multifractal image analysis

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    Eur. Phys. J. B 15, 739764 (2000) THEEUROPEANPHYSICALJOURNAL B

    cEDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2000

    A wavelet-based method for multifractal image analysis.II. Applications to synthetic multifractal rough surfaces

    N. Decoster1, S.G. Roux1,2, and A. Arneodo1,a

    1 Centre de Recherche Paul Pascal, Avenue Schweitzer, 33600 Pessac, France2 Climate & Radiation Branch, NASAs Goddard Space Flight Center, Greenbelt, Maryland 20771, USA

    Received 10 August 1999

    Abstract. We apply the 2D wavelet transform modulus maxima (WTMM) method to synthetic randommultifractal rough surfaces. We mainly focus on two specific models that are, a priori, reasonnable candi-dates to simulate cloud structure in paper III (S.G. Roux, A. Arneodo, N. Decoster, Eur. Phys. J. B 15,765 (2000)). As originally proposed by Schertzer and Lovejoy, the first one consists in a simple power-lawfiltering (known in the mathematical literature as fractional integration) of singular cascade measures.The second one is the foremost attempt to generate log-infinitely divisible cascades on 2D orthogonalwavelet basis. We report numerical estimates of the (q) and D(h) multifractal spectra which are in verygood agreement with the theoretical predictions. We emphasize the 2D WTMM method as a very efficienttool to resolve multifractal scaling. But beyond the statistical information provided by the multifractaldescription, there is much more to learn from the arborescent structure of the wavelet transform skeleton ofa multifractal rough surface. Various statistical quantities such as the self-similarity kernel and the space-scale correlation functions can be used to characterize very precisely the possible existence of an underlyingmultiplicative process. We elaborate theoretically and test numerically on various computer synthetizedimages that these statistical quantities can be directly extracted from the considered multifractal functionusing its WTMM skeleton with an arbitrary analyzing wavelets. This study provides algorithms that arereadily applicable to experimental situations.

    PACS. 47.53.+n Fractals 05.40.-a Fluctuations phenomena, random processes, noise, and Brownianmotion 07.05.Pj Image processing 68.35.Bs Surface structure, and topography

    1 Introduction

    Multiplicative cascade models have enjoyed increasing in-terest in recent years as the paradigm of multifractal ob-jects [15]. The notion of cascade actually refers to aself-similar process whose properties are defined multi-plicatively from coarse to fine scales. In that respect, itoccupies a central place in the statistical theory of tur-

    bulence [4,6,7]. Since Richardsons famous poem [8], theturbulent cascade picture has been often invoked to ac-count for the intermittency phenomenon observed in fullydeveloped turbulent flows [611]: energy is transfered fromlarge eddies down to small scales (where it is dissipated)through a cascade process in which the transfer rate at agiven scale is not spatially homogeneous, as supposed inthe theory developed by Kolmogorov [12] in 1941, but un-dergoes local intermittent fluctuations [6,7,10,11]. Overthe past fourty years, refined models including the log-normal model of Kolmogorov [13] and Obukhov [14], mul-tiplicative hierarchical cascade models like the random -model [15], the -model [16], the p-model [17] (for a re-

    view, see Ref. [4]), the log-stable models [1820] and morea e-mail: [email protected]

    recently the log-infinitely divisible cascade models [2124]with the rather popular log-Poisson model advocated byShe and co-workers [22,25], have grown in the literatureas reasonable models to mimic the energy cascading pro-cess in turbulent flows. On a very general ground, a self-similar cascade is defined by the way the scales are refinedand by the statistics of the multiplicative factors at eachstep of the process [4,5,20]. One can thus distinguish dis-crete cascades that involve discrete scale ratios leading to

    log-periodic corrections to scaling (discrete scale invari-ance [2628]), from continuous cascades without prefer-able scale factor (continuous scale invariance). As far asthe fragmentation process is concerned, one can specifywhether some conservation laws are operating or not [5];in particular, one can discriminate between conservative(the measure is conserved at each cascade step) and nonconservative (only some fraction of the measure is trans-fered at each step) cascades. More fundamentally, thereare two main classes of self-similar cascade processes: de-terministic cascades that generally correspond to solvablemodels and random cascades that are likely to providemore realistic models but for which some theoretical care

    is required as far as their multifractal limit and some ba-sic multifractal properties (including multifractal phase

