Two-dimensional Multifractal Detrended Fluctuation Analysis Research Articles

Impact of surface-roughness and fractality on electrical conductivity of SnS thin films

Mono- and multi-fractal geometry have been used to explore the surface characteristics of scanning electron microscopy (SEM) micrographs of the SnS films with thicknesses of 100 nm (SnS1) to 600 nm (SnS4), respectively. For this investigation, the SnS thin films have been grown on fluorine-doped tin oxide (FTO)-coated glass substrate through the thermal evaporation route, and surface morphologies are captured by SEM. Two-dimensional multi-fractal detrended fluctuation analysis (MFDFA) based on the partition function is used to examine whether the surfaces have a multi-fractal nature or not. The partition function is applied to extract the generalized Hurst exponent from the segment size. It has been found that surfaces with higher surface roughness induce substantial nonlinearity and a wider width of the multi-fractal spectrum. The multi-fractal spectrum acquired from the analysis of the geometry and shape of the singularity spectrum is used to quantify the irregularity and complexity of surfaces. Minkowski functionals (MFs) parameters such as volume, boundary, and connectivity were measured for each thin film. Moreover, we tried to correlate the electrical conductivity with the mono- and multi-fractal parameters such as fractal dimension (Df), singularity strength function (Δα), singularity spectrum Δf(α), and it is observed that the conductivity of a thin film decreases with decreasing fractal dimension. The minimum (maximum) resistivity (conductivity) was observed for the surface having a larger fractal dimension. The present investigation suggests that such SnS surfaces, having minimal resistivity and maximum conductivity on the roughest surface, indicate enhanced light trapping capacity and can be utilized as active layers for advanced optoelectronics devices.

Multifractal and Long-Term Memory of Impervious Surface Spatial Patterns in a Coastal City in China

An understanding of the multifractal and long-range dependence of impervious surfaces (IS) spatiotemporal patterns is helpful for regional environmental assessment and urban planning. Linear spectral-mixture analysis has been applied to compute IS in the coastal city of Xiamen, China, based on Landsat TM/OLI/TIRS images, and then the long-term trends and multifractal characteristics of IS patterns have been investigated using two-dimensional multifractal detrended fluctuation analysis. The IS spatial distribution displayed similar positive long-range correlations in study areas during the 1994–2015 period. IS have a long-memory characteristic within a certain spatial range, with an increase in the value of a pixel likely to cause an increment in the value of its neighbors. The multifractality of the IS distribution increased in Xiamen City during 1994–2015, but was lower than those in Xiamen Island. The multifractal spectra in Xiamen City vary in shape between years, capturing its evolution from right truncation with a long left tail to left truncations and long right tails and then a symmetrical shape. The fractal structure for Xiamen Island exhibits similar patterns of long right tails and left truncations. Economic and political considerations coupled with natural geographic conditions dominate the long-range trends in the IS spatial patterns.

Alumina abrasive grains mediated grinding induced glass surface and fractal analysis of SiO2 glass

Fabrication of optical components such as passive optics (mirrors, prisms, lenses) or active optics (polarizer’s, laser gain media, adaptive optics) starts from a bulk material and reaches to the required size and shape. To achieve the required specification of the component, a series of process has to be followed from grinding to polishing stage. Initially, in grinding stage, material removal is faster, with more surface damage and less geometric control. During polishing stage, material removal is slower, with no surface damage but with greater geometric control. Hence, to achieve good surface quality, the component or work piece has to be controlled from grinding stage. We present an advanced surface and fractal analysis study on fused silica samples processed with different grit sizes such as 3, 5, 12, and 25 μm aluminium oxide (Al2O3) abrasives. The maximum sub surface damage (SSD) with respect to size of the abrasive grain ranged between 30 μm to 5 μm in empirical module. In theoretical module it varied from 45 μm to 5.4 μm for fused silica glass, mediated with loose abrasive aluminium oxide powder of 25 μm and 3 μm respectively. An experimental investigation is reported on the increasing effect of Al2O3 grit sizes on the surface topography such as average roughness (Ra), which varied from 57 nm to 847 nm for 3 μm and 25 μm Al2O3 abrasives respectively. Three-dimensional image analysis was captured through Phase Shift Interferometry (PSI) and Coherence Correlation Interferometer (CCI) technique. Field Emission Scanning Electron Microscope (FESEM) technique was utilized to characterize the Al2O3 powders and the processed fused silica samples. Later, the images have been analysed using two-dimensional multifractal detrended fluctuation analysis (2D-MFDFA) to understand and confirm the multiple fractal nature on fused silica samples caused by varying grit size of Al2O3 abrasives.

