Algorithm to estimate the Hurst exponent of high-dimensional fractals

PHYSICAL REVIEW E 76, 056703 共2007兲

Algorithm to estimate the Hurst exponent of high-dimensional fractals Anna Carbone Physics Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy 共Received 19 June 2007; published 13 November 2007兲 We propose an algorithm to estimate the Hurst exponent of high-dimensional fractals, based on a generalized high-dimensional variance around a moving average low-pass filter. As working examples, we consider rough surfaces generated by the random midpoint displacement and by the Cholesky-Levinson factorization algorithms. The surrogate surfaces have Hurst exponents ranging from 0.1 to 0.9 with step 0.1, and different sizes. The computational efficiency and the accuracy of the algorithm are also discussed. DOI: 10.1103/PhysRevE.76.056703

PACS number共s兲: 05.10.⫺a, 05.40.⫺a, 05.45.Df, 68.35.Ct

I. INTRODUCTION

The scaling properties of random curves and surfaces can be quantified in terms of the Hurst exponent H, a parameter defined in the framework of the fractional Brownian walks introduced in 关1兴. A fractional Brownian function f共r兲 : Rd 2 → R, is characterized by a variance ␴H ,

␴H2 = 具关f共r + ␭兲 − f共r兲兴2典 ⬀ 储␭储␣

with

␣ = 2H,

with r = 共x1 , x2 , . . . , xd兲, ␭ = 共␭1 , ␭2 , . . . , ␭d兲, = 冑␭21 + ␭22 + ¯ + ␭2d; a power spectrum SH, S H ⬀ 储 ␻ 储 −␤

with

␤ = d + 2H,

and

共1兲 储␭储 共2兲

with ␻ = 共␻1 , ␻2 , . . . , ␻d兲 the angular frequency, 储␻储 = 冑␻21 + ␻22 + ¯ + ␻2d; and a number of objects NH of characteristic size ⑀ needed to cover the fractal, NH ⬀ ⑀−D

with

D = d + 1 − H,

共3兲

D being the fractal dimension of f共r兲. The Hurst exponent ranges from 0 to 1, taking the values H = 0.5, H ⬎ 0.5, and H ⬍ 0.5, respectively for uncorrelated, correlated, and anticorrelated Brownian functions. The application of fractal concepts, through the estimate of H, has been proven useful in a variety of fields. For example, in d = 1, heartbeat intervals of healthy and sick hearts are discriminated on the basis of the value of H 关2,3兴; the stage of financial market development is related to the correlation degree of return and volatility series 关4兴; coding and noncoding regions of genomic sequences have different correlation degrees 关5兴; and climate models are validated by analyzing long-term correlation in atmospheric and oceanographic series 关6,7兴. In d 艌 2 fractal measures are used to model and quantify stress induced morphological transformation 关8兴; isotropic and anisotropic fracture surfaces 关9–13兴; static friction between materials dominated by hard core interactions 关14兴; diffusion 关15,16兴 and transport 关17,18兴 in porous and composite materials; mass fractal features in wet or dried gels 关19兴 and in physiological organs 共e.g., lung兲 关20兴; hydrophobicity of surfaces with hierarchic structure undergoing natural selection mechanism 关21兴 and solubility of nanoparticles 关22兴; and digital elevation models 关23兴 and slope fits of planetary surfaces 关24兴. A number of fractal quantification methods—based on Eqs. 共1兲–共3兲 or on variants of these relationships—such as rescaled range analysis 共R / S兲, detrended fluctuation analysis 1539-3755/2007/76共5兲/056703共7兲

