Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
A wavelet-based method for multifractal analysis of 3D random fields: application to turbulence simulation data Pierre Kestener and Alain Arneodo Laboratoire de Physique, Ecole Normale Sup´erieure de Lyon 46 all´ee d’Italie, 69364 Lyon c´edex 07, France Keywords: Fractals, multifractal formalism, scale invariance, singularity, H¨older exponent, fractional Brownian motion, multifractal cascade models, 3D continuous wavelet transform, 3D Wavelet Transform Modulus Maxima (WTMM) method, fully developed turbulence, Direct Numerical Simulations (DNS), intermittency, 3D dissipation field. Abstract: We generalize the so-called wavelet transform modulus maxima (WTMM) method to multifractal analysis of 3D data. This method has been originally designed to describe statistically the roughness fluctuations of fractal signals like turbulent 1D signals and 2D rough surfaces. The 3D WTMM method consists in performing a multi-scale edge detection. This can be achieved using the 3D continuous wavelet transform provided one chooses an appropriate analyzing wavelet. The filtering step takes advantage from recursive filter techniques which, for 3D data analysis, are much less time consuming that FFT algorithms. After linking across scales the local maxima of the WT modulus, one obtains the WT skeleton which provides a space-scale partitioning of the considered 3D data. From this skeleton one computes, at each scale , some partition functions that are at the heart of the multifractal analysis. Then from the scaling behavior of these partition functions, one extracts the so-called multifractal spectra including the singularity spectrum. We report results of test applications of the 3D WTMM method on synthetic 3D monofractal Brownian fields and on 3D multifractal realizations of singular cascade measures as well as fractionally integrated singular cascades for which analytic multifractal properties are known. Then the 3D WTMM method is applied to the dissipation field issue from 3D isotropic turbulence simulation data. The departure from monofractality and the intermittent nature of the dissipation field is clearly shown and quantified. A comparative analysis with the results obtained with 3D box-counting techniques reveals that these classical techniques indeed fail to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master non-conservative cascade measures. We conclude by emphasizing the wide range of potential applications of the 3D WTMM method, e.g. geophysical data, molecular cloud structure in astrophysics, medical imaging as well as many other areas of fundamental and applied sciences.
1
Introduction
Since the late 70’s and the propagation of fractal ideas throughout the scientific community [1], there have been numerous applications of the concepts of scale invariance, self-similarity, longrange dependence in many areas of physics, chemistry, biology, geology, meteorology, economy, 1
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
social and material sciences [2–10]. Various methods were developed to quantify scale-invariance properties through the computation of the fractal dimension for self-similar objects or the roughness exponent for self-affine fractals [1, 6, 11–15]. Unfortunately and are global quantities that do not account for the possibility of point-to-point fluctuations of the scaling properties of a fractal object. The multifractal formalism was introduced in the mid-eighties to provide a statistical description of the fluctuations of regularity of singular measures that are found in chaotic dynamical systems [16–18] or in modelling of the energy cascading process in turbulent flows [19–22]. Box-counting and correlation algorithms were successfully adapted to resolve multifractal scaling for isotropic self-similar fractals by computation of the generalized fractal dimensions [23–25]. As to self-affine fractals, Parisi and Frisch [26] proposed, in the context of the analysis of fully-developed turbulence velocity data, an alternative multifractal description based on the investigation of the scaling behavior of the so-called structure functions [8, 27]: ( integer ), where is an increment of the recorded signal over a distance . Then, after reinterpreting the roughness exponent as a (power-law behavior), the singularity spectrum local quantity [26, 28–30]: is defined as the Hausdorff dimension of the set of points where the local roughness (or H¨older) exponent of is . In principle, can be attained by Legendre transforming the structure function scaling exponents [26, 28–30]. Unfortunately, as noticed by Muzy et al [31], there are some fundamental drawbacks to the structure function method. Indeed, it generally fails to fully singularity spectrum since only the strongest singularities of the function characterize the itself (and not the singularities present in the derivatives of ) are a priori amenable to this analysis.
! # "%$ &
+, 6
('
? *+,
A
8;:= 6
) *+,-./0+2135467+, 98;:= 6 +
In previous work, Arneodo and collaborators [28–31] have shown that there exist a natural way of performing a unified multifractal analysis of both singular measures and multi-affine functions, which consists in using the continuous wavelet transform [32–36]. By using wavelets instead of boxes, one can take advantages of the freedom of the choice of these “generalized oscillating boxes” to get rid of possible smooth behavior that might either mask singularities or perturb the estimation of their strength . The other fundamental advantage of using wavelets is that the skeleton defined by the wavelet transform modulus maxima (WTMM) [37, 38], provides an adaptative space-scale partitioning from which one can extract the singularity spectrum via the scaling exponent of some partition functions defined from the WT skeleton. The so-called WTMM method [28–31] therefore gives access to the entire spectrum via the . We refer the reader to Refs. [39, 40] for usual Legendre transform rigorous mathematical results and to Ref. [41] for the theoretical treatment of random multifractal functions.
B C 6DFEHGJIK LCL64 B C
6 6
Applications of the WTMM method to 1D signals have already provided insight into a wide variety of problems, e.g. the validation of the cascade phenomenology of fully developped turbulence, the characterization and the understanding of long-range correlations in DNA sequences, the demonstration of the existence of a causal cascade of information from large to small scales in financial time series, the discovery of a Fibonacci structural ordering in 1D cuts of diffusion limited aggregates (DLA) (see Refs. [9, 42–44] and references therein for details). In a recent work, the canonical WTMM method has been generalized from 1D to 2D with the specific goal to achieve multifractal analysis of rough surfaces with fractal dimension anywhere between 2 and 3 [45–48]. The new 2D algorithms have been already successfully applied to characterize the intermittent nature of satellite images of the cloud structure [46, 49, 50] and to assist in the diagnosis in digitized mammograms [50, 51].
2
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX Our aim here is to generalize the WTMM method from 2D to 3D, starting from the original theoretical concepts, following the same methodological steps but using recursive filter techniques [52, 53] as an alternative to FFT algorithms in order to improve the performances of our 3D wavelet-based softwares. Beyond some test applications to synthetic random monofractal and multifractal 3D fields, we report the preliminary results of the application of the 3D WTMM method to the dissipation field computed from direct high resolution numerical simulations (DNS) of the incompressible Navier-Stokes equations at a Reynolds number around 1000 ( ) [54]. When comparing the so-obtained multifractal spectra to those obtained with the 3D box-counting techniques, we illustrate the insufficiencies of these classical techniques to achieve a correct multifractal description because of their intrinsic inability to account for the possible non-conservativity of the underlying multiplicative cascade structure.
