Research On The Singularity Spectrum Analysis Of The Stochastic Multifractal Signal And The Related Applications

This paper has done researches on the multifractal analysis methods of singularity spectrum(multifractal spectrum)of the random multifractal signal,the reconstruction theory of the random multifractal signal and the applications of these methods in the target detection and related application fields.Fractal and multifractal theory provides new methods for the radar signal analysis and target detection in the background of sea.In many practical problems with irregular distribution of physical quantity in the space,the quantitative representation of the inhomogeneous distribution of the physical quantity can be realized by means of the calculation of the singularity spectrum.This paper has done some comparative studies on the multifractal analysis and multifractal cross-correlation methods that are easy to be implemented and done some researches on the imp lications in the target detection within sea clutter and the applications in the analysis of the actual image(texture images and electron microscope pictures).The main researches are listed as follows:1.This paper introduces the theory concept of the fractal,self-similar fractal,self-affine fractal and multifractal,the commonly used calculation steps of the analysis methods of fractal dimension and spectrum of singularities(multifractal spectrum)at first.On this basis,this paper has done researches on the methods of multifractal spectrum of one-dimensional random multifractal signal.This paper has done a comparative study of the one-dimensional multifractal detrended fluctuation analysis(MFDFA)and multifractal detrended moving average(MFDMA)method,which are easy to be implemented.By applying the two methods to the series generated from the binomial multiplicative cascades(BMC),we systematically do the comparative analysis to get the advantages,disadvantages and the applicability of the two algorithms 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 MFDFA and MFDMA,and how to make the schemes of the parameter setting of the two algorithms when dealing with specific signals in practical applications.On this basis,this paper has done a lot of numerical simulations on the IPIX radar sea clutter data by using the Qth-order moment structure partition function(QMSPF),MFDFA,MFDMA methods and wavelet Leaders(WL),and used the singularity exponent corresponding to the top point of the singularity spectrum,the singularity spectrum width and the difference between the two end points of the singularity dimension of the singularity spectrum as the detection parameters to do target detection,which provides a valuable reference for the one-dimensional signal target detection using conventional multifractal singularity spectrum.2.This paper has done researches on the methods to calculate the multifractal cross-correlation spectrum between two one-dimensional random multifractal signals.In this paper,we compare the algorithm models and the requirements of the multifractal cross-correlation analysis based on the partition function approach(MFXPF),multifractal detrended fluctuation analysis methods based on detrended fluctuation analysis(MFXDFA)and detrended moving average(MFXDMA)analysis.Many numerical experiments are carried out to study the performances of different methods such as MFXDFA,MFXDMA,MFXPF to the well-established mathematical models(BMC and Cantor sets),which have known analytical expressions.These methods are applied to the IPIX radar sea clutter data and unveil intriguing multifractality in the cross correlations of the sea clutters with and without target.This paper has used a pure sea clutter sequence to do multifractal cross-correlation analysis with 14 groups of sea clutter sequences with and without target respectively,and used the difference of the singularity dimension between the two ends of the multifractal cross-correlation spectrum as the detection parameter to do target detection effectively.The simulation results show the feasibility of using the multifractal cross-correlation spectrum between two sequences to detect the target detection.3.This paper has done researches on the commonly used two-dimensional multifracatl analysis methods.By applying the two-dimensional multifractal analysis based on the detrended fluctuation analysis(2D-MFDFA)and two-dimensional multifractal analysis based on the detrended moving average(2D-MFDMA)algorithm 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 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.Based on this,we use 2D-MFDFA to deal with the real texture images,the results show that it is possible to distinguish composite texture images from the general texture images by using the singularity spectrum width.The simulation results verify the feasibility of texture detection and classification by using singularity spectrum.4.We apply 2D-MFXPF,2D-MFXDFA and 2D-MFXDMA methods to pairs of two-dimensional multiplicative cascades(2D-MC)to do a comparative study.We can see that when the generators of the two 2D-MC signals are similar to each other,the estimated results may fit nicely with the "theoretical" curves.But when the generators of the two 2D-MC signals are unlike with each other,the estimated results may deviate greatly from the "theoretical" curves.Based on this,we apply the 2D-MFXDFA method to real images and unveil intriguing multifractality in the cross correlations of the material structures(the electron microscope pictures downloaded from internet).The multifractal cross-correlation spectrum between the two images with similar texture structure is similar to the"theoretical" curve,which is the average of the two singularity spectra of the two images calculated separately.In the converse condition,it will deviate from the "theoretical"curve.The multifractal cross-correlation spectrum width,the difference between the singularity dimension of the two ends of the multifractal cross-correlation spectrum,the error between the multifractal cross-correlation spectrum and the "theoretical" curve can be used as the detection parameter.The simulation results provide a valuable reference on how to choose the two-dimensional multifractal cross-correlation algorithms in the potential applications in the field of real image classification and detection.5.We do researches to the commonly used one-dimensional and two-dimensional random multifractal signals by these multifractal methods and compare the characteristics of the singularity spectra of these signals.We find that if we construct the regular multifractal signals based on BMC,Cantor and multifractal Brownian motion(MBM)firstly and do fractional calculus to them,then estimate the singularity spectrum by MFDFA to check the similarity with the given singularity spectrum(the spectrum width??=1,the top point of the singularity spectrum f(?)max=1).Simulation results prove the feasibility of the construction method.