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This paper was presented in the 1st IEEE EMBS conference on Neural Engineering, March, 2003.

Wavelet Based Estimation of the Fractal

Dimension in fBm ImagesCarlos Parra, Khan Iftekharuddin, David Rendon

Abstract Fractional Brownian Motion (fBm) has beensuccessfully exploited to model an important number ofphysical phenomena and non-stationary processes suchas medical images. These mathematical models closelydescribe essential properties of natural phenomena, such asself-similarity, scale invariance and fractal dimension. Theuse of wavelet analysis combined with fBm analysis mayprovide an interesting approach to compute key values forfBm processes, such as the fractal dimension D. In thispaper we propose two approaches to calculate the HurstCoefficient H (and hence D) for both one-dimensionaland two-dimensional signals. The first approach is basedon statistical properties of the signals, while the secondone is based on their spectral characteristics. A formalextension to 2-D processes is developed and implemented,and a comparison of both is presented. The proposedalgorithms are tested using tomographic brain tumor data.Our simulation experiments offer promising results in thelocation of such lesions.

I. INTRODUCTION

In the analysis of tomographic medical images, the

presence of randomness in the acquired data, derived

from the imaging modality procedure, noise, and thenature of the analyzed tissue itself, has been modeled

in terms of texture. This problem is common to other

sciences in which the description of the specific study

object requires the consideration of self similar structures

and ruggedness. The fundamental theory that support

this models is owed fundamentally to B. B. Mandelbrot,

whose work with J. W. Ness on Fractional Brownian

Motions in [1] and his formulation of the fractal theory

concepts [2], provided the work frame required for the

modeling of such problems. One of the most usual

applications of this theory was the creation of algorithms

leading to the simulation of different fractal motions,

being the Brownian motion the closest in the graphicalsimulation of structures normally seen in nature [4],

or in biomedical imaging. From the signal processing

analysis point of view, an extensive research work has

been leaded by Flandrin [5][6], particularly in the field

of spectrum computation and parameter estimation of

stochastic processes obeying fractional power laws.

Texture analysis of natural scenes described through

fBm, was greatly enhanced with the development of

both Wavelet Analysis as well as the Mallats Wavelet

Multiresolution Analysis [7][8], whose concepts are very

close in nature to the descriptions of fractal science, and

allowed scientists and engineers to derive a common

signal processing ground, in which the statistical and

spectral and properties of the fBms can be exploited

to estimate fractal parameters analyze an so model and

measure the texture content of a specific image. This

joint study of fractals and wavelets has lead to the

development of methods and models to analyze frac-

tional power law processes. An important contribution

in the formulation of a 2-D model is proposed by

Heneghan [9], who describes both the spectral properties

and correlation function of an fBm, and proposes a

method to estimate the FD using the statistical properties

of the Continuous Wavelet Transform (CWT) of an fBm.

Wornell [10] gives a detailed demonstration on how

1/f processes can be optimally represented in terms of

orthonormal wavelet bases, which opens the possibility

to the use of discrete wavelet transforms in the present

article.

Given the non-stationary character of fBm processes,alternative methods have been developed to estimate

the spectral. In [13], WEN, C. et al. propose a 1-D

approach to estimate H. Grassin and Garello provide

also a 2-D model using Wigner-Ville methods and a

discrete version for the Discrete Time Frequency Wigner

Ville Distribution (WVD) as a mathematical support for

experiments with Synthetic Aperture Radar. These au-

thors describe the problems related to interference terms

or cross-terms generated in the WVD algorithm, but a

tentative solution of these is presented by Min Y. et al

[14]. Although the numerical analysis of measurements

using WVD tend to suggest the inappropriateness of the

traditional FFT spectral estimation as a valid methodfor the study of fBm processes, the high computational

efficiency this algorithm is exploited and a meaningful

results are derived from it.

The fundamental goal of this study is the formulation

and evaluation of algorithms leading to an accurate and

reliable method for identification and location of brain

tumors, considering the texture measurements associated

to different areas of a given tomographic image, in terms

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of the Hurst parameter H or the Fractal Dimension

D. For the implementation of this system, synthetic

images are generated and to test the algorithms. The

formal development of a 2-D model for the spectral

estimation of these values is derived presented, as well as

experimental results regarding particular image modali-

ties. A comparison of spectral estimation methods suchas Wigner-Ville Description and traditional spectrum

estimation is also considered, as well as a 2-D statistical

method. Tomographic brain images are analyzed with

each algorithm and a comparison is finally performed.