MONQPSR)T
2
'
The 3D wavelet transform modulus maxima method
This section is devoted to a detailed description of the 3D WTMM method. An overview of the main steps of this methodology starting from the 3D original data up to the computation of the and multifractal spectra is illustrated in Fig. 2.1.
B CL
6
2.1 3D continuous wavelet transform for multi-scale edge detection Edges in image processing are often the most important features for pattern recognition tasks. Hence, in computer vision, a large class of edge detectors look for points where the gradient of the data has a modulus which is locally maximun in its direction. As originally noticed by Mallat and collaborators [37, 38], with an appropriate choice of the analyzing wavelet, one can reformalize the Canny’s multi-scale edge detector [55] in terms of a 2D wavelet transform for image processing. This idea was further exploited to extend the WTMM method in 2D [45–48]. Here we will keep following the same strategy to generalize this methodology in 3D. The main steps consist in starting by smoothing the discrete 3D field data by convolving it with a filter and then in computing the gradient on the smoothed signal.
+ U
W +,X U X V Y[Z 0+,X U X V ]\ W +,+ X U Z X V X \
Let us consider three analyzing wavelets that are, respectively, the partial derivative with respect to , and of a 3D smoothing function :
with
V
+ Z Q+KX U
or
V
for
W
G^ R `X _
(2.1)
+a U V '
or 3 respectively .
We will assume that is a well localized (around ) isotropic function that depends on only. In this work, we will mainly use the gaussian function:
b c/b
W 0 +KX U X V d(egf :==op eqfsr tr h*o%p X
(2.2)
as well as the isotropic mexican hat:
W 0 +KX U X V duv4 c p eqf/r tr h*o%p!w 3
(2.3)
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
x +,YvX U ~ X VY zy|{,Y p ( } ), the wavelet transform , p and } can be expressed as a vecs7` x? f } g} c Y ~ f ~ c 4
x c s x? f } }gc Y p f ~ c 4
x c w s;?h x? f } }gc Y f ~ c 4
x c } (2.4)
For any function (WT) with respect to tor [] :
x? J
dX [
~
d x }gc W ff ~ c 4 4 d
g} c W ~ c 4
x x }gc2W f c
3D original data
3D Continuous Wavelet Transform
Then, after a straightforward integration by parts, one gets:
` x J
dX f } ` g
c c c
X
x 0+,X U X V W f ~ c W ?x *
dX b x J
dX b X p ~ x 1 pp ?x 1 p x? ~ o%p w }
WTMM surfaces
(2.5)
which can be rewritten as:
~ 4
d X x? *
X d f ;} f x c c } c2W s x J
dX X W? xs w
Computing the WT modulus maxima:
Detecting the local maxima of WT modulus along WTMM surfaces: WT3M points
(2.6)
The 3D continuous wavelet transform is then defined [56] as the gradient vector of once smoothed by dilated versions of the filter . The modulus of the wavelet transform is simply defined by: (2.7)
Linking WT3M across scales: WT skeleton
Multifractal analysis: Computation of the partition functions and
2.2 Continuous wavelet transform and convolution algorithms
τ (q) and D(h) spectra
Figure 2.1: A flow chart representation of the 3D WTMM method.
In the fields of image processing and computer vision, a large amount of research has focused on multi-resolution techniques [36]. The main idea of the Continuous Wavelet transform (CWT) is to convolve data with operators of increasing width. In our work, we only consider the gaussian filtering operator and the mexican hat operator so that only the gaussian function and its first, second and third partial derivatives are involved. As time computing is a limiting factor in these multi-scale computations, two kinds of algorithms were previously used in the literature, namely recursive filtering and FFT. In the present work, we follow the implementation of recursive filtering proposed by Deriche [57]. This technique is computationally efficient and requires much less computational steps than sampling the frequency domain using Fast Fourier Transform. The key point with this approach is to design in the spatial domain an exponentially based filter familly that well approximates, in a mean square sense, the Gaussian filters. The 1-D Gaussian filter given by : (2.8)
@ 0+K[(e f < h o%p h X 4
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
+ ' , by the following IIR (Infinite Impulse Response) operators: 3 +,( g¡[¢ £)¤ 0¥ ¡ + q1§ ¡7¤q¨ © 0¥ ¡ + Lª`«¬ f * ® 1! ~ ¢ £)¤ 0¥O~ + q1§i~ ¤q¨ © 0¥~ + Lª`«¬ f ® X (2.9) ¦ ¦ ¦ ¦ X ~¯X`§ ¡ X`§¯~¯X ¥ ¡ and ¥°~ are output of an once for all optimization routine where the parameters @¡ and do not need to be recomputed at each scale. The corresponding transfer function (i.e. the V transform of the positive part of this operator) is then [57]: ~ 1²I f p°1²I f I ² 1 d I ~ ~ f ~ ¡ ¡ ± V f - 1 ~~ f ~ 1 V f p p1 p V f }1 } V } f X (2.10) R V pp V }} V } g³³ V ³ I Z´Z and ZµZ are simply related to those of 0+K . This finally leads to a stable where the coefficients R X w#w·w Xi¸ ): fourth-order recursive difference equation for the input-output filtering process ( ¶ U;¹ .I ¡¡ + ¹ 1vId~~J+ ¹ f ~ 1º1^I pp + ¹ f p 1»I }} + ¹ f } 4 ~~ U;¹ f ~ 4 pp U;¹ f p 4 }} U;¹ f } 4 `³³ U;¹ f ³ w (2.11) is well approximated, for
Extension of these equations to deal with non-causal impulse response, i.e. symmetrical filters, is straightforward. This technique is well adapted to 3D filters with separated variables, since it reduces to three consecutive 1D filtering steps for each component of the 3D CWT for the Gaussian smoothing : function
¼ 0 +KX U X V d ¡ 0+, ¡ U ¡ V ½4 ¿¾
k~ 0+K ¡ ¡ 0+K ~i U ¡ 0+K ¡ U U
¡ V ¡ V w (2.12) ~¯ V +,X U X V d(uO4 b c/b )p ¼ 0 +KX U X V : However, it is slightly more complex for the 3D Mexican Hat À +, ¡ ¡ Â1 ~ 0+K ¡ Ã1 ~ +, ¡ ½4 } +, ~i U ¡ V Â1 ¡ 0+K p U ¡ V Ã1 ¡ +, ~¯ U p V X ²Á (2.13) p +, ¡ U ~¯ V Â1 0+K } U ~i V Ã1 ¡ +, ¡ U p V p p p U V U V U } V @Ä 0+K ¡ +, where is the n-th derivative of the 1D gaussian function . For the 3D Gaussian smoothing function, recursive filters turns out to be much more time computing saving than FFT without significative loss of quality. However they can not be used for implementing the Mexican Hat in the 3D CWT without acceptable loss of quality. For this particular case we will use the FFT algorithm.