共DFA兲, detrending moving average analysis 共DMA兲, and spectral analysis, have been thus proposed to accomplish accurate and fast estimates of H in order to investigate correlations at different scales in d = 1. A comparatively small number of methods able to capture spatial correlations, operating in d 艌 2, have been proposed so far 关25–32兴. This work is addressed to develop an algorithm to estimate the Hurst exponent of high-dimensional fractals and thus is intended to capture scaling and correlation properties over space. The proposed method is based on a generalized highdimensional variance of the fractional Brownian function around a moving average. In Sec. II, we report the relationships holding for fractals with arbitrary dimension. It is argued that the implementation can be carried out in directed or isotropic mode. We show that the detrending moving average 共DMA兲 method 关30–32兴 is recovered for d = 1. In Sec. III, the feasibility of the technique is proven by implementing the algorithm on rough surfaces—with different size N1 ⫻ N2 and Hurst exponent H—generated by the random midpoint displacement 共RMD兲 and by the Cholesky-Levinson factorization 共CLF兲 methods 关33,34兴. The generalized variance is estimated over subarrays n1 ⫻ n2 with different size 共“scales”兲 and then averaged over the whole fractal domain N1 ⫻ N2. This feature reduces the bias effects due to nonstationarity with an overall increase of accuracy—compared to the two-point correlation function, whose average is calculated over all the fractal. Furthermore—compared to the twopoint correlation function, whose implementation is carried out along one-dimensional lines 共e.g., for the fracture problem, the two-point correlation functions are measured along the crack propagation direction and the perpendicular one兲, the present technique is carried out over d-dimensional structures 共e.g., squares in d = 2兲. In Sec. IV, we discuss the accuracy and range of applicability of the method. II. METHOD

In order to implement the algorithm, the generalized vari2 is introduced as follows: ance ␴DMA N −m

2 ␴DMA =

N −m

N −m

2 2 d d 1 1 1 ¯ 兺 关f共i1,i2, . . . ,id兲 兺 兺 N i1=n1−m1 i2=n2−m2 id=nd−md

− ˜f n1,n2,. . .,nd共i1,i2, . . . ,id兲兴2 ,

共4兲

where f共i1 , i2 , . . . , id兲 = f共i兲 is a fractional Brownian function

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defined over a discrete d-dimensional domain, with maximum sizes N1 , N2 , . . . , Nd. It is i1 = 1 , 2 , . . . , N1, i2 = 1 , 2 , . . . , N2 , . . ., id = 1 , 2 , . . . , Nd. n = 共n1 , n2 , . . . , nd兲 defines the subarrays ␯d of the fractal domain with maximum values n1 max = max兵n1其, n2 max = max兵n2其 , . . .. nd max = max兵nd其; m1 = int共n1␪1兲, m2 = int共n2␪2兲 , . . ., md = int共nd␪d兲 and ␪1, ␪2, . . .␪d are parameters ranging from 0 to 1; N = 共N1 − n1 max兲共N2 − n2 max兲 ¯ 共Nd − nd max兲. The function ˜f n1,n2,. . .,nd共i1 , i2 , . . . , id兲 =˜f is given by ˜f n1,n2,. . .,nd共i1,i2, . . . ,id兲 1 = n 1n 2 ¯ n d

N −m

n1−1−m1 n2−1−m2

兺 兺 k =−m k =−m 1

1

2

2 ␴DMA =

¯ 2

nd−1−md

⫻

兺 k =−m d

will be further clarified in Sec. III, where the algorithm is implemented over fractal surfaces with different sizes. For 2 each subarray ␯d, the corresponding value of ␴DMA is calculated and finally plotted on log-log axes. To elucidate the way the algorithm works, in the following we consider its implementation for d = 1 and d = 2. The case d = 1 reduces to the detrending moving average 共DMA兲 method already used for long-range correlated time series 关30–32兴. One-dimensional case. By posing d = 1 in Eq. 共4兲, one obtains