I I . FRACTIONA B ROWNIAN M OTIONBACKGROUND

Fractional Brownian Motions (fBm) are part of the set

of 1/f processes. They are non-stationary zero-mean

Gaussian random functions, defined as [1], [2], [3]

BH(0) = 0

BH(t) BH(s) = 1

(H+ 0.5) (1)

0

(t s)H0.5 sH0.5

dB(s)

+

0

(t s)H0.5 dB(s)

where BH(s) is an ordinary Brownian motion and the

Hurst coefficient H is the parameter that characterizesfBm 1, with the restriction 0 < H

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rBH (u , v) = E[BH(u)BH(v)] (7)

= VH

2

|u |2H + |v |2H |u v |2H

.

The increments of the processB(u ) = B(u +u )B(u ) form a stationary, zero-mean Gaussian process.

The variance of the increments B(u ) depends only

on the distance u=

u2x+ u2y, so

E|BH(u )|

2 uH. (8)

These properties are central to the definition of a pro-

cedure based on wavelet analysis, as described in the

following subsections.

A. 1-D Case

The essential cause of non-stationarity in fBms is con-

centrated in the low frequencies [6]. When the fBm is

decomposed using a multiresolution analysis, the low

and high frequency components can be separated in

a non-stationary approximation and a stationary detail

part, given the low-pass and band-pass character of the

respective analyzing wavelets.

For a specific approximation resolution 2J, the multires-

olution representation [7] of an fBm process is

BH(t) = 2J/2

n

aJ[n](2Jt n) (9)

+j

2j/2n

ndj [n](2jt n),

with j = J, ..., and n = ,...,. The basicwavelet satisfies the admissibility condition

(t)dt= 0 (10)

and the orthonormal wavelet decomposition of the fBm

at resolution j is [6]:

dj[n] = 2j/2

BH(t)(2jt n)dt. (11)

Considering the definition of E[B(t)B(s)] in (3) andalso the admissibility condition in (11), it can be said that

the variance of the detail wavelet coefficients is related

to the specific analyzing wavelet and the H coefficientof the fBm processes as [6]:

E|dj [n]|2

=VH

2

V(H)(2j)2H+1 (12)

whereV(H)is the scale-independent and depends onlyon the selected mother wavelet (t) and the value ofH[5]. Defining =t s,

V(H) =

(t)(s)dt ||2Hd (13)

which is an inner product that generates a constant value.

By applying logarithm in both sides of equation (12),

the following linear equation is obtained [11]

log2E|dj [n]|

2

= (2H+ 1)j+ C1 (14)

with

C1= log2VH

2 V(H). (15)

The Hurst coefficient (and the dimension) of a fBm

process can be calculated from the slope of this variance,

plotted as a function of the resolution in a log-log plot.

B. 2-D Case

The wavelet filter used to obtain the details equation

for the high frequency part, both in x and y, and at anspecific resolution j, corresponds to Theorem 4 in [7],which is presented in Appendix ??.

Taking equation (11) as a reference to extend the defini-

tion of the detail coefficients, at resolution j , to the 2-Dcase, it can be shown that for (n, m) Z2,

D3

2j [n, m] =

2j

BH(x, y)

32j

(x 2jn, y 2jm)dxdy

(16)

or equivalently

D32j

[] =

2j

BH(u) 32j

(u 2j)du

(Z2)

(17)

where corresponds to the position [n, m] and 32j

satisfies the admissibility condition

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32j

(x, y)dxdy = 0. (18)

The variance function of the detail coefficients in equa-

tion (17) is obtained following a similar process to the

continuous wavelet approach described by Heneghan in

[9] (see Appendix B ??):

ED3

2j[]

2 = 22ju

v

32j

(u 2j ) (19)

32j

(v 2j)E[B(u)B(v)]

du dv .

Considering the definition of the covariance function of

a process in (7), the previous equation can be written as

E

D32j

[]2

= VH2

22j u

v 32j

(u 2j)32j

(v 2j) |u |2Hdu dv+

u

v 32j (u 2

j)32j (v 2

j) |v |2H

du dv+u

v

32j

(u 2j)32j

(v 2j) |u v |2H du dv

(20)

or equivalently

E

D32j

[]2

= VH2

22j u

32j

(v 2j)dv v

32j

(u 2j) |u |2Hdu+u

32j

(u 2j)du v

32j

(v 2j) |v |2Hdv+u

v

32j

(u 2j) 32j

(v 2j) |u v |2H du dv

.

