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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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