f共i1 − k1,i2 − k2, . . . ,id − kd兲,

共5兲

d

which is an average of f calculated over the subarrays ␯d. Equations 共4兲 and 共5兲 are defined for any value of n1 , n2 , . . . , nd and for any shape of the subarrays, however, it is preferable to choose subarrays with n1 = n2 = ¯ · = nd to avoid spurious directionality in the results. The generalized 2 varies as 共冑n21 + n22 + ¯ + n2d兲2H as a consevariance ␴DMA quence of the property 共1兲 of the fractional Brownian functions. Upon variation of the parameters ␪1 , ␪2 , . . . , ␪d in the range 关0,1兴, the indexes i1 , i2 , . . . , id and k1 , k2 , . . . , kd of the sums in Eqs. 共4兲 and 共5兲 are accordingly set within ␯d. In particular, 共i1 , i2 , . . . , id兲 coincides, respectively, with 共a兲 one of the vertices of ␯d for ␪1 = ␪2 = ¯ = ␪d = 0 and ␪1 = ␪2 = ¯ = ␪d = 1 or 共b兲 the center of ␯d for ␪1 = ␪2 = ¯ = ␪d = 1 / 2. It is worthy of note that the choice ␪1 = ␪2 = ¯ = ␪d = 1 / 2 corresponds to the isotropic implementation of the algorithm, while ␪1 = ␪2 = ¯ = ␪d = 0 and ␪1 = ␪2 = ¯ = ␪d = 1 correspond to the directed implementation. For example, in d = 2, the isotropic implementation implies that the variance defined by Eq. 共4兲 is referred to a moving average ˜f calculated over squares n1 ⫻ n2 whose center is 共i1 , i2兲. Conversely, the directed implementation implies that the function ˜f is calculated over squares n1 ⫻ n2 with one of the vertices in 共i1 , i2兲. The directed mode is of interest to estimate H in fractals with preferential growth direction, e.g., biological tissues 共lung兲, epitaxial layers, and crack propagation in fracture 共anisotropic fractals兲. If the fractal is isotropic and the accuracy is a priority, the parameters ␪1 , ␪2 , . . . , ␪d should be preferably taken equal to 1 / 2 to achieve the most precise estimate of H. The dependence of the algorithm on ␪ for d = 1 has been discussed in 关32兴. In order to calculate the Hurst exponent, the algorithm is implemented through the following steps. The moving average ˜f is calculated for different subarrays ␯d, by varying n1 , n2 , . . . , nd from 2 to the maximum values n1 max , n2 max , . . . , nd max. The values n1 max , n2 max , . . . , nd max depend on the maximum size of the fractal domain. In order to minimize the saturation effects due to finite size, it should be n1 max Ⰶ N1; n2 max Ⰶ N2 ; . . . ; nd max Ⰶ Nd. These constraints

1 1 1 兺 关f共i1兲 − ˜f n1共i1兲兴2 , N1 − n1 max i1=n1−m1

共6兲

where N1 is the length of the sequence, n1 is the sliding window, and n1 max = max兵n1其 Ⰶ N1. The quantity m1 = int共n1␪1兲 is the integer part of n1␪1 and ␪1 is a parameter ranging from 0 to 1. The relationship 共6兲 defines a generalized variance of the sequence f共i1兲 with respect to the function ˜f n1共i1兲 as follows: ˜f 共i 兲 = 1 n1 1 n1

n1−1−m1

兺

f共i1 − k1兲,

共7兲

k1=−m1

which is the moving average of f共i1兲 over each sliding window of length n1. The moving average ˜f n1共i1兲 is calculated for different values of the window n1, ranging from 2 to the 2 is then calculated maximum value n1 max. The variance ␴DMA according to Eq. 共6兲 and plotted as a function of n1 on loglog axes. The plot is a straight line, as expected for a power2 on n1 as follows: law dependence of ␴DMA 2 ␴DMA ⬃ n2H 1 .