2.3 Characterizing local regularity properties with the 3D continuous wavelet transform Fractal functions generally display multi-affine properties in the sense that their roughness (or regularity) fluctuates from point to point. Then, the Hurst exponent [1], which quantifies the global regularity of a function , has to be replaced by a local quantity, namely the H¨older exponent [26, 28–30]. The purpose of this section is to illustrate the ability of the 3D continuous wavelet transform to characterize the pointwise H¨older regularity of a scalar function from to . A rigorous definition of the Ho¨ lder exponent (as the strength of a singularity of the function
x
x
5
}
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
c ¡ Ç' Æ c c¡ Ä x c Â4¿È c 4 c ¡ É Ê Æ b c 4 c ¡ b 8;: t > w I IfÄ x is times continuous differentiable at the point c ¡ , then one can use for polynomial È c 4 c ¡ , the order-n Taylor series of x at c ¡ and thus prove that c ¡ the I . Thus c ¡ measures how irregular the function x is at the point c ¡ . The higher the exponent c ¡ , the more regular the function x . In this work, we will mainly consider fractal functions of three variables which possess only cusp-like singularities. This exponent corresponds to the intuitive idea of x
¡ IÅ] c c ¡
at the point ) is given by the largest exponent such that there exists a polynomial of degree and a constant , so that for any point in the neighborhood of one has [29, 30, 37, 38, 47]: (2.14)
smoothness [58, 59], i.e., we loose one degree when differentiating and win one degree when integrating a function (oscillating singularities are left apart). This local singularity exponent can be estimated via a wavelet-based scaling relation, provided the analyzing wavelet be appropriately (Eq. (2.1)) are of order : chosen. Indeed, if the three analyzing wavelets
I Ë Z X-GO R Xi_)Xiu Ì Ì Y[Z +,X X d ' X,ÎÏ~/XsÏ ÂX Ï y ' X I 4 X Y Z + + Ì p } ÀÍÌ Ì h Ì U V R¯ (2.15) U V U hV then one can show that for a function x from } to with an isolated isotropic singularity of y [ I X Ð I 1 ¡ ¡ H¨older exponent c R located at the point c , its wavelet transform behaves as : Ñ if I c ¡ Ò /ÔÓk x? c ¡ X 8)Ä?:Õ t > X ½ '' j XGO R X`_@XiuDX (2.16) X Ñ if I ( c ¡ Ò /ÔÓk x? c ¡ X ½ j XGO R X`_@Xiu w (2.17) c¡
Thus, provided the number of vanishing moments of the analyzing wavelet be sufficiently high, the local regularity of the function at the point is characterized by the scaling behavior of the wavelet transform modulus:
x
~ ?x c ¡ X [ } s ÔÓm x c ¡ X p o%p ×Z Ö ~
8;: t > X ½ ' j w
(2.18)
c ¡ c ¡ Ø1!Ù ), a scaling behavior Ä?Ú ~ x? c ¡ X d } sÔÓm x c ¡ X p o%p X ½ 'jw (2.19) ×Z Ö ~ Following the extension from 1D to 2D analysis [47], to recover the H¨older exponent c ¡ of a function x from } to , one needs to study the behavior of the wavelet transform modulus 4 c ¡ b Æ in the (space-scale) half space [47, 59]. According to Mallat and inside a cone b c x
( Let us remark that if is infinitely differentiable at point still exists but it is governed by the shape of the analyzing wavelet :
collaborators [37, 38], a very efficient way to perform point-wise regularity analysis is to use the wavelet transform modulus maxima (WTMM). In the spirit of Canny edge detection [55], at a given scale , the WTMM are defined as the points where the wavelet transform modulus (Eq. (2.7)) is locally maximum along the gradient direction given by the direction of the wavelet transform vector (Eq. (2.6)). As illustrated in the example just below, these WTMM lie on connected surfaces hereafter called maxima surfaces. In theory, one only needs to record the position of the local maxima of along the maxima surfaces together with the value of and the WT vector direction at the corresponding locations. At each scale , our wavelet analysis thus reduces to store those WTMM maxima (WTMMM or WT3M) only. They indicate
?x *
dX ?x
6
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX locally the direction where the signal has the sharpest variation. These WTMMM are disposed along connected curves across scales called maxima lines [46, 47] living in a fourth-dimensional space . We will define the WT skeleton as the set of maxima lines that converge to the -hyperplane in the limit . This WT skeleton is likely to contain all the information concerning the local H¨older regularity properties of the function under consideration [47].
+,X X X +,X U X V U V
½ 'j
x
2.4 Simple example of an isotropic singularity interacting with a localized smooth structure
x c y } :
Let us illustrate the above definitions on the simple example of the function
¡qÝ x c [Ûe f : tft > h o%p h 1¿Ü b c 4 c ¡ b } w (2.20) This function is D Æ Þ ¡ ßeverywhere ' w u except at c c ¡ where x is isotropically singular with a H¨older exponent c . Some isosurfaces (colored opened half-shells) of the 3D graph of this function are shown in Figure 2.2(a). Figure 2.2(b) is a three-dimensional representation of x 0+KX U X V à+K ; note that the center of the+asingularity c ¡ and the center of the gaussian part c ~ are located in the plane of cartesian equation V.
(a)
(b)
á
Figure 2.2: (a) Isosurfaces of the 3D graph of the function defined in Eq. (2.20); the isosurfaces are cut by the plane of cartesian equation ; the values of are rescaled between and ; the blue half-shell (respectively green, yellow and orange) corresponds to the isosurface (respectively , , ). The singularity S is located at , and the gaussian G at in the grid. The paremeter values are and . (b) Three-dimensionnal representation of the function (Eq. (2.20)) along the plane . In (a) the straight line passing through and was added to guide the eye.