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    transitions) are concerned [5]. As a notable member of thelater class, the independent random cascades introducedby Mandelbrot (commonly calledM-cascades [9,29]) asa general model of random curdling in fully developed

    turbulence, have a special status since they are the maincascade model for which deep mathematical results havebeen obtained [30,31]. Note that most of these 1D cascademodels have natural generalization in 2D and higher di-mensions. Since realistical turbulent systems (e.g. atmo-spheric turbulence [32]) are likely to display anisotropicscaling, full realism requires the possibility of introducingspatial anisotropy [18,20,3234]. Moreover, as pointed outby Schertzer and Lovejoy [33], this first generation ofcascade models is static in the sense that it accounts forthe multiplicative hierarchical structure of the data in thespatial domain only [20]. With the specific goal to modeltemporal evolution, these authors have proposed a sec-ond generation of space-time cascade models that take

    into account both the scaling anisotropy between spaceand time, and the breaking of the mirror symmetry alongthe temporal axis, i.e., causality [20,33,34]. For the ba-sic framework necessary to handle space-time anisotropicscaling, we refer the reader to references [20,34], where theconcepts of Generalized Scale Invariance (GSI) and space-time multifractals have been introduced and explored.

    However, in physics as well as in other applied sci-ences, fractals appear not only as singular measures butalso as singular functions [1,7,32,3545]. For instance infully developed turbulence, directly observable quantitiesare the velocity field or the temperature field rather than

    the dissipation field [6,7,10,11]. Paradoxically, if there isa plethora of mono and multifractal cascade models inthe literature that generate deterministic as well as ran-dom singular measures in the small-scale limit, there arestill only a handful of distinct algorithms for synthesizingrough functions of a single variable with multifractalstatistics. Beyond the problem of the multifractal descrip-tion of singular functions that has been solved with theWTMM method [4650], there is thus the practical is-sue of defining in any concrete way how to build a mul-tifractal function. Schertzer and Lovejoy [18] suggested asimple power-law filtering (fractional integration) of sin-gular cascade measures. So, this model combines a mul-tiplicative procedure with an additive one reminiscent of

    some algorithms to generate fractional Brownian motion(fBm) [1,51,52]. In the same spirit, the bounded cas-cade model of Marshak et al. [53] consists in acting onthe multiplicative weights during the cascade in physicalspace, to recover continuity in the small-scale limit. Inreferences [54,55], the midpoint displacement techniquefor building fBm was generalized to generate determinis-tic or random multi-affine functions. The same goal wasachieved in references [47,48] by combining fractional orordinary integration with signed measures obtained by re-cursive cascade like procedures. Several other attemptsto simulate synthetic turbulence that shares the inter-mittency properties of turbulent velocity data have par-

    tially succeeded [5659]. More recently, the concept ofself-similar cascades leading to multifractal measures has

    been generalized to the construction of scale-invariant sig-nals using orthonormal wavelet basis [6063]. Instead ofredistributing the measure over sub-intervals with mul-tiplicative weights, one allocates the wavelet coefficients

    in a multiplicative way on the dyadic grid. This methodhas been implemented to generate multifractal functionsfrom a given deterministic or probalistic multiplicativeprocess. From a mathematical point of view, the conver-gence of theseW-cascades and the regularity propertiesof the so-obtained deterministic or stochastic functionshave been discussed by one of us (A.A.) and co-workers inreference [64].

    Coming to rough surfaces generated by fractal func-tions of two variables, we are aware of various algorithmsfor fBm surface generation [51,52,6567]. We have listedthese algorithms in paper I [68] and shown that the roughsurfaces simulated with those additive processes display

    homogeneous monofractal scaling properties as describedby a unique scaling exponent, namely the Hurst expo-nent. To the best of our knowledge, only two algorithmsfor multifractal functions of two variables have been doc-umented, to some extend, in publications. Both were de-veloped with atmospheric applications in mind and corre-spond to straightforward generalizations in 2D of the frac-tionally integrated singular cascade models [18,20,69,70]and the bounded cascade model [71,72] mentioned justabove. But our purpose here is far from being restricted toturbulence and to possible geophysical applications (thisissue will be addressed in paper III [73]). Actually ourfirst main objective is to explore ways of generating multi-

    fractal rough surfaces with prescribed multifractal statis-tics along the path open by the randomW-cascades onwavelet dyadic trees in references [6064]. The idea is tointroduce a new class of square integrable (finite energy)functions over the real plane R2, using a separable 2Dwavelet orthogonal basis [7476]. The multifractal scalingproperties are recovered from the multiplicative processused to generate the wavelet coefficients: these functionsare built recursively in the orthogonal wavelet space-scalerepresentation, cascading from an arbitrary given largescale towards small scales. These 2D randomW-cascademodels present two crucial advantages as far as future re-search is concerned. From a theoretical