(21)

Given that 32j

meets the admissibility condition (18),

v

32j

(v 2j)dv =

u

32j

(u 2j)du = 0 (22)

so, (19) can be written as

E

D32j

[]2

= VH2

22j

32j

(u 2j)

u

v

32j

(v 2j) |u v |2H du dv

.

(23)

By substituting p = u v and q = v 2j ,Eq. (19) can be stated as

E

D32j

[p , q]2

= VH2

2j(2H+2)p

q

32j

(p + q )

32j

(q) |p |2Hdp dq

(24)

showing an integrand independent of the scale 2j . LetV32 be defined as [11]

V32=

p

q

32j

(p + q) 32j

(q) |p |2Hdp dq (25)

so, the integral in q is the wavelet transform of thewavelet itself at a resolution j, in the same way that(13). The variance ofD3

2j[n, m]is hence a power law of

the scale2j and can be used to calculate H in a similarway to (14)

log2ED3

2j[n, m]

2= (2H+ 2)j+ C2 (26)with

C2= log2VH

2 V3

2j(H). (27)

The dimension of the fBm can be extracted from the

slope of equation (26) in a log-log plot.

C. Algorithm

VARIANCE ALGORITHM FOR THE CALCULATION

OF HURST COEFFICIENT

V arianceF D(M0,N,Wavelet)

M0 is the matrix that corresponds to theinput image (formats .jpg, .tif, .gif) with

an appropriate colormap.

Wavelet is the analyzing wavelet filter.N is the desired level of Multi-Resolutiondecomposition steps.

1) for j = 1 :N {a) Compute a multiresolution

decomposition of the signal at severalresolutions 2j according to Eqs. (11)and (16).

b) Compute the variance for theD32j

matrix at each resolution 2j.

c) Compute the base-2 logarithm of theprevious result.

}

2) Compute the slope of the resultingequations using the pairs

j, log2

var

D32j

obtained from Eqs. (12)

and (14).

3) Derive the value of H from the slope

values of the previous step.

IV. POWERS PECTRUMM ETHOD

Some of the most frequently seen structures in fractal

geometry, generally known as 1/f processes, show apower spectrum obeying the power law relationship

S() k

|| (28)

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where corresponds to the spatial frequency, and =2H+1 . This kind of spectrum is associated to statisticalproperties that are reflected in a scaling behavior (self-

similarity), in which the process is statistically invariant

to dilations or contractions, as described in the equation

S() = |a|SX(a). (29)

A. 1-D Case

Another approach to the computation of the fractal di-

mension of an fBm uses the wavelet representation of its

power spectrum. In [5], Flandrin shows that the spectrum

of an fBm follows the power law of fractional order

shown in (28), using either a time-frequency description

or a scale-time description.

If the frequency signalS()is filtered with a wavelet fil-ter(u), the resulting spectrum at the specific resolutionis [7]

S2j () = S() 2j2 (30)

with

() = ei H(+ ) (31)

where H() corresponds to the Discrete-Time FourierTransform of the corresponding scaling function (x),and defined in terms of its coefficients h(n):

H() =

n=

h(n)ein. (32)

Using the sampling for the discrete detail description of

a function f in [7],

f(u), 2j (u 2jn)= (f(u) (u)) 2jn ,(33)

or,

D2j =

(f(u) 2j (u))

2jn

(34)

which contains the coefficients of the high frequency

details of the function, the spectrum of the discrete detail

signal can be written as [7]

Sd2j

() = 2jk=S2j (+ 2j2k). (35)

The energy of the detail function at an specific resolution

j is defined as [7]

22j

=2j

2

2j2j

Sd2j

()d. (36)

This equation describes the support of the wavelet in

the frequency domain [8], for an specific resolution j.Finally, it can be shown that the solution of the integral

leads to an expression that relates the energy content in

two consecutive resolution filtering operations [7]:

22j = 2

2H22j+1

. (37)

From this expression, the Hurst coefficient H can be

derived, using the expression

H=1

2log2

2

2j

22j+1

. (38)

B. 2-D Case

Extending the analysis to the two-dimensional case, the

same steps of the previous section are followed. For a

2-D fBm, the power spectral density assumes the form

[9]

S() = S(x, y) K

(2x+ 2y)

2H+22

. (39)

For this part of the study,S()is computed as the powerspectral density of a stationary signal, which doesnt

apply strictly to non-stationary processes, as is the case.