â6ãzä
ëìgê ë%ígî égègï æqòãQåóëêgô)ñ¯ëêgô)ñ¯ëê`ôç
égêgìõ æð æò
á
7
Ôá åæqç è égê`ê á3ã(ëêgè æ;ð!ã¿å*ëègé;ñkëègé;ñkëègéç ö.ãQëi÷ è ù ø ãûúKè;÷ égê âaã¿ä
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
á
Figure 2.3: Wavelet transform (Eq. (2.4)) of the function shown in Figure 2.2, with a first-order analyzing wavelet ( is the isotropic Gaussian function) using the recursive filter technique. Each scene is made of streamlines of and two isosurfaces of wavelet transform modulus maxima (Eq. (2.7)). The colors along the streamlines are given by the value (from blue ( ) to red ( )). The scale parameter is (a), (b) and (c) in units where (pixels) is the characteritic size of at the smallest resolved scale.
ü
égêgè Øþÿ á ×å sñ )ç
ã
c ~ W ¡ _ ¦ Xi_ ¦ (
ýþÿ=á å sñ )ç mâ þ
þ
þÿ á ×å sñ )ç éò é
ãÐé ð
x
The 3D wavelet transform (Eq. (2.4)) of with a first-order analyzing wavelet (the smoothing function is the isotropic Gaussian function) is shown in Figure 2.3 for three given scales and , where is the width (in pixel units) of the analyzing wavelet
_`p ¦
¦
?x *
X
at the smallest scale (or the highest resolution accessible to our WT microscope). For each scale, are displayed along with the streamlines of the veccharacteristic isosurfaces of tor field . In Figure 2.3(a), at small scale, one can clearly distinguish a beam of streamlines (mainly blue-green) converging towards the center of the Gaussian G, and another one (mainly yellow-red) pointing towards the location of the singularity S. As colors of streamlines are mapped onto local values, from blue ( ) to red ( ), Figure 2.3(a) illustrates that the data under study have sharper variations around the singularity S than around G. When looking at larger scales (Figures 2.3(b) and (c)), one can no longer distinguish two streamlines beams because the more the width of the smoothing function increases, the more strealines seem to converge towards some point located in between S and G. In addition, let us notice that characteristic isosurfaces of can be disconnected at small scale (Figure 2.3(a)) and have increasing mean radius along with a more spherical shape when looking at larger of our WT microscope, one is scales (Figure 2.3(c)). When decreasing the magnification no longer able to distinguish the singularity S and the Gaussian G.
x J
dX
?x J
dX
EHG*I
?x J
dX
_¡¦
_¯p Ý ¦
E +
R
In Figure 2.4 are shown WTMM surfaces for various values of the scale parameter ranging from (Fig. 2.4(a)) to (Fig. 2.4(f)). At small scale, there exist mainly two maxima surfaces. One is a closed surface surrounding where the singularity S is located. The other one is an open surface which partially surrounds G. On each of these maxima surfaces, one finds only one WTMMM, whose corresponding gradient vector is represented by a black segment whose length is proportional to and direction is along the local gradient vector.
c¡
?x *
dX
8
Í x J
dX
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
á
Figure 2.4: Maxima surfaces defined by the WTMM of the function represented in Figure 2.2 (Eq. values from blue (2.20)). The colors along the WTMM surfaces are mapped onto local ( ) to red ( ) . The local maxima of (WTMMM) along these surfaces are shown with a black segment whose length is proportional to . The scale parameters are (a), (b), (c), ! (d), " (e) and (f) in units. Same first-order analyzing wavelet as in Figure 2.3.
Øþ 2ã²é ð
mâ þ éð éò
éò
þ[ÿ=á ×å sñ )ç
é
é
9
þ =ÿ á ×å sñ )ç þÿ=á ×å sñ )ç
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
þ[ÿ=á
å @ç
Figure 2.5: Evolution of when following, from large scale to small scale, the maxima lines #'$ & and & pointing respectively to the singularity S and to the localized smooth structure G. Same first-order analyzing wavelet as in Fig. 2.3.
#%$
å @ç
When increasing the scale parameter , the maxima surfaces evolve; in particular the closed maxima surface around S swells (its characteristic size behaves like ) until it connects with the maxima surface associated with G (Fig. 2.4(e)) to form a single closed surface surrounding both S and G (Fig. 2.4(f)). As previously indicated in the flow chart (Fig. 2.1), one then proceeds to a step by step, as continuously as possible, linking procedure of the WTMMM points from small to large scales. The set of these maxima lines is called the WT skeleton of the function . One of these maxima lines points to the singularity S, in the limit . As shown in Figure 2.5, along this maxima line (( ), the wavelet transform modulus behaves as [37, 38]
x
½ 'j
t
8): t > X ½ ' j X
x ( t
' wu [ ¡ where c
c¡
(2.21)
is the H¨older exponent of S. Moreover, along this maxima line, gives the direction of the largest variation of around .
x
Ð x ( t
c¡
From the maxima line, one thus gets the required amplitude as well as directional informations to characterize the local H¨older regularity of at . Note that along the other maxima line which points to where is located the smooth localized structure G, the wavelet transform ( modulus behaves as (Fig. 2.5):
t
where
x
c~
I R
Ä;Ú
x ( t %
X ½ 'j X
(2.22)
is the order of the analyzing wavelet.
Through this example we have seen that the maxima lines defined from the linking of the WTMMM across scales are likely to contain all the information about the point-wise H¨older regularity of any function from to . From the power-law behavior of versus along a maxima line pointing to a singularity, one can estimate its H¨older exponent . Let us note that very much like what has been done in 2D in Ref. [47], one can use this 3D WTMM methodology to detect and characterize anisotropic as well as self-affine singularities in 3D.
x c ¡
}
10
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
2.5 The wavelet transform modulus maxima methodology for 3D multifractal analysis
6
As recalled in the introduction, multifractal analysis aimed at computing the singularity spectrum of a fractal function , here from to :
x
}
6 *) c y } °X c ( w 6 associates with any , the Hausdorff dimension of 8 , the set of all points c ¡ H¨older exponent of x at c ¡ is h.