共8兲

Equation 共8兲 allows one to estimate the scaling exponent H of the series f共i1兲. Upon variation of the parameter ␪1 in the range 关0,1兴, the index k1 in ˜f n1共i1兲 is accordingly set within the window n1. In particular, ␪1 = 0 corresponds to the average f n1共i1兲 over all the points to the left of i1 within the window n1; ␪1 = 1 corresponds to the average f n1共i1兲 over all the points to the right of i1 within the window n1; ␪1 = 21 corresponds to the average f n1共i1兲 with the reference point in the center of the window n1. Two-dimensional case. For d = 2, the generalized variance defined by Eq. 共4兲 writes 2 ␴DMA =

1 共N1 − n1 max兲共N2 − n2 max兲 N1−m1

⫻

兺

N2−m2

兺

i1=n1−m1 i2=n2−m2

with ˜f n1,n2共i1 , i2兲 given by

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关f共i1,i2兲 − ˜f n1,n2共i1,i2兲兴2 ,

共9兲

ALGORITHM TO ESTIMATE THE HURST EXPONENT OF …

1 ˜f n1,n2共i1,i2兲 = n 1n 2

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n1−1−m1 n2−1−m2

兺

兺

f共i1 − k1,i2 − k2兲. 共10兲

k1=−m1 k2=−m2

The average ˜f is calculated over subarrays with different size n1 ⫻ n2. The next step is the calculation of the difference f共i1 , i2兲 −˜f n1,n2共i1 , i2兲 for each subarray n1 ⫻ n2. A log-log plot 2 , of ␴DMA 2 ␴DMA ⬃ 关冑n21 + n22兴2H ⬃ sH

共11兲

s = n21 + n22,

yields a straight line with slope H. as a function of Depending upon the values of the parameters ␪1 and ␪2, entering the quantities m1 = int共n1␪1兲 and m2 = int共n2␪2兲 in Eqs. 共9兲 and 共10兲, the position of 共k1 , k2兲 and 共i1 , i2兲 can be varied within each subarray. 共i1 , i2兲 coincides with a vertex of the subarray if 共i兲 ␪1 = 0, ␪2 = 0; 共ii兲 ␪1 = 0, ␪2 = 1; 共iii兲 ␪1 = 1, ␪2 = 0; and 共iv兲 ␪1 = 1, ␪2 = 1 共directed implementation兲. The choice ␪1 = ␪2 = 1 / 2 corresponds to take the point 共i1 , i2兲 coinciding with the center of each subarray n1 ⫻ n2 共isotropic implementation兲 关41兴. III. RESULTS

In order to test feasibility and robustness of the proposed method, synthetic rough surfaces with assigned Hurst exponents have been generated by the random midpoint displacement 共RMD兲 and by the Cholesky-Levinson factorization 共CLF兲 algorithms 关33,34兴. The widespread use of the RMD algorithm is based on the trade-off of its fast, simple, and efficient implementation to the limited accuracy, especially for H Ⰶ 0.5 and H Ⰷ 0.5. Conversely, the Cholesky-Levinson factorization algorithm is one of the most accurate techniques to generate 1D and 2D fractional Brownian functions, at the expense of a more complex implementation structure 关42兴. 2 as a function of s are In Fig. 1, the log-log plots of ␴DMA shown for the synthetic fractal surfaces generated by the RMD 共circles兲 and by the CLF method 共squares兲. The surfaces have Hurst exponents Hin ranging from 0.1 to 0.9 with step 0.1. The domain sizes are, respectively, N1 ⫻ N2 = 256 ⫻ 256 共a兲, N1 ⫻ N2 = 1024⫻ 1024 共b兲, and N1 ⫻ N2 = 4096 ⫻ 4096 共c兲. The dashed lines show the behavior that should be exhibited by variances varying exactly as sHin over the 2 as a function s are entire range of scales. The plots of ␴DMA in good agreement with the behavior expected on the basis of Eq. 共11兲. The quality of the fits is higher for the surfaces generated by the CLF method, confirming that the RMD algorithm synthesizes less accurate fractals. By comparing the results of the simulation 共symbols兲 to the straight lines corresponding to full linearity over the whole range 共dashed兲, deviations from the full linearity can be observed especially for the small surfaces at the extremes of the scale. A plot of the slopes for the fractal surfaces generated by the CLF algorithm is shown in Fig. 2 for different sizes of the fractal domain. A detailed discussion of the origin of the deviations at low and large scales is reported in the Sec. IV. Finally, we show three examples of digital images currently mapped to fractal surfaces with reference to the color intensity, i.e., to the level of red, green, and blue 共RGB兲. The