However, this approach leads to good results.

S(x, y) = |F F T(image)|2 . (40)

If the frequency-domain signal is filtered with a wavelet

filter , the resulting spectrum at the specific resolution is[7]

S2j ( ) = S( )

32j

2j(x, y)

2 (41)where

|(x, y)|2

= |(x)|2 |(y)|

2 . (42)

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The discrete version of this spectrum can be written as

[7]

Sd2

j (x, y) = 22j

l=

k=S2j x+ 2j2k,y+ 2

j2l .(43)

Fig. 1. Dyadic frequency allocation for each wavelet filter in a MRdecomposition.

Fig. 2. Fourier Transform of the high frequency 2-D wavelet,3(x, y), for Daubechies 6.

The energy of the details function at an specific reso-

lutionj can be calculated by integration in the supportof3j() of the chosen wavelet filter [8], as shown infigure 1. This is described by the equation

22j

=22j

42

2j2j

2j2j

Sd2j

(x, y)dxdy. (44)

After an appropriate change of variable it can be said

that the previous equation can be written as [7]

22j = 2

2H22j+1

(45)

so, also in the case of 2-D signals, the ratio of the

energy corresponding to the detail signals at successive

resolutions, provides a solution for the computation of

H, in the same way than Eq. (38).

H=1

2log2

22j

22j+1 . (46)

C. Algorithm

POWER SPECTRAL DENSITY ALGORITHM FOR THE

CALCULATION OF HURST COEFFICIENT

SpectralFD(M0,N,Wavelet)

M0 is the M M matrix that corresponds to theinput image (formats .jpg, .tif, .gif) adjusted

to a standardized gray level number.

Wavelet is the analyzing wavelet filter.N is the desired level of Multi-Resolutiondecomposition steps.

1) Compute the 2-D FFT of the image.

2) Compute Power Spectral Density (PSD) of

the image according to Eq. (40).

3) For j = 1 :N {

a) Compute the magnitude of the M M

2-D filter, corresponding to the

separable wavelet

3(x, y) according

to Eq. (42).

b) Multiply (frequency domain) the PSD of

the image with the filter, as in Eq.

(41).

c) Sum all the elements of the resulting

matrix.

d) Divide the resulting matrix by 2 2j

to obtain energy(j).

}

4) Estimate H(the Hurst coefficient)

according to Eq. (45) using the values of

energy(j) for all the values of j.

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V. WIGNER-V ILLE D ESCRIPTION (WVD) METHOD

Previous research has been developed in the modeling

and estimation of fBm parameters[8], [13] and have

demonstrated that WVD is a valid method for the

analysis of this kind of processes. For a 1-D process,

the WVD is given by

Wf(t, ) =

f

t +

t02

f

tt0

2

exp(jt0)d0

(47)

or in frequency domain

WF(t, ) =

F

+0

2

F

02

exp(j0t)dt0

(48)

with F() corresponding to the Fourier Transform off(t) and Wf(t, ) = WF(t, ). The properties of thisdistribution can be found among others in [14].

The extension to 2-D [15] is described with the equation

Wf(x,y,u,v) =R2

f

x + 2

, y+ 2

f

x 2

, y 2

exp(2j(u + v)) dd

(49)

whose discrete time-frequency version can be

defined[16], for an image of size NX NY, as

W(n1, n2, m1, m2) = 14NX NY

NX1l1=0

NY1l2=0

f(l1, l2)f(n1 l1, n2 l2)

expj

m1NX (2l1 n1) + m

2NY (2l2 n2)

.

(50)

Letfbe the original image sample both in time (space)and frequency domains. The WVD for this signal is

defined as

Wf(x,y,u,v) = 1

XY

n1,n2,m1,m2

W(n1, n2, m1, m2)

x n1X2

y n2Y2

u m12NxX

v m22NyY

.

(51)

Once the WVD of the fBm process has been completed,

the local power spectrum can be computed simply as

PBH (t, ) = |WBH (t, )| (52)

VI . RESULTS

Two algorithms were implemented to synthesize 2-D

fractional Brownian motion[4]. The first is based on

the midpoint displacement algorithm and the other one

is based in the spectral properties of the fractional

Brownian Motion. The measurements derived from the

images generated with power spectrum method were

closer to the theoretical values used to generate each

fBm. For the variance method, the result of averaging

a specified number of measurements for 20 uniformly

distributed values ofH, between 0 and 1, is summarizedin figures 3 and 4.