(2.23) so that the
Mapping the methodology developed in Refs [28–30, 46–48], for multifractal analysis of 1D and 2D irregular landscapes, we use the space-scale partioning given by the wavelet transform skeleton (set of WTMMM lines which point to the singularities of the function under study) to define the following partition functions:
2CLX d +
Cºy
,.-/,
?x c X 6 1¤ 0 ¬ * , : > t : > /2 32545
X
(2.24)
Y
where and ( is the set of maxima lines of the WT skeleton. As compared to classical box-counting techniques [1, 6, 11, 12, 23–25], the analyzing wavelet plays the role of a generalized “oscillating box”, the scale defines its size, while the WTMMM skeleton indicates how to position these oscillating boxes. From a deep analogy that links the multifractal formalism to statistical thermodynamics [16–18, 20, 30], one can define the exponent from the power-law behavior of the partition function:
B C
2C X +
C
B CL
"7
: > X ½ ' j X
(2.25)
where and play respectively the role of the inverse temperature and the free energy. The main result of the wavelet-based multifractal formalism is that in place of the energy and the entropy (i.e. the variables conjugated to and ), one has the H¨older exponent (Eq. (2.14)) and the singularity spectrum (Eq. (2.23)). This means that the singularity spectrum of can be determined from the Legendre transform of the partition function scaling exponents :
C
6
B
6
698 ¨ © CL24 B CL w
x B CL
(2.26)
Í \ B \ C
From the properties of the Legendre transform, it is easy to convince oneself that homogeneous (monofractal) fractal functions that involve singularities of unique H¨older exponent , : are characterized by a spectrum which is a linear function of . On the contrary, a nonlinear curve is the signature of nonhomogeneous functions that display multifractal properties, in is a fluctuating quantity that depends upon the spatial the sense that the H¨older exponent position (in other words, the local roughness exponent is fluctuating from point to point).
B CL
B CL
c
C
c
6
From a practical point of view, one can avoid performing a Legendre transform by considering the quantities and as mean quantities defined in a canonical ensemble, i.e. with respect to their Boltzmann weights computed from the WTMMM [28–30, 47] : ;
¬ X ?x CLX ( X 1¤ 0 : t 2 > -*, +22 4 CLX x c 6 X 11
(2.27)
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Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
2CLX
+ where is the partition function defined in equation (2.24). Then one computes the expectation values:
C X [
6C X [
and
from which one extracts
and therefore the
3
,.-*,
6
: >
2 45 ¤1-/0 , ¬
©
,=-/,
: >
;
x C X ( X < © ;
;
x? CLX ( X [X
x C X ( X X
C [ < ¨ 8 ¡ ? CLX < ©D X 6CL "< > ¨ 8 ¡ ? 6 CLX < º© X " >
(2.28)
(2.29)
(2.30) (2.31)
singularity spectrum.
Test-applications of the 3D WTMM method to monofractal and multifractal synthetic 3D data
3.1 Fractionnal brownian fields Since its introduction by Mandelbrot and Van Ness [60], the fractional Brownian motion (fBm) has become a very popular model in signal and image processing [1–11, 14]. FBm’s are homogenous random self-affine functions that have been specifically used to calibrate both the 1D [28–30] and the 2D WTMM methodology [46, 47]. This section is devoted to a test application of the 3D WTMM methodology described in section 2, on several realizations of 3D fBms.
Ü c c 0 +KX U X V y
The generalization of Brownian motion to more than one dimension was first considered by Levy [61]. A three-parameter fractional Brownian motion (3D fBm) ) , , indexed by , is a stochastic process with stationary zero-mean Gaussian increments and whose covariance function is given by:
' y XR
}
where
w`wiw
Ü ) &@-Ü ) BA- ¦ _ p b @ b p ) 1 b A b p ) 4 b @Ð4CA b p ) X
(3.1)
represents the ensemble mean value. The variance of such a process is
Ü ) c ¦ p b c7b p ) X (3.2) Ü ~ o%p c ° ¦ p b c7b for uncorrelated Brownian from which one recovers the classical behavior var _ . Indeed, the demonstration that the increments of a fBm, i.e., Ü ) D c ^ motion with Ü ) c 1CE Â4¿Ü ) c R with E,+,X` Xi U V , are stationary is rather straightforward. var
The 3D fractional Brownian motion is a self-affine process that is statistically invariant under isotropic dilations:
Ü ) c ¡ 1GFIHÃ4¿Ü ) c ¡ KJLF ) Ü ) c ¡ 1CHdÂ4¿Ü ) c ¡ X (3.3) 12 Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
H J where is a unitary vector and stands for the equality in law. The 3D fBm exhibits a power 'M spectral density , where the spectral exponent N is related to the Hurst exponent . Almost all realizations of the fBm process are continuous, everywhere nondifferentiable, isotropically scale-invariant as characterized by a unique H¨older exponent , [1, 14]. Thus fBm fields are the representation of homogeneous stochastic fractal functions characterized by a singularity spectrum which reduces to a single point
Îc
0¥°
R ¥
_ 1 u
c °
6 ( u ß X if (3.4) (4!Ù if P O w 6 according to equation (2.26), one gets the following By inverse Legendre transforming
expression for the partition function exponent (Eq. (2.25)):
B CL(C 4¿u w
(3.5)
B CL is a linear function of C , the signature of monofractal scaling, with a slope given by the index of the fBm.
Let us test the 3D WTMM method on fBm fields generated by the so-called Fourier transform filtering method [14]. It simply amounts to a fractional integration of a 3D “white noise” and therefore it is expected to reproduce quite faithfully the expected isotropic scaling invariance properties (Eqs (3.3)–(3.5)). First we have wavelet transformed Q ( Q ) data fields of ) with an isotropic first-order analyzing wavelet. To master edge effects, we mainly restrain our analysis to the Q central part of the wavelet transform of each volume.
R _ T q}
' R }
Ü Ö ~ o p
In Figure 3.1 is illustrated the computation of the maxima surfaces and of the WTMMM for an individual volume at two different scales. A mid-range value iso-surface representation of the original data is shown in Figure 3.1(a). Figure 3.1(b) displays a grey-scale coding of ) in R . Streamlines of the vector field are represented in Figthe plane of equation and respectively. According ure 3.1(c) and 3.1(d) for the scale parameter values to the definition of the wavelet transform modulus maxima, the maxima surfaces correspond to and well defined edges of the smoothed image (Figs. 3.1(e) and 3.1(f) for scales resp.). The local maxima of along these surfaces are located at the points where the sharpest intensity variation is observed. The corresponding black segment clearly indicate that locally, the gradient vector points in the direction of maximum change of the intensity field data. corroborating the isotropy of fBm Statistically these gradient vectors cover all the directions in fields. When going from large scale (Fig. 3.1(f)) to small scale (Fig. 3.1(e)), the characteristic average distance between two nearest neighbour WTMMM decreases like . This means that the number of WTMMM and in turns, the number of maxima lines, proliferates across scales like . As maxima lines live in the 4D (x,y,z,scale)-space, a projected version of the corresponding WT skeleton (projected onto the 3D (x,y,scale)-space) is shown in Figure 3.2. As confirmed just below, when extrapolating the arborescent structure of this skeleton to the limit , one recovers the theoretical result that the support of the singularities of a 3D fBm has a dimension , i.e. ) is nowhere differentiable [1, 14, 61].
