraints. Using the first arrival phase as the basis phase, this study combines a polarization filter including rectilinearity functions, linear polarization, phase difference and directionality with GA to detect polarized S-wave of seismograms. Finally, the technique was applied to teleseismic data and near-field motion to verify the accuracy and wide applicability of this method. To conclude, this proposed method, an efficient and brand-new method of associating signal processing technique with physical wave motion properties, may be of importance in finding P-wave and S-wave phase arrivals accurately using three-component seismograms.

Simulation of crystallization process of iron with globular graphite

Special features of distribution of graphite inclusions in cast iron with globular graphite (IGG) are studied. The method of fractal analysis of images is used for determining the regular features of the distribution of eutectic cells during crystallization of iron. The dependences obtained are used for developing a fractal model of cluster crystallization of IGG.

Effects of Plant Population Density on Fractal Dimension of Corn Based on Image Processing

Using fractal analysis method and image process, corn cultivar Qiangsheng 49 was analyzed the effects of plant population density on fractal dimension (FD) of plant skeletal images. When the population density increased from 2.4 to 7.8 plants per square meters by 1.2 increment, the FD value decreased gradually from 1.192 to 1.163. Under all population density conditions, the FD values increased till early kernel filling and then decreased. The FD value was correlated with plant height, photosynthetic leaf sources and biomass to some extents. FD can be used widely in describing corn plant architecture as a new meaningful indicator.

On separated shear layers and the fractal geometry of turbulent scalar interfaces at large Reynolds numbers

This work explores fractal geometrical properties of scalar turbulent interfaces derived from experimental two-dimensional spatial images of the scalar field in separated shear layers at large Reynolds numbers. The resolution of the data captures the upper three decades of scales enabling examination of multiscale geometrical properties ranging from the largest energy-containing scales to inertial scales. The data show a −5/3 spectral exponent over a wide range of scales corresponding to the inertial range in fully developed turbulent flows. For the fractal aspects, we utilize two methods as it is known that different methods may lead to different fractal aspects. We use the recently developed method for fractal analysis known as the Multiscale-Minima Meshless (M3) method because it does not require the use of grids. We also use the conventional box-counting approach as it has been frequently employed in various past studies. The outer scalar interfaces are identified on the basis of the probability density function (p.d.f.) of the scalar field. For the outer interfaces, the M3 method shows strong scale dependence of the generalized fractal dimension with approximately linear variation of the dimension as a function of logarithmic scale, for interface-fitting reference areas, but there is evidence of a plateau near a dimension D ~ 1.3 for larger reference areas. The conventional box-counting approach shows evidence of a plateau with a constant dimension also of D ~ 1.3, for the same reference areas. In both methods, the observed plateau dimension value agrees with other studies in different flow geometries. Scalar threshold effects are also examined and show that the internal scalar interfaces exhibit qualitatively similar behaviour to the outer interfaces. The overall range of box-counting fractal dimension values exhibited by outer and internal interfaces is D ~ 1.2–1.4. The present findings show that the fractal aspects of scalar interfaces in separated shear layers at large Reynolds number with −5/3 spectral behaviour can depend on the method used for evaluating the dimension and on the reference area. These findings as well as the utilities and distinctions of these two different definitions of the dimension are discussed in the context of multiscale modelling of mixing and the interfacial geometry.

Fractal research on AFM images of polycrystalline aluminum surface with adsorption film of SDS inhibitor

We propose a new method of fractal analysis of the phase images obtained by the atomic-force microscopy (AFM) to study the properties of molecular adsorption films of corrosion inhibitors on the coarse polycrystalline surface, which can hardly be distinguished by using AFM height images due to the roughness of the surface. The system of polycrystalline aluminum and SDS (Sodium Dodecyl Sulfate) inhibitor is chosen as the research object. The electrochemical measurements of aluminum electrodes precoated with SDS films are carried out in HCl solutions, and the results show that the inhibition efficiency increases with the concentration of SDS in the prefilming solution and then remains nearly constant as the CMC (critical micelle concentration) of SDS is exceeded. The fractal analysis of the AFM images of prefilmed aluminum surfaces obtained in SDS solutions is performed and demonstrates that, as the SDS concentration increases, no obvious changes are observed for the fractal dimensions of height images, whereas the phase images present regular changes in the values of fractal dimensions, thus displaying the consistent alteration rule as revealed by electrochemical tests. The results prove the feasibility of the fractal analysis of AFM phase images in characterizing the adsorption behavior of the corrosion inhibitor on polycrystalline coarse surfaces.