Fig. 3. Estimation of Fractal Dimension from a set of synthesizedimages for 20 values of H between 0 and 1. The images wereobtained using the Spectral Synthesis approach. The difference of eachestimation regarding the theoretical estimation line is shown in thesecond graph.

Fig. 4. Estimation of Fractal Dimension from a set of synthesizedimages for 20 values of H between 0 and 1. The images weresynthesized using the Midpoint approach. The difference of eachestimation regarding the theoretical estimation line is shown in thesecond graph.

In Fig. 4, there is an average difference of 0.24 for the

theoretical and estimated values of fractal dimension.

However, the estimated fractal dimension has a linearbehavior that can be used to compute the actual value.

This is a topic for future research.

On the other hand, the Spectral Method of estimation

of Fractal Dimension gave far more consistent results,

close to the theoretical values and coherent in all the

detail scales. The next figures describe respectively the

worst and best cases for this approach. The lines, from

lower to upper, describe the ratio of energy content (see

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Eq. 46) between resolutions 1 - 2, 2 - 3, 3 - 4, and 4

- 5. Daubechies 2 showed a relatively poor result, but

the rest of the Daubechies family and the other families

of wavelets, had a consistent behavior similar to the one

shown in Fig. 6.

Fig. 5. Spectral estimation of Fractal Dimension using Daubechies 2wavelet.

Fig. 6. Spectral estimation of Fractal Dimension using a biorthogonalwavelet.

The algorithm is applied to a brain CT (see Fig. 7),

and the objective is to estimate the fractal dimension

in different areas of the image, delimited by a grid. For

this study, 16 and 32 pixel grids were used to analyze

the tomographic images (CTs and MRIs).

Fig. 7. Source image for which the local fractal dimension wasestimated .

A. Variance Method

Although the localization of certain areas of the image

associated to tumor tissue is consistent, the results for

this method depend heavily on the analyzing wavelet.

Another drawback of this method is seen in the relation-

ship between observations of different grain. Theres not

a direct correspondence between different grid sizes, as

can be seen in Figs. 8 and 9. This method is efficient

in terms time of execution, as it performs a wavelet

decomposition and the computation of the variance of

the detail coefficients.

Fig. 8. Estimation of Fractal Dimension, using Variance Method, thewavelet Daubechies 5, and a grid of 32 pixels.

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Fig. 9. Estimation of Fractal Dimension, using Variance Method, the

wavelet Daubechies 5, and a grid of 16 pixels.

It is possible to locate some features comparing different

grids as in figures 8 and 9. In the 32 pixel grid, for

instance, the positions (3, 4)and (4, 3) appear as a highvalues of Fractal Dimension, almost with every wavelet

family. The same happens with positions(6, 3), or(7, 5).When a finer grid is considered, it is possible to see

some details corresponding to some of the previously

mentioned example areas.

B. Spectral Method

Contrary to Variance Method, the spectral method shows

far more independence from the selected analyzing

wavelet, in the sense that almost the same features are

identified and located, no matter which wavelet is used

to perform the filtering, both for 16 and 32 pixel grids.

It was also observed that the computation times are

considerably higher for this algorithm, compared to the

Variance Method.

Fig. 10. FFT Power Spectrum estimation of local Fractal Dimensionvalues, using the wavelet Daubechies 5, and a grid of 32 pixels.

Fig. 11. FFT Power Spectrum estimation of local Fractal Dimensionvalues, using the wavelet Daubechies 5, and a grid of 16 pixels.

In this experiment, the position (4, 3) presented the

highest fractal dimension. This can be clearly seen inpositions(7, 8)and (8, 5). The same situation is seen in(7, 5) for a 32 pixel grid, and the corresponding lowvalues in (13, 10) and (13, 11) for a 16 pixel grid.

C. Wigner-Ville Description Method

The graphical results for the 32-pixel grid are very

similar to the ones obtained in the previous methods.

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Fig. 12. WVD Method estimation of local Fractal Dimension values,

using the wavelet Daubechies 5, and a grid of 32 pixels.