V R_
Ü Ö ~ %o p
H x ~ J
dX ¡ (_ ¦ _ ¦
_ ~¦
_¯p ¦
}
f}
(u
½ 'j
Ü Ö ~ o p c
13
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
ø
ò åæqç
Figure 3.1: 3D wavelet transform analysis of TS:U V . W is a first-order radially symmetric analyzing function. (a) Mid-range value isosurface plot of the original field. (b) grey-scale coding " and (d) of the central YX portion of the original image in the plane . In (c) " are shown the streamlines of the wavelet transform field . In (e) " and " are shown the wavelet transform modulus maxima surfaces; from the local maxima (f) (WTMMM) of along these surfaces originates a black segment whose length is proportional to and direction is along the gradient vector. The colors along the WTMM surfaces are mapped onto local values with the same coding as in Figures 2.3 and 2.4.
ãÐé ò ãé þ
ë%ìgè ëìgè
ôgé ä°ãûëégí ãÐé ð ò ý þ å sñ )ç ãHé
þ
þ[ÿ=á å sñ )ç
14
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
(a)
(b)
ø
ò
Figure 3.2: (a) Wavelet transform skeleton of the 3D Brownian field ZS:U [V shown in Figure 3.1(a), projected onto the 3D space (x,y,scale). This skeleton is defined by the set of maxima lines obtained after linking the WTMMM detected at different scales. (b) Characterizing the local H¨older regularity of %S:U V from the behavior of the WTMMM along the maxima lines. \^] _ vs \^]*_ along three arbitrarly chosen maxima lines. Same analyzing wavelet as in Figure 3.1.
ø
þ
ò
c ¡
Ü Ö ~ %o p
The local scale invariance properties of a 3D ) are investigated in Figure 3.2. When looking at the behavior of along some maxima lines belonging to the WT skeleton (Fig. 3.2(a)), despite some superimposed fluctuations, one observes a rather convincing power-law decrease with an exponent which does not seem to depend upon the spatial location . Moreover, the theoretical value for the H¨older exponent provides a rather good fit of the vs (Eqs. (2.18) and (2.21)). slopes obtained at small scale in a logarithmic representation of
c¡ c ¡ R _ CL and 6 spectra using the In Figure 3.3 are reported the results of the computation of the B +ºCLX 3D WTMM method. As shown in Figure 3.3(a), the annealed average partition functions Ü ~ (over 16 3D-realizations of op c ) display a remarkable scaling behavior over 3 octaves when plotted versus in a logarithmic representation (Eqs. (2.24) and (2.25)). Moreover, for a wide Cy 4¿_@X`u , the data are in good agreement with the theoretical B CL spectrum range of values of (Eq. (3.5)). When proceeding to a linear regression fit of the data over the first three octaves, C spectrum shown in Figure 3.3(c) with a slope which slightly underestimates the one gets the B R _ Hurst exponent. Let us point out that a few percent underestimates has corresponding been also reported when performing similar analysis of 1D [28–30, 62] and 2D [47] fBm. As C X vs < £ ` p (Eqs. (2.28) and (2.30)), the theoretical seen in Figure 3.3(b), when plotting R _ provides an excellent fit of the limiting slope of the data at the smallest Hurst exponent ¦ cb bed¦ ) and this independently of the value of Cy 4¿_@X`u . available scales ( a
15
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
/å gç
2å 5ç
ø
ò
Figure 3.3: Determination of the f hg and i &j spectra of 3D kS:U V with the 3D WTMM &g n vs \^] _ ; the solid lines correspond to linear regression fits. (b) j &g method. (a) \l] _ am 3p . vs \^]*_ & (Eqs. (2.28) and (2.30)); the solid lines correspond to the theoretical slope o (c) f hg vs g ; the solid line corresponds to the theoretical linear spectrum (Eq. (3.5)). (d) i &j vs j as obtained from the scaling behavior of j hg vs \l] _ (Eqs. (2.28) and (2.30)) and i hg vs \^]*_ (Eq. (2.29) and (2.31)). These results correspond to annealed averaging over ( ) qS:U V fields. is expressed in units.
/å gç
å )ç
ø
6CLX
ò
å iñ @ç
5å iñ )ç ã ë é 2å 5ç 2å `ñ )ç ëì é`êgìOõ
å `ñ )ç
£ p
6 R _@! u w '@' ù' w ' _ B CL 6
C
In Figure 3.3(d) are reported the corresponding estimates of from a linear regression fit of < vs ` (Eqs. (2.29) and (2.31)) again at small scale. Independently of the value of , sr one gets quantitatively comparable values . Let us emphasize that similar quantitative estimates of the and spectra have been obtained for 3D fBm’s and . The 3D WTMM method can thus be considered as having ) for successfully passed the test of homogenous monofractal fBm stochastic fields.
Ü
R u
_ u
3.2 3D multifractal p-model and fractionally integrated singular cascades Generating multifractal measures using multiplicative cascades is a well documented subject still very active [19–22]. The so-called p-model (or “binomial model”), originally proposed in its 1D version by Meneveau and Sreenivasan [63], is a multiplicative cascade process designed to model the statistical scaling properties of the dissipation field in fully developped turbulence. Its multifractal properties are very simple and analytically known. As originally proposed by Schertzer and Lovejoy [64, 65], a way to synthetize multifractal functions consists in fractionally integrating a multifractal singular measure. This can be done, for example, by using the Fractionally Integrated Singular Cascade (FISC) algorithm [64, 65] designed for multifractal geophysical field modeling and which amounts to a simple Fourier space filtering over realizations of the p-model. 16
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX Following what was done in 2D by Decoster et al [48], we apply the 3D WTMM method to synthetic random fields generated by the the p-model and the FISC algorithm to show to which extent one retrieves the multifractal properties from their wavelet transform decomposition. The 3D version of the p-model (or singular cascade) consists in starting with some 3D spatial domain, let say a cube of characteristic size , in which a measure t tvu is uniformly distributed. At the first step of the construction, the initial cube breaks into eight smaller cubes of size , each receiving a fraction of the original measure as defined by a random variable with a certain probability distribution . By repeating the same procedure recursively at smaller scales using independent realizations of the random variable , one generates a random singular measure xw yw over the cubic domain:
{
{ ²{ ²{
À
È! À t
Ä cz
{ _
À
Ä
t=u
Ä ' Z f X ¿ { ( _ ½ w ×Z Ö ~ À
(3.6)
~ À : >ß &
À : pq> CL R 4 & B w < £ ` L C ( 4 O C ¿ 1 ) _ Â 4 à 1 4 p (3.7) B | & R & _ Z X G^and the } eight Thus generating a T Qg} realizations of the p-model requires only ' w#w R resulting meaZ ~ f steps f sure is poorly quantized since its can only takes nine values ³ . Nevertheless ³
At each step, one selects at random among the 8 sub-cubes, the four which will receive a fraction { d , the four others receiving the fraction d of the measure at the previous step. A straightforward computation [63] of the spectrum yields:
we demonstrate that our multifractal analysis can actually be applied provided one uses the third order analyzing wavelet which turns out to be much more adapted to characterize the fluctuations of such poorly quantized measures [48].