AUTOMATED CLASSIFICATION FOR BRAIN MRIS BASED ON 2D MF-DFA METHOD

Magnetic resonance image (MRI) is an important tool to diagnose human diseases effectively. It is very important for research and clinical application to classify the normal and abnormal human brain MRI images automatically. In this paper, an accurate and efficient technique is proposed to extract features of MRIs and classify these images into normal and abnormal categories. We use two-dimensional multifractal detrended fluctuation analysis (2D MF-DFA) to obtain image features. These features are the local generalized Hurst exponents calculated by 2D MF-DFA. In this regard, the values of Hurst exponents are given as the training input vector and are taken to the classifiers. We use [Formula: see text]-nearest neighbor ([Formula: see text]-NN) and support vector machine (SVM) to classify a specific brain MRI as normal or glioma affected. For SVM, we apply the leave-one-out cross-validation method for experimental verification. The 2D MF-DFA-SVM system achieved accuracy, sensitivity, and specificity of [Formula: see text] ±[Formula: see text], [Formula: see text], and [Formula: see text] ±[Formula: see text], respectively. The 2D MF-DFA-[Formula: see text]-NN system achieved accuracy, sensitivity, and specificity of [Formula: see text], [Formula: see text], and [Formula: see text], respectively. We find that when performing binary classification for brain MRIs, the SVM is superior to [Formula: see text]-NN. In addition, our experimental results indicate that the proposed 2D MF-DFA-SVM achieved excellent outcomes compared to those of the previous works. The proposed system is a promising system to clinical use.

Combining MF-DFA and LSSVM for retina images classification

Diabetic retinopathy is the main cause of blindness in adults. Early diagnosis of diabetic retinopathy is essential for avoiding deterioration of illness and vision loss. The use of computer technology to identify diabetic retinopathy images provides significant means to reduce the risk of deterioration. In this paper, we propose a new approach for retina images detection and classification by using a hybrid system which is constructed by two-dimensional multifractal detrended fluctuation analysis (2D MF-DFA) and least square support vector machines (LSSVM). In the proposed method, we applied 2D MF-DFA to compute the local generalized Hurst exponents which are the multifractal features of the diabetic retinopathy image, and these values are recorded as LHq. Then, the Hurst exponents are taken as the training input vector for the training in LSSVM. Finally, we classified a specific retina image as healthy or lesion image. We present experimental verification to investigate the efficiency and robustness of the proposed system. The results show that the proposed system yields a classification accuracy with 99.01% ± 0.0074, sensitivity with 99.03% ± 0.0051, and specificity with 97.73% ± 0.0075. When the performance was compared with state-of-the-arts, the solution indicated that the MF-DFA-LSSVM system outperforms most of others in terms of all the classification sensitivity, accuracy, and specificity. The proposed method will be useful for clinical medicine.

Two dimensional multifractal detrended fluctuation analysis of low coherence images for diagnosis of cervical pre-cancer

We report detection of cervical pre-cancer through their low coherence images by applying two dimensional multifractal detrended fluctuation analysis. Low coherent backscattered images of pre-cancerous cervical tissue sections were captured using a common path interferometric setup. The captured images contain both depth and lateral information of the spatial variation in refractive index (RI) occurring with progression of cervical pre-cancer. A two-dimensional multifractal detrended fluctuation analysis (2D MFDFA) was applied on these low coherent images to study the variations occurring in their fractal nature. Long-range correlations were observed in the RI fluctuations and the strength of multifractality was found to be stronger for higher grades of cervical pre-cancer. A combination of derived multifractal parameters, namely, the generalized Hurst exponent and width of singularity spectrum showed clear differences among the different grades of pre-cancers. Normal, CIN-I and CIN-II were clearly discriminated by application of support vector machine (SVM) using radial Bessel function (RBF) kernel. The specificities and sensitivities between normal and CIN-I, CIN-I and CIN-II and normal and CIN-II were found to be 94%, 88% and 93%, 96% and 98%, 100% respectively.