2 FIG. 1. 共Color online兲 Log-log plot of ␴DMA for fractal surfaces, respectively, with size N1 ⫻ N2 = 256⫻ 256 共a兲, N1 ⫻ N2 = 1024 ⫻ 1024 共b兲, and N1 ⫻ N2 = 4096⫻ 4096 共c兲. The data refer to fractal surfaces generated by the RMD 共black circles兲 and by the CLF 共blue squares兲 methods. The Hurst exponent Hin—input of the RMD and CLF algorithms—varies from 0.1 to 0.9 with step 0.1. The results correspond to the isotropic implementation, i.e., with the parameters ␪1 = ␪2 = 1 / 2 in the Eq. 共9兲. The dashed lines represent the behavior expected for full linearity, i.e., the log-log plot of curves varying as sHin. It is worthy of note that the CLF results are much closer to the full-linear behavior compared to the RMD ones.

Hurst exponents estimated by the proposed method are, respectively, H = 0.1 共a兲, H = 0.5 共b兲, and H = 0.9 共c兲 for the images in Fig. 3. IV. DISCUSSION

The proposed algorithm is characterized by short execution time and ease of implementation. By considering the case d = 2, the function ˜f n1,n2共i1 , i2兲 is indeed simply obtained by summing the values of f共i1 , i2兲 over each subarray n1 ⫻ n2. Then the sum is updated at each step by adding the last and discarding the first row 共column兲 of each sliding array n1 ⫻ n2. For higher dimensions, the sum is updated at each

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FIG. 2. 共Color online兲 Plot of the values of H obtained by linear fit of the data shown in Figs. 1共a兲–1共c兲 obtained by implementing the proposed algorithm on the fractal surfaces generated by the Cholesky-Levinson factorization method. The dashed line represents the ideal behavior: H = Hin.

step by adding and discarding a d − 1-dimensional set of each array n1 ⫻ n2 ⫻ ¯ ⫻ nd. The algorithm does not use arbitrary parameters, the computation simply relying on averages of f. We will now argue on the origin of the deviations at small and at large scales.

FIG. 3. 共Color online兲. Cloudy sky images, respectively, with Hurst exponent H = 0.1 共a兲, H = 0.5 共b兲, and H = 0.9 共c兲. Such heterogeneous surfaces can be represented as fractals by mapping their color intensity in terms of RGB content.

FIG. 4. HT共␻1 , ␻2兲.

共Color online兲 Plot of the transfer function

Deviations occurring at large scales. The deviations from the linearity at large scales, leading to the saturation of the 2 ␴DMA values, are due to finite size effects. The small surfaces do not contain enough data to make the evaluation of the scaling law over the subarrays statistically meaningful. By comparing the data in Figs. 1共a兲–1共c兲, one can note that the saturation effect reduces upon increasing the size N1 ⫻ N2 of the fractal surface. The finite size effects become negligible when the conditions n1 max Ⰶ N1; n2 max Ⰶ N2 ; . . . ; nd max Ⰶ Nd are fulfilled. Deviations occurring at small scales. The deviations occurring at low scales are related to the departure of the lowpass filter from the ideality. This problem also occurs with one-dimensional fractals 共time series兲 resulting in the quite generally reported overestimation of H in anticorrelated signals and underestimation of H in correlated signals 关35–39兴. We will discuss the origin of these deviations by means of the filter transfer function HT共␻兲 关40兴. The algorithm is based on a generalized variance of the function f with respect to ˜f . The function ˜f is the output of a low-pass filter driven by f, with impulse response a box-car function. In the Appendix, the transfer function HT共␻兲 of ˜f is explicitly calculated and shown in Fig. 4 for d = 2. For an ideal low-pass filter, the transfer function should be one or zero, respectively, at frequencies lower or higher than the cutoff frequency. However, in real low-pass filters, at frequencies lower than the cutoff frequency, all the components of the signal suffer some attenuation but ␻ = 0. The cutoff frequencies of HT共␻兲 are ␻i = ␲ / ␶i, i.e., the first zeroes of the functions sin ␻i␶i / ␻i␶i in the Eq. 共A4兲. Moreover, in real filters, at frequencies higher than ␲ / ␶i, due to the presence of the sidelobes, components of the signals lying in the bands 共␲ / ␶i , 2␲ / ␶i兲 ; 共2␲ / ␶i , 3␲ / ␶i兲 ; . . .., are not fully filtered out. As a result, the function ˜f contains 共a兲 less components with frequency lower than ␻i = ␲ / ␶i and 共b兲 more components with frequency higher than ␻i = ␲ / ␶i compared to what would be expected with an ideal low-pass filter. The lack of low-frequency components depends on the central lobe, while the excess of high-frequency components depends on the sidelobes. The excess of high-frequency components results in the decrease of the difference f −˜f , i.e., in the de2 and, finally, in the increase of the slope of the crease of ␴DMA log-log plot. Conversely, the lack of low-frequency compo-