Ä c Ä Ä ~ ' x c [ c b c7b f : f > X R w B CL spectrum of these multifractal random funcThis leads to the following expression for the
As a mean of introducing continuity, one further performs (FISC algorithm) a fractional integration of the measure t which simply reduces to a low-pass power-law filtering in Fourier space: )Y~ t (3.8) tions [45, 48, 66]:
CLÂ1àC X B C [( B | (3.9) 4_v4àCL R 4 Â4 < £` p & 1¿ R 4 & w C ' , one finds B ' 6 4! 4!u , i.e., those fractionally integrated Let us note that for C R , one gets BI R » 4ùu , which gives for the fractal fields are singular everywhere. For
dimension of the graph of this function: 8
¯« u@X R 8
¯«/u@X d d 4 X u ( i.e., a fractal dimension in between
4 *B R X 4 X R ) and d (
(3.10)
'
6
). The singularity spectrum of this function is just a shifted version along the h-axis of the one of the corresponding singular measure generated with the p-model:
d( | 24 w 17
(3.11)
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
égêgìÔõ
Figure 3.4: 3D wavelet transform analysis of a realization of the p-model with parameter value . W is the first-order radially symmetric analyzing wavelet. (a) Mid-range value isosurface plot of original field data. (b) grey-scale coding of a cut along the plane in the original data. In (c) and (d) are shown the streamlines of the wavelet transform field . In (e) and (f) are shown the wavelet transform modulus maxima surfaces; from the local maxima (WTMMM) of along these surfaces originate a black segment whose length is proportional to and direction is along the gradient vector. The colors along the values with the same coding as in Figure 2.3 WTMM surfaces are mapped onto local and 2.4.
ã è;÷ ôgé
ºã²é ð ò ý^þdå sñ )ç ºãzé
ôgé
2ãaé ò ºãzé þ
äã¿ë%égí
þ
þÿ á ×å sñ )ç 18
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
7å gç 2å 5ç
ã è;÷ ìgô`í 5å iñ )ç
ã è;÷ ô`é
) and its Figure 3.5: Determination of the f &g and i &j spectra of the 3D p-model ( ) with the 3D WTMM method (third-order analyzing fractionally integrated version ( oP wavelet). (a) \^]*_ &g vs \^] _ ; (b) j &g vs \^] _ (Eqs. (2.28) and (2.30)); the solid lines m correspond to linear regression fit for the p-model ( ) and the FISC ( ) over the first two and a half octaves. (c) f &g vs g ; the solid lines correspond to the theoretical predictions (Eqs. (3.7) and (3.9)). (d) i hj vs j as obtained from the scaling behavior of i hg vs \l] _ (Eqs. (2.29) and (2.31)). ) images. is expressed in units. These results correspond to annealed averaging over (
¯õ å iñ )ç
2å ç
/å gç
2å `ñ )ç ëì gé êgì/õ
_ T @}
' & w u)_
In Figure 3.4 is illustrated the computation of the maxima surfaces and the WTMMM for a Q . The analyzing wavelet realization of the p-model with the following parameter values: is the third-order radially symmetric wavelet generated from the Mexican hat smoothing function (Eq. (2.3)). In Figure 3.4(a) is shown the mid-range isosurface plot of the 3D data and in Figure R . Figures 3.4(c) and 3.4(b) a grey-scale coding of a cut along the plane of equation for scale parameter and 3.4(d) are streamlines plots of the vector field t in units. From the WTMMM (black segments) defined on the maxima surfaces (Figures 3.4(e) and 3.4(f)), one constructs the WT skeleton and then computes annealed average of the + . In Figure 3.5(a) are shown the partition functions corresponding to partition functions + vs in a logarithmic 16 ( Q ) realizations of the p-model. Actually we have plotted . representation in order to extract from linear regression fits, the scaling exponents | Hence, for conservative singular cascades ( ), Equation (3.7) shows that the so-called cancellation exponent [67, 68] must vanish :
H s *
X
¦
_T }
ºCLX
V R_
} 2C X
_¡
_~
B C )1 C/1 _)
& ~K1 & p R ' (3.12) B | C R Ã1¿u B| = C R X CL spectrum computed with the 3D WTMM method can be rewhere the definition of the B | C lated to the B | = spectrum computed by “standard” box-counting algorithms via the simple relationship [50] : (3.13) B | CL B | : CLÃ4¿u@C(C^4 R ^4àu)CX 19 Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
C
4!_@X
where the ’s are the so-called generalized fractal dimensions [23–25]. Notice the good scaling d for which behavior observed over the first three octaves and for values of in the interval statistical convergence turns out to be achieved. The corresponding spectrum is displayed in Figure 3.5(c) along with the theoretical prediction (Eq. (3.7)). The agreement is quite remarkable. The so-obtained | spectrum unambiguously deviates from a linear spectrum, the hallmark of multifractal scaling. Note that the data for | provide experimental evidence for the vanishing of the cancellation exponent (Eq. (3.12)), as expected for the conservative p-model. As shown in Figure 3.5(d), the computation of the | spectrum via the scaling behavior of the partition function (Eqs. (2.28) and (2.30)) and (Eqs. (2.29) and (2.31)) yields similar good quantitative agreement with the theoretical spectrum. Let us point out that this | singularity spectrum is directly related to the commonly used | spectrum [16–18, 20, 22, 29] obtained by Legendre transforming the scaling exponent | : computed by box-counting techniques : (3.14) /| |
B C
B C¿ R
C X
B CL
6C X
x CL Ï, B
x Ï, 6(Ï4¿u)[X 6 spectrum on the right by (u , one recovers the x | Ï which means that by translating the
spectrum.