Experimental and numerical study of multicomponent nitride coatings based on TiAlSiY fabricated by CA-PVD method

We report technical parameters of the deposition of multielement nitride protective coatings based on TiAlSiY by CA-PVD method and fractal analysis of their surfaces that are presented by the scanning electronic microscopy (SEM) images. Numerical analysis of the digital images was performed within two-dimensional multifractal detrended fluctuation analysis. Numerical models of surfaces of studied samples were built from their SEM images and generalized Hurst exponents and multifractal spectrums were calculated for all obtained structures. The performed calculations showed the widest range of Hurst exponent for TiAlSiY coating, which can indicate the highest relative roughness of its surface.

A comparative study of two-dimensional multifractal detrended fluctuation analysis and two-dimensional multifractal detrended moving average algorithm to estimate the multifractal spectrum

Multifractal detrended fluctuation analysis (MFDFA) and multifractal detrended moving average (MFDMA) algorithm have been established as two important methods to estimate the multifractal spectrum of the one-dimensional random fractal signal. They have been generalized to deal with two-dimensional and higher-dimensional fractal signals. This paper gives a brief introduction of the two-dimensional multifractal detrended fluctuation analysis (2D-MFDFA) and two-dimensional multifractal detrended moving average (2D-MFDMA) algorithm, and a detailed description of the application of the two-dimensional fractal signal processing by using the two methods. By applying the 2D-MFDFA and 2D-MFDMA to the series generated from the two-dimensional multiplicative cascading process, we systematically do the comparative analysis to get the advantages, disadvantages and the applicabilities of the two algorithms for the first time from six aspects such as the similarities and differences of the algorithm models, the statistical accuracy, the sensitivities of the sample size, the selection of scaling range, the choice of the q-orders and the calculation amount. The results provide a valuable reference on how to choose the algorithm from 2D-MFDFA and 2D-MFDMA, and how to make the schemes of the parameter settings of the two algorithms when dealing with specific signals in practical applications.

Detrended fluctuation analysis of spatial patterns on urban impervious surface

This paper analyzes satellite data from downtown Shanghai, China, to investigate the long-range dependence of spatial patterns on urban impervious surfaces (UIS) using two-dimensional detrended fluctuation analysis (DFA). The UIS fraction is estimated from Landsat Thematic Mapper and Enhanced Thematic Mapper Plus data from 1997 to 2010 using linear spectral mixture analysis. The results indicate that the spatial distribution of the UIS exhibits a positive spatial dependence, as revealed by Moran’s index, capturing the evolution from aggregation to dispersion and back to aggregation during the study period. The use of two-dimensional DFA reveals a strong long-range power-law dependence in the UIS spatial pattern during the study period. The DFA scaling exponent can be seen as a measure of the uniformity of the UIS spatial distribution, and exhibits an approximate 1/f behaviour. Two-dimensional multifractal detrended fluctuation analysis (MFDFA) confirmed that the UIS spatial pattern is not multifractal in nature, meaning that only a single scaling exponent is required to disclose the long-range dependence in the UIS spatial pattern. The application of two-dimensional DFA to UIS patterns should be viewed as a complementary tool to existing techniques, such as Fourier analysis, wavelets, and structure function, that can provide additional information about the structure of UIS patterns, including its 1/f behaviour.

Two-dimensional multifractal detrended fluctuation analysis for plant identification.

BackgroundIn this paper, a novel method is proposed to identify plant species by using the two- dimensional multifractal detrended fluctuation analysis (2D MF-DFA). Our method involves calculating a set of multifractal parameters that characterize the texture features of each plant leaf image. An index, I0, that characterizes the relation of the intra-species variances and inter-species variances is introduced. This index is used to select three multifractal parameters for the identification process. The procedure is applied to the Swedish leaf data set containing leaves from fifteen different tree species.ResultsThe chosen three parameters form a three-dimensional space in which the samples from the same species can be clustered together and be separated from other species. Support vector machines and kernel methods are employed to assess the identification accuracy. The resulting averaged discriminant accuracy reaches 98.4% for every two species by the 10 − fold cross validation, while the accuracy reaches 93.96% for all fifteen species.ConclusionsOur method, based on the 2D MF-DFA, provides a feasible and efficient procedure to identify plant species.