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TABLE I. Slopes HI, HII, HIII, and relative errors ⌬HI, ⌬HII, ⌬HIII of the curves plotted in Fig. 1共b兲 共squares兲 with N1 ⫻ N2 = 1024⫻ 1024. The slopes have been calculated by linear fit, respectively, over the ranges 10艋 s 艋 100 共HI兲, 10艋 s 艋 1000 共HII兲, and 10艋 s 艋 10000 共HIII兲. Hin

HI

⌬HI

HII

⌬HII

HIII

⌬HIII

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1346 0.2272 0.3233 0.4205 0.5178 0.6171 0.7185 0.8207 0.9253

+3.46⫻ 10−1 +1.36⫻ 10−1 +7.77⫻ 10−2 +5.12⫻ 10−2 +3.56⫻ 10−2 +2.85⫻ 10−2 +2.64⫻ 10−2 +2.58⫻ 10−2 +2.81⫻ 10−2

0.1073 0.2050 0.2995 0.3970 0.4973 0.5973 0.6956 0.7999 0.8999

+7.30⫻ 10−2 +2.50⫻ 10−2 −1.67⫻ 10−3 −7.50⫻ 10−3 −5.40⫻ 10−3 −4.50⫻ 10−3 −6.29⫻ 10−3 −1.25⫻ 10−4 −1.11⫻ 10−4

0.0718 0.1700 0.2716 0.3691 0.4752 0.5617 0.6770 0.7659 0.8679

−2.822⫻ 10−1 −1.500⫻ 10−1 −9.467⫻ 10−2 −7.725⫻ 10−2 −4.960⫻ 10−2 −6.383⫻ 10−2 −3.286⫻ 10−2 −4.263⫻ 10−2 −3.567⫻ 10−2

nents results in the increase of the difference f −˜f , i.e., in the 2 and, finally, in the decrease of the slope of increase of ␴DMA the log-log plot. The two effects are more relevant with smaller values of the scales, when the filter nonideality is greater. Moreover, as one can deduce from Eqs. 共2兲 and 共A6兲, the effect of the sidelobes dominates in high-frequency rich fractals with H ⬍ 0.5, while the effect of the central lobe is dominant in low-frequency rich fractals with H ⬎ 0.5. In order to gain further insight in the above theoretical arguments, we report in Table I the slopes HI, HII, and HIII of the curves 共squares兲 plotted in Fig. 1共b兲 over different ranges. The slopes have been calculated by linear fit, respectively, over the ranges 10艋 s 艋 100 共HI兲, 10艋 s 艋 1000 共HII兲, and 10艋 s 艋 10000 共HIII兲. The relative errors ⌬H = 共H − Hin兲 / Hin are given, respectively, in the third, fifth, and seventh columns. The slope HI is greater than the expected value Hin. The slope HII is overestimated for H = 0.1 and H = 0.2 and underestimated for H ⬎ 0.2. The slope HIII is underestimated since the effects of the finite size of the fractal domain play a dominant role. We address the question if the artifacts due to the filter nonideality described above might be corrected somehow. In the remaining of this section, we will thus consider the use of windows whose general effect is to increase the width of the central lobe while reducing those of the sidelobes of the function HT共␻兲 共a detailed description of these methods can be found in 关40兴兲. By restricting our discussion to the present technique, the correction is performed by using the following variant of the relationship 共5兲: ˜f 쐓 n ,n