_ T `}
In Figure 3.5 are also reported the results of the 3D WTMM method for the FISC data set with the same statistical sample of 16 ( Q ) realizations as before. Again the numerical results for + , and are found in quite good agreement with the theoretical predictions (Eqs. (3.9) and (3.11) respectively). The singularity curve is found identical to the | curve up to a translation to the right by (Eq. (3.11)). This is the numerical demonstration that our 3D WTMM methodology paves the way from multifractal analysis of singular measures to continuous multi-affine functions.
} 2CLX B C
4
3D dissipation field from Direct Numerical Simulations of the incompressible Navier-Stokes equations
This section is devoted to the application of the 3D WTMM method to the dissipation field from DNS of isotropic turbulence carried out by Meneguzzi [69] with the same numerical code as previously developed in Ref. [54] but at a higher resolution. The DNS were performed using mesh points in a 3D periodic box and a viscosity of . A statistically steady state was obtained by forcing low Fourier modes in a deterministic way. The Taylor microscale Reynolds number is Q . Here we will examine only one snapshot of the dissipation 3D spatial field. We will mainly proceed to a comparative multifractal analysis when using classical 3D boxcounting techniques and the 3D WTMM method. The corresponding and spectra will result from an annealed averaging over the 8 ( Q ) sub-domains in a cube.
T)R _ }
' TR f ³
MN _ R
B C _ T@R }
_T }
Since Kolmogorov’s founding work in 1941 (K41) [70], fully developed turbulence has been intensively studied theoretically, numerically and experimentally [8, 27] mainly through the statistical analysis of the fluctuations of the velocity increments over a distance : 3 e (4.1) where is an arbitrary unit vector. For instance, investigating the scaling properties of the longitudinal structure functions: & 3 (4.2)
- c Xi O [ c 1¿ Â4 [ c X
Ô !( w - c Xi ! "%$ X 20
&
('
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
w#w·w
A
where stands for ensemble average, leads to a spectrum of scaling exponents which has been widely used as a statistical characterization of turbulent fields [8,27,71,72]. Based upon assumptions of statistical homogeneity, isotropy and constant mean energy dissipation per unit mass , K41 asymptotic theory predicts the existence of an inertial range C for which the structure functions behave as: (4.3)
Ô
{
o o X } }
{
where is the Kolmogorov dissipative scale and the so-called integral scale. Although these assumptions are usually considered to be correct, there has been increasing numerical [8, 54, 73] and experimental [8, 27, 71, 72] evidence that deviates substantially from the K41 prediction , at large . To take into account the local intermittent fluctuations of the energy dissipation rate, one often considers the so-called Kolmogorov’s refined similarity hypothesis (RSH) [74, 75], which amounts to rewrite the velocity structure functions as :
~ A }&
A
&
* c o } o } X 7n : o } > j o } X
Ô
× c
}
(4.4)
where is the spatial average of the energy dissipation over a ball of radius centered at the point and of volume : 1 6 6 (4.5) Let us recall that the dissipation rate is defined as ( ):
c
c } c w GmX R X`_@Xiu _ \ Z 1 \ Z p)w (4.6) Z K are likely to be related to those of According to Eqs. (4.3) and (4.4), the scaling exponents of J c : A B &{ u@1 &{ u w (4.7) * c [ R
In this section, we use the 3D WTMM method to analyze the multifractal properties of the DNS dissipation field with the specific goal to revisit the results obtained with classical box-counting techniques [22]. In Figure 4.1 are illustrated the main steps of our 3D wavelet transform computations. Figures 4.1(a) and 4.1(b) depict the original data via some isosurface plot (Fig. 4.1(a)) and a grey-level plot of a typical 2D cut in the 3D volume (Fig. 4.1(b)). In Figures 4.1(c) and 4.1(d) streamlines at two different scales. Figures 4.1(e) and are illustrated the vector field 4.1(f) show the WTMM surfaces along with the WTMMM points.
x? J
dX
Figure 4.2 shows the results of a comparative multifractal analysis of this 3D dissipation field using our 3D WTMM methodology and the classical box-counting techniques. The results correspond to the computation of an annealed average of the partition functions over the 8 ( Q ) sub-domains. In Figure 4.2(a) and 4.2(b), rather good scaling properties are observed over the b b range of scales and for values of between and d for both the partition func+ tions (Eqs. (2.24) and (2.25)) and (Eqs. (2.28) and (2.30)). When proceeding to linear regression fit of the data into this range of scales in Figure 4.2(b), one gets that the slope unambiguously depends on , the hallmark of multifractality. This is confirmed in Figure 4.2(c) where the corresponding spectra data fall on nonlinear curves. But surprisingly, the spectrum obtained with our 3D WTMM methodology ( ) significantly differs from the one estimated with box-counting techniques ( ).
CL B C
} 2CLX
_ ¡LÝ
_}
C CLX
_ T 5}
4_
C
B C
Ñ
21
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX
Figure 4.1: 3D Wavelet transform analysis of the dissipation field from DNS [69]. W is the firstorder radially symmetric analyzing wavelet. (a) Characteristic isosurface plot of the original dissipation field. (b) grey-scale coding of a cut along the plane in original data. In (c) and (d) are shown streamlines of wavelet transform field . In (e) and (f) , are shown the wavelet transform modulus maxima surfaces; from the local maxima (WTMMM) of along these surfaces originate a black segment whose length is proportional to and direction is along the gradient vector. The colors along the WTMM n values with the same coding as in Figs. 2.3 anf 2.4. surfaces are mapped onto local
6ã.é ð ò Êã é
ôgé ò ãàé Êã¿é þ
äÊã ëégí
ý þå sñ )ç
þ þ[ÿ=á ×å /ñ @ç
22
Copyright c 2003 by PSFVIP-4
Proceedings of PSFVIP-4 June 3-5, 2003, Chamonix, France. FXXXX Actually, most of the difference comes from the fact that our 3D WTMM algorithm reveals that the cancellation exponent (Eq. (3.12)) is definitely different from zero : r (4.8)
' ¿' ' ' B C R Â1¿u!(4 w _ R w 2u w
C B 4!_É CºÉ
Note that this is some evidence for the possible existence of an underlying non-conservative multiplicative structure. Indeed, as illustrated in Figure 4.2(c), the spectrum issued from our 3D d , by the theoretical spectrum of a WTMM method is rather nicely fitted, at least for non-conservative p-model :
~ 1 [X