Generation of super smooth surface in compression of aluminum

In order to obtain fundamental information about generation of super smooth surface in metal forming, a simple compression test of aluminum strips is carried out. The tool and specimen surfaces are observed on the nanometer scale by atomic force microscope (AFM). It is found that the specimen surface asperities are smoothed to the almost same surface roughness as the tool on the nanometer scale at about 20% compressive strain. The maximum height in scanning range 20×20 μm2 and 500×500 nm2 are attained to 8 nm and 1 nm at the smoothest position of the specimen profiles, respectively. The tool and specimen surfaces are examined by the power spectrum method and the zeroset method of fractal analysis.

SIZE MEASURE RELATIONSHIP METHOD FOR FRACTAL ANALYSIS OF SIGNALS

The fractal dimension of signals represents a powerful tool for analyzing the irregular behavior and state of the dynamical systems. Analysis of waveforms has been used to identify and distinguish specific complex patterns. A variety of algorithms are available for the computation of fractal dimension of waveforms. In this paper we evaluate the performance of our algorithm based on size measure relationship method, quantifying the synthetic waveforms and electroencephalographic signals. Compared to Katz's, Higuchi's and Petrosian's algorithm advantages of this method include greater speed and not affected by noise. The computation time for the algorithm suggested in this paper is much less than the other methods.

Methods of fractal and cross-correlation analysis of EEG as a means of objectification of preclinical epileptogenesis and evaluation of effectiveness of prophylactic pharmacotherapy of epilepsy

Methods of fractal and cross-correlation analysis of EEG as a means of objectification of preclinical epileptogenesis and evaluation of effectiveness of prophylactic pharmacotherapy of epilepsy

Fractal analysis of surface electromyography signals: A novel power spectrum-based method

This paper presents a novel power spectrum-based method for fractal analysis of surface electromyography signals. This method, named the bi-phase power spectrum method, provides a bi-phase power-law which represents a multi-scale statistically self-affine signal. This form of statistical self-affinity provides an accurate approximation for stochastic signals originating from a strong non-linear combination of a number of similar distributions, such as surface electromyography signals which are formed by the summation of a number of single muscle fiber action potentials. This power-law is characterized by a set of spectral indicators, which are related to distributional and geometrical characteristics of the electromyography signal’s interference pattern. These novel spectral indicators are capable of sensing the effects of motor units’ recruitment and shape separately by exploiting the geometry of the interference pattern. The bi-phase power spectrum method is compared to geometrical techniques and the 1/ f α approach for fractal analysis of electromyography signals. The extracted indicators using the bi-phase power spectrum method are evaluated in the context of force and joint angle and the results of a human study are presented. Results demonstrate that the bi-phase power spectrum method provides reliable information, consisting of components capable of sensing force and joint angle effects separately, which could be used as complementary information for confounded conventional measures.

Automated detection of proliferative retinopathy in clinical practice

Timely intervention for diabetic retinopathy (DR) lessens the possibility of blindness and can save considerable costs to health systems. To ensure that interventions are timely and effective requires methods of screening and monitoring pathological changes, including assessing outcomes. Fractal analysis, one method that has been studied for assessing DR, is potentially relevant in today's world of telemedicine because it provides objective indices from digital images of complex patterns such as are seen in retinal vasculature, which is affected in DR. We introduce here a protocol to distinguish between nonproliferative (NPDR) and proliferative (PDR) changes in retinal vasculature using a fractal analysis method known as local connected dimension (D(conn)) analysis. The major finding is that compared to other fractal analysis methods, D(conn) analysis better differentiates NPDR from PDR (p = 0.05). In addition, we are the first to show that fractal analysis can be used to differentiate between NPDR and PDR using automated vessel identification. Overall, our results suggest this protocol can complement existing methods by including an automated and objective measure obtainable at a lower level of expertise that experts can then use in screening for and monitoring DR.

Fractal analysis of river flow fluctuations

We use some fractal analysis methods to study river flow fluctuations. The result of the Multifractal Detrended Fluctuation Analysis (MF-DFA) shows that there are two crossover timescales at s 1 × ∼ 12 and s 2 × ∼ 130 months in the fluctuation function. We discuss how the existence of the crossover timescales are related to a sinusoidal trend. The first crossover is due to the seasonal trend and the value of second one is approximately equal to the well-known cycle of sun activity. Using Fourier Detrended Fluctuation Analysis, the sinusoidal trend is eliminated. The values of Hurst exponents of the runoff water of rivers without the sinusoidal trend show a long-range correlation behavior. For the Daugava river, the value of Hurst exponent is 0.52 ± 0.01 and also we find that these fluctuations have multifractal nature. Comparing the MF-DFA results for the remaining data set of Daugava river to those for shuffled and surrogate series, we conclude that its multifractal nature is almost entirely due to the broadness of probability density function.

A comparison of different fractal analyses for VHF electromagnetic emissions and their self-organization for the off-sea Miyagi-prefecture earthquake

Abstract. The VHF electromagnetic noise intensity data at several stations in the Tohoku area of Japan during the period of a rather large (with magnitude of 7.2) earthquake (Miyagi-ken oki earthquake) taken place on 16 August 2005, are analyzed by means of different fractal analysis methods, including (1) spectral slope estimation, (2) multi-fractal detrended fluctuation analysis and (3) multi-fractal wavelet transform modulus maxima method. It seems to the authors that there is no definite analysis method for the analysis of any seismogenic phenomenon, so that the only way we have to take, is to apply different methods to the same data for the detailed comparison of the results. This comparison enables us to deduce the properties commonly observed by the above methods. Because the most important feature common to these three methods, is that significant changes in fractal scaling characteristics are observed just during the earthquake (mainly before the earthquake) only at one station of Kunimi. Finally, we can come to the definite conclusion on the self-organization of VHF emissions only at one station in the present case.

Variability and Fractal Analysis of Middle Cerebral Artery Blood Flow Velocity and Arterial Blood Pressure in Subarachnoid Hemorrhage

Higher biologic systems operate far from equilibrium resulting in order, complexity, fluctuation of inherent parameters, and dissipation of energy. According to the decomplexification theory, disease is characterized by a loss of system complexity. We analyzed such complexity in patients after subarachnoid hemorrhage (SAH), by applying the standard technique of variability analysis and the novel method of fractal analysis to middle cerebral artery blood flow velocity (FV) and arterial blood pressure (ABP). In 31 SAH -patients, FV (using transcranial Doppler sonography) and direct ABP were measured. The standard deviations (s.d.) and coefficients of variation (CV=relative s.d.) for FV and ABP time series of length 2(10) secs were calculated as measures of variability. The spectral index beta(low) and the Hurst coefficient H(bdSWV) were analyzed as fractal measures. Outcome was assessed 1 year after SAH according to the Glasgow Outcome Scale (GOS). Both FV (beta(low)=2.2+/-0.4, mean+/-s.d.) and ABP (beta(low)=2.3+/-0.4) were classified as nonstationary (fractal Brownian motion) signals. FV showed significantly (P<0.05) higher variability (CV=7.2+/-2.5%) and Hurst coefficient (H(bdSWV)=0.26+/-0.13) as compared with ABP (CV=5.5+/-2.7%, H(bdSWV)=0.19+/-0.11). Better outcome (GOS) correlated significantly (P<0.05) with higher s.d. of FV (Spearman's r(s)=0.51, r(s)(2)=0.26) and ABP (r(s)=0.57, r(s)(2)=0.32), as well as with a higher Hurst coefficient of ABP (r(s)=0.46, r(s)(2)=0.21). Cerebral vasospasm reduced CV of FV, but left H(bdSWV) unchanged. FV and ABP fluctuated markedly despite homeostatic control. A reduced variability of FV and ABP might indicate a loss of complexity and was associated with a less favorable outcome. Therefore, the decomplexification theory of illness may apply to SAH.

Reduced Long-range Correlations Of Gait Cycle Relate To Declines In Physical Ability In The Elderly

Some researchers have reported that long-range correlations in the fluctuations of gait cycle duration brake down in the elderly gait, and functional impairment of the central nervous system (CNS) has been thought to be responsible for the reduced long-range correlations. PURPOSE: To reveal the origin of deteriorated long-range correlations, we hypothesized that declines in physical ability as well as alterations in CNS were responsible for altered long-range correlations of gait in the elderly. METHODS: 15 healthy old adults aged 64–79 yr were recruited. The subjects walked around a 34.2 m oval walkway in ‘fast’ speed for six minutes. Heel contact timing of each stride was obtained from foot switches attached to the insole in both shoes. The subjects stepped on the same spot for six minutes on force platforms. Foot contact timing was measured from vertical force on the platforms. Fractal scaling exponents (?) which evaluate long-range correlations in gait and stepping were calculated with a fractal analysis method, detrended fluctuation analysis (DFA). Range of motion (ROM) of neck, waist, hip, knee, and ankle joints were measured. Isokinetic joint torque of hip, knee, and ankle joints were measured with Cybex Dynamometer. Flection and extension torques of each joint were measured. Functional reach test was used for measuring subjects' ability to maintain balance. RESULTS: Reduced α in gait was significantly (p < 0.05) correlated with reduced ROM in hip extension, knee extension, ankle external and internal rotation, and marginally correlated with reduced ROM in waist rotation (p = 0.056). Reduced α in stepping was significantly correlated with reduced ROM in neck lateroflection, hip external and internal rotation, abduction, and flection. It also significantly correlated with joint torque in knee flection and extention, and ankle dorsiflection. Reduced functional reach range was significantly correlated with reduced α in both gait and stepping. CONCLUSIONS: Deteriorated long-range correlations in both gait and stepping were correlated with declined physical abilities, and additionally, each correlated with different items of physical ability decline. The result implies a strong relationship existing between declined physical ability and deteriorated long-range correlations of gait and stepping cycle.

Fractal Analysis of Mammographic Parenchymal Patterns in Breast Cancer Risk Assessment

To evaluate fractal-based computerized image analyses of mammographic parenchymal patterns in the task of differentiating between women at high risk and women at low risk for developing breast cancer. The fractal-based texture analyses are based on a box-counting method and a Minkowski dimension, and were performed within the parenchymal regions of normal mammograms. Four approaches were evaluated: 1) a conventional box-counting method, 2) a modified box-counting technique using linear discriminant analysis (LDA), 3) a global Minkowski dimension, and 4) a modified Minkowski technique using LDA. These fractal based texture features were extracted from regions of interest to assess the mammographic parenchymal patterns of the images. Receiver operating characteristic analysis was used to evaluate the performance of these features in the task of differentiating between the two groups of women. Receiver operating characteristic analysis yielded an A(z) value of 0.74 based on the conventional box-counting technique and an A(z) value of 0.84 based on the global Minkowski dimension in the task of distinguishing between the two groups. By using LDA to assess the characteristics of mammograms, A(z) values of 0.90 and 0.93 were obtained in differentiating the two groups, for the modified box-counting and Minkowski techniques, respectively. Statistically significant improvement was achieved (P < .05) with the new techniques compared to the conventional fractal analysis methods. A simulation study, which used the slope and intercept extracted from the least square fit of the experimental data with the LDA approaches, yielded A(z) values similar to those obtained with the conventional approaches in the task of differentiating between the two groups. The proposed LDA approach improved significantly the separation between the two groups based on experimental data. Because this approach was used as a linear classifier rather than as a regression function, it combined the fractal analysis with the knowledge of the high- and low-risk patterns, and thus better characterized the multifractal nature of the parenchymal patterns. We believe that the proposed analyses based on the LDA technique to characterize mammographic parenchymal patterns may potentially yield radiographic markers for assessing breast cancer risk.

Fractal analyses for ‘short’ time series: A re-assessment of classical methods

The aim of this study was to evaluate the performances of some classical methods of fractal analysis with short time series. We simulated exact fractal series to test how well methods estimate the Hurst exponent. We successively tested power spectral density analysis, detrended fluctuation analysis, rescaled range analysis, dispersional analysis, maximum likelihood estimation, and two versions of scaled windowed variance methods. All methods presented different advantages and disadvantages, in terms of biases and variability. We propose in conclusion a systematic step-by-step procedure of analysis, based on the performances of each method and their appropriateness regarding the scientific aims that could motivate fractal analysis.

Osteophytes, juxta-articular radiolucencies and cancellous bone changes in the proximal tibia of patients with knee osteoarthritis

To determine differences in tibial cancellous bone organisation in knee osteoarthritis (OA) between the central weight-bearing region and juxta-articular radiolucencies adjacent to small, medium or large marginal osteophytes. Patients with medial compartment OA (n = 60; F = 39), mean (SD) age 60.0 (9.7) years, and non-OA reference subjects (n = 21; F = 5), mean (SD) age 36.8 (11.5) years, had x4 macroradiographs digitised by laser scanner. Using a modified Osteoarthritis Research Society (OARS) Atlas, right and/or left knees were graded according to marginal osteophyte size into those with small (n = 30), medium (n = 30) or large (n = 27) marginal osteophytes, identified as OPH1, OPH2 and OPH3, respectively. Non-OA knees (n = 30) were anatomically normal. Computerised method of Fractal Signature Analysis (FSA) quantified differences in cancellous bone structure between non-OA and osteophyte subgroups at two regions of interest (ROIs); central weight-bearing and tibial margin. Compared to non-OA, vertical trabecular number increased significantly (P < 0.05) in all osteophyte subgroups (width range 0.12-1.14 mm) within both ROIs. In OPH3, this increase was significantly (P < 0.05) greater compared to OPH2 in the central ROI, and to OPH2 and OPH1 in the marginal ROI at most trabecular widths (0.12-1.14 mm). In the marginal ROI, compared to non-OA, horizontal trabeculae number decreased in all osteophyte subgroups. This decrease was significantly greater in OPH3 compared to OPH2 and OPH1 at small to medium trabecular widths (0.12-0.54 mm). Compared to disease associated bone loss at the central ROI of the tibia, the extent of juxta-articular bone loss appears to be associated with the size of the marginal osteophytes.

A 2 yr longitudinal radiographic study examining the effect of a bisphosphonate (risedronate) upon subchondral bone loss in osteoarthritic knee patients

To determine whether risedronate (RIS) slows down trabecular bone loss in the medial compartment of the proximal tibia, a characteristic of patients with progressive knee osteoarthritis (OA). Initially, 100 patients were randomly selected from each treatment group (each N approximately 300) comprising placebo and RIS 5 mg/day, 15 mg/day and 50 mg/week from a double blind, multi-centre, placebo-controlled, 2 yr investigation of OA knee patients in North America. Using fluoroscopic semi-flexed standard radiography, baseline and exit knee radiographs were digitized by laser scanner. Following computerized measurement of minimum medial compartment joint space width, each group was subdivided into joint space narrowing (JSN) non-progressor or JSN-progressor (JSN >or=0.6 mm measured at any point post-baseline). Computerized method of fractal signature analysis (FSA) quantified longitudinal changes separately in horizontal and vertical trabeculae in region of interest (three-fourth width of tibial compartment x 6 mm height) in the medial compartment. Following the initial study, all JSN-progressor knees within the entire patient cohort (N = 1232) were similarly analysed. OA knees in JSN non-progressor group had a slight decrease in FSA for vertical and horizontal trabeculae and showed no drug effect. In JSN-progressor knees, bone loss was greater in both placebo and RIS 5 mg/day groups compared with those in RIS 15 mg/day group in which trabeculae were retained, and in the RIS 50 mg/week group in which the vertical trabecular number increased significantly (P < 0.05). This preliminary study showed that patients with marked cartilage loss (JSN>or=0.6 mm) receiving RIS 15 mg/day retained vertical trabecular structure, and those receiving RIS 50 mg/week increased vertical trabecular number, thereby preserving the structural integrity of subchondral bone in knee OA.

Identification of hair cycle-associated genes from time-course gene expression profile using fractal analysis

Microarray technology permits one to monitor thousands of processes going on inside the cell. This tool has been used to study gene expression profiles associated with the hair-growth cycle. We provide a novel method called the fractal analysis method to identify hair-growth cycle associated genes from time course gene expression profiles. Fractal analysis is a much better method than the computational method used by Lin et al. (2004). The fractal dimension obtained by fractal analysis process also indicates the irregularity in hair-growth pattern. The computational method used by Lin et al. (2004) was unable to make any inference about the hair-growth pattern.

Comparison of three methods of fractal analysis applied to soot aggregates from wood combustion

The morphological and dynamic properties of fractal-like particles produced from the combustion of wood are studied in this work. Particles with electrical mobility diameters of 200, 275, 350 and 450 nm were deposited on filters and imaged using a high-resolution scanning electron microscope (SEM). The soot particles consisted of aggregated primary spheres with mean radius a p = 25.5 ± 3.5 nm and standard deviation σ g = 1.27 ± 0.09 . The fractal dimension of the aggregates, D f , was derived from three techniques: (1) D f = 1.84 ± 0.05 from the projected surfaces in the SEM images; (2) D f = 1.83 ± 0.05 from the relationship between number of primary particles per aggregate N, and radius of gyration R g ; (3) D f = 1.80 ± 0.13 from the relationship of mobility and N. Thus, there is agreement between the three techniques with an average D f of 1.83. The dynamic shape factors of the soot particles were between 1.5 and 2.5 and increasing with N. In the regime 0.2 < Kn < 0.7 (Knudsen number, Kn = 2 λ / d mob ), the mobility diameter d mob was observed to be proportional to the radius of gyration with a ratio d mob / 2 R g = 0.81 ± 0.07 . The specific surface area of the aggregates was determined to be 70 ± 10 m 2 / g , based on SEM image analysis.