1 2,. . .,nd

values are summed together and weighted by ␣. In Fig. 5, we 2 / sHin obtained by implementing the alshow the ratio ␴DMA gorithm, respectively, with the function ˜f 共solid lines兲 and ˜f 쐓 共dashed lines兲 in the range 10艋 s 艋 100. To avoid the overlap in the scaled variance obtained for different values of Hin, the 2 ␴DMA / sHin has been shifted. The plots are shown in logarithmic scales to allow the correction by direct comparison with 2 / sHin is the data plotted in log scales in Fig. 1. The ratio ␴DMA noticeably closer to a constant when the implementation is performed with ˜f 쐓, with a corresponding reduction of two orders of magnitude in the relative error ⌬HI of the slope HI. V. CONCLUSION

We have put forward an algorithm to estimate the Hurst exponent of fractals with arbitrary dimension, based on the 2 defined by Eq. high-dimensional generalized variance ␴DMA 共4兲. The methods currently used to estimate the Hurst exponent of high-dimensional fractals are based on: 共i兲 1-d twopoint correlation and structure functions operated along dif-

共i1,i2, . . . ,id兲 = 共1 − ␣兲f n1,n2,. . .,nd共i1,i2, . . . ,id兲 + ␣˜f n1,n2,. . .,nd共i1 − 1,i2 − 1, . . . ,id − 1兲, 共12兲

where ␣ = n1n2 , . . . , nd / 关共n1 + 1兲共n2 + 1兲 ¯ 共nd + 1兲兴. Equation 共12兲 reduces for d = 1 to the exponentially weighted moving average 共EWMA兲. In practice, the difference between Eq. 共5兲 and Eq. 共12兲 is that the function ˜f 쐓 places more importance to the data around the point i1 , i2 , . . . , id. This is achieved by assigning to the function a weight 共1 − ␣兲, while all the other

2 FIG. 5. Plot of the function ␴DMA / sHin, respectively, with ˜f defined by the Eq. 共4兲 共solid lines兲 and ˜f 쐓 defined by the Eq. 共12兲 共dashed lines兲. The data refer to fractal surfaces generated by the CLF algorithm with N1 ⫻ N2 = 1024⫻ 1024. It can be noted that the deviations from the constant behavior at small scales are reduced with ˜f 쐓 implying a corresponding reduction of the relative error ⌬HI in the slope HI.