Fig. 13. FFT Power Spectrum WVD Method estimation of localFractal Dimension values, using the wavelet Daubechies 5, and a gridof 16 pixels.

The results show clearly that for 32-pixel grid, the areain the position (4, 3) shows a high fractal dimension.Regarding the 16-pixel grid analysis, low H areas suchas(13, 10)are clearly visible using the three algorithms.Numeric values obtained from the three algorithms,

specifically for the 32-bit grid position (3, 4). Theseresults are summarized in the following table.

The values obtained from the variance method are in

general terms above the expected values, in an average

TABLE I

COMPARISON OF THEF D VALUES FROM THE THREE METHODS.

POSITION(4,3) IN THE3 2- BIT GRID

General daubechies biorthogonal coifflet

variance 3.08 3.09 3.13 3.03

fft spectrum 2.52 2.53 2.49 2.54

wvd 2.50 2.57 2.42 2.55

of 0.4. On the other hand, FFT method values tend to

be below this value. WVD are closer to the expected

results. However, all three approaches show coherence

and can be used to derive FD values after some simple

processing and particularly show relative differences that

can be exploited for the spotting of different FD tissues.

VII. CONCLUSION

Three mathematical procedures to compute the Hurst

coefficient of fractional Brownian motions have been

presented, both for one and two-dimension cases. From

the experimental results of this study, it can be concluded

that lesions can be associated to localized high or low

values of the fractal dimension. It can be noted that

different anomalies in brain images are perceived as

drastic differences of the fractal dimension in adjacent

areas of a given grid. On the other hand, the frequency-

domain processing of tomographic images provides a

more consistent framework for the multiresolution anal-

ysis, although the time required for this computation isgreater than in the statistical processing of the image.

VIII. FUTURE W ORK

From the results obtained in this study, we are inter-

ested in developing classification algorithms based on

the three techniques shown in this paper, as well as

a comparison of these in terms of computational effi-

ciency and accuracy, and the validation of results using

sets of tomographic images corresponding to differentbrain pathologies. Future work also includes also the

development of a more general fBm model known as

multifractional Brownian motion (mBm) in which the

Hurst coefficient H is modeled as a function of timeor position H(t) as a fundamental part of a moredetailed tissue recognition algorithm, based probably in

Neural Networks techniques. The implementation of fast

algorithms for FFT spectrum and WVD estimations are

also part of this research work.

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I X. ACKNOWLEDGMENT

The authors wish to thank the Whitaker Foundation

for partially supporting this work through a Biomedical

Engineering Research Grant (RG-01-0125).

APPENDIX

Let (V2j )jZbe a separable multiresolution approxima-tion ofL2(R). Let (x, y) = (x)(y) be the associ-ated two-dimensional scaling function. Let (x) be theassociated one-dimensional wavelet associated with the

scaling function (x). Then the three wavelets

1(x, y) = (x)(y) (53)

2(x, y) = (x)(y)

3(x, y) = (x)(y)

are such that

2j1

2j

x 2jn, y 2jm

,

2j22j

x 2jn, y 2jm

,

2j32j

x 2jn, y 2jm

(n,m)Z2

(54)

is an orthonormal basis O2j and

2j1

2j x 2jn, y 2jm ,2j22j x 2jn, y 2jm ,

2j32j

x 2jn, y 2jm

(n,m,j)Z3

(55)

is an orthonormal basis ofL2(R2).

Considering the expected value of the square of the

wavelet transform (which is equal to its variance since

the expected value of the transform itself is zero), we

obtain:

E

CW TB(a,

b)

2

=

1a2u

v

u

ba

v

ba

E[B(u)B(v)] du dv(56)

Using the expression (3) in conjunction with (18) leads

to

E

CW TB(a,b)2

=

C2a2

u

v

u

b

a

v

b

a

|u v | du dv

(57)

The substitutions lead to

E

CW TB(a,b)2

=

C2a2H+2

p

q

(p + q) (q) |p |2H dp dq(58)

which can be conveniently rewritten as

E

CW TB(a,b)2

=

C2a2H+2

p CW T

(1,

p) |p |2Hdp

(59)

where CW T represents the wavelet transform of thewavelet itself. Since the integrand in (58) is independent

ofa, the variance ofCW TB(a,b) varies as a power

law in a, and can be used to estimate the self-similarityparameter of the process. This approach therefore pro-

vides an alternative to the power spectral density for

estimating H.

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