Relationships of exponents in two-dimensional multifractal detrended fluctuation analysis

Multifractal detrended fluctuation analysis (MF-DFA) is a generalization of the conventional multifractal analysis. It is extended from the detrended fluctuation analysis (DFA) which is developed for the purpose of detecting long-range correlation and fractal property in stationary and nonstationary time series. The MF-DFA and some corresponding relationships of the exponents have been subsequently extended to the two-dimensional space. We reexamine two extended relationships in this study and demonstrate that: (i) The invalidity of the relationship h(q)≡H for two-dimensional fractional Brownian motion, and h(q=2)≡H between the Hurst exponent H and the generalized Hurst exponent h(q) in the two-dimensional case. Two more logical relationships are proposed instead as h(q=2)=H for the stationary surface and h(q=2)=H+2 for the nonstationary signal. (ii) The invalidity of the expression τ(q)=qh(q)-D(f) stipulating the relationship between the standard partition-function-based multifractal exponent τ(q) and the generalized Hurst exponent h(q) in the two-dimensional case. Reasons for its invalidity are given from two perspectives.

Numerical analysis of the morphological and phase changes in the TiN/Al_{2}O_{3} coating under high current electron beam modification

A modification of the surface structure of the hybrid coating TiN/Al_{2}O_{3} with a low-energy high-current electron beam (NCEB) is performed. The surface roughness is considered as a function of beam current. Surfaces of the obtained samples are investigated within the two-dimensional multifractal detrended fluctuation analysis (MF-DFA). The multifractal spectrum of the surface is calculated as a quantitative parameter of the roughness. It is shown that with an increase of the beam energy, the surface becomes more regular and uniform.

Fractal and multifractal analysis of LiF thin film surface

Fractal and multifractal analysis is performed on the atomic force microscopy (AFM) images of the surface morphologies of the LiF thin films of thickness 10nm, 20nm, and 40nm, respectively. Autocorrelation function, height–height correlation function, and two-dimensional multifractal detrended fluctuation analysis (MFDFA) are used for characterizing the surface. It is found that the interface width, average roughness, lateral correlation length, and fractal dimension of the LiF thin film increase with the thickness of the film, whereas the roughness exponent decreases with thickness. Thus, the complexity and roughness of the LiF thin films increases as thickness increases. It is also demonstrated that the LiF thin films are multifractal in nature. Strength of the multifractality increases with thickness of the film.

Multifractal analysis of the fracture surfaces of foamed polypropylene/polyethylene blends

The two-dimensional multifractal detrended fluctuation analysis is applied to reveal the multifractal properties of the fracture surfaces of foamed polypropylene/polyethylene (PP/PE) blends at different temperatures. Nice power-law scaling relationship between the detrended fluctuation function F q and the scale s is observed for different orders q and the scaling exponent h ( q ) is found to be a nonlinear function of q, confirming the presence of multifractality in the fracture surfaces. The multifractal spectra f ( α ) are obtained numerically through Legendre transform. The shape of the multifractal spectrum of singularities can be well captured by the width of spectrum Δ α and the difference of dimension Δ f . With the increase of the PE content, the fracture surface becomes more irregular and complex, as is manifested by the facts that Δ α increases and Δ f decreases from positive to negative. A qualitative interpretation is provided based on the foaming process.

Multifractal detrended fluctuation analysis of combustion flames in four-burner impinging entrained-flow gasifier

On a laboratory-scale testing platform of an impinging entrained-flow gasifier with four opposed burners, the flame images for diesel combustion and gasification process were recorded with a single charge coupled device (CCD) camera. The two-dimensional multifractal detrended fluctuation analysis was employed to investigate the multifractal nature of the flame images. Sound power-law scaling relations in the annealed average of detrended fluctuations were unveiled when the order q > 0 , and the multifractal feature of flame images was confirmed. Further analyses identified two multifractal parameters, the minimum and maximum singularities α min ⁡ and α max ⁡ , serving as characteristic parameters of the multifractal flames. These two characteristic multifractal parameters vary with respect to different experimental conditions.

Detrended fluctuation analysis for fractals and multifractals in higher dimensions

One-dimensional detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MFDFA) are widely used in the scaling analysis of fractal and multifractal time series because they are accurate and easy to implement. In this paper we generalize the one-dimensional DFA and MFDFA to higher-dimensional versions. The generalization works well when tested with synthetic surfaces including fractional Brownian surfaces and multifractal surfaces. The two-dimensional MFDFA is also adopted to analyze two images from nature and experiment, and nice scaling laws are unraveled.