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ferent directions, and 共ii兲 high-d Fourier and wavelet transforms 关9–13,29兴. The advantage of the methods 共i兲 is the ease of implementation. Their drawback is the limited accuracy due to biases and nonstationarities, being these functions are calculated over the entire fractal domain. The methods 共ii兲 are more accurate, however, their implementation is complicated especially for data set with limited extension. 2 is scaled, meaning that it is The generalized variance ␴DMA calculated over subarrays of the whole fractal domain by means of the function ˜f . The scales are set by the size of the subarrays n1 ⫻ n2 ⫻ ¯ . ⫻ nd. Therefore, the proposed method exhibits at the same time 共a兲 ease of implementation, being based on a variancelike approach and 共b兲 high accuracy, being calculated over scaled subarrays rather than on the whole fractal domain. A further important feature of the proposed algorithm is that it can be implemented “isotropically” or in “directed” mode to accomplish estimates of H in fractals having preferential growth direction, e.g., biological tissues, epitaxial layers, or in crack propagation in fracture. The isotropic implementation is obtained by taking ␪1 = ␪2 = ¯ = ␪d = 1 / 2 in Eq. 共4兲. This choice implies that the reference point 共i1 , i2 , . . . , id兲 of the moving average lies in the center of each subarray n1 ⫻ n2 ⫻ ¯ ⫻ nd and thus ˜f is calculated by summing the values of f around 共i1 , i2 , . . . , id兲. Conversely, to implement the algorithm in a preferential direction 共directed implementation兲, the reference point must be coincident with one of the extremes of the segment n1, or with one of the vertices of the square grid n1 ⫻ n2 or of the d-dimensional array n1 ⫻ n2 ⫻ ¯ ⫻ nd. The directed implementation can be performed by choosing, for example, ␪1 = ␪2 = ¯ = ␪d = 0. Further generalizations of the proposed method can be envisaged for application to the analysis of time-dependent spatial correlations in d 艌 2.

␪1 = ␪2 = , . . . , = ␪d = 0. Equation 共A1兲 can be rewritten as a convolution integral as follows: ˜f 共x ,x , . . . ,x 兲 = 1 2 d

⫻ −

⫻

冕

x2

x2−␶2

冕

x1−␶1

dx2⬘ ¯

dx1⬘

冕

xd

xd−␶d

⬁

dx*1U

−⬁

⬁

dx*dU

冉冊 x*d

␶d

冉 冊冕 x*1 ␶1

⬁

dx*2U

−⬁

冉冊

x*2 ¯ ␶2

f共x1 − x*1,x2 − x*2, . . . ,xd

x*d兲,

共A2兲

with the convolution kernels given by the boxcar function as follows: U共x*i /␶i兲 =

再

1 for 0 ⬍ x*/␶i ⬍ 1 0 elsewhere.

冎

The transfer function can be calculated as follows: H T共 ␻ 1, ␻ 2, . . . , ␻ d兲 =

1 ␶ 1␶ 2 ¯ ␶ d ⫻

冕

␶d

0

冕 冕 ␶1

␶2

dx1

0

0

dx2 ¯

dxd exp关− i2␲共␻1x1 + ␻2x2 + ¯

+ ␻dxd兲兴

共A3兲

that can be written as d

H T共 ␻ 1, ␻ 2, . . . , ␻ d兲 = 兿 i=1

sin ␻i␶i ␻ i␶ i

共A4兲

that is thus d times the function sin ␻i␶i / ␻i␶i. ˜ of the function ˜f can be obtained The Fourier transform F by means of the following relationship: ˜ 共␻ , ␻ , . . . , ␻ 兲 = H 共␻ , ␻ , . . . , ␻ 兲F共␻ , ␻ , . . . ␻ 兲, F 1 2 d T 1 2 d 1 2 d

The function ˜f , defined by Eq. 共5兲, corresponds to the discrete form of the integral as follows: x1

冕

冕

−⬁

APPENDIX: TRANSFER FUNCTION OF ˜f

1 ˜f 共x ,x , . . . ,x 兲 = 1 2 d ␶ 1␶ 2 ¯ ␶ d

1 ␶ 1␶ 2 ¯ ␶ d

dxd⬘ f共x1⬘,x2⬘, . . . ,xd⬘兲, 共A1兲

共A5兲 where F共␻1 , ␻2 , . . . , ␻d兲 is the Fourier transform of the function f共x1 , x2 , . . . , xd兲. The power spectrum ˜S of the function ˜f is given by ˜S共␻ , ␻ , . . . , ␻ 兲 = 兩H 共␻ , ␻ , . . . , ␻ 兲兩2S共␻ , ␻ , . . . , ␻ 兲, 1 2 d T 1 2 d 1 2 d 共A6兲

where for the sake of simplicity we have considered the case

where S共␻1 , ␻2 , . . . , ␻d兲 is the power spectrum of the function f共x1 , x2 , . . . , xd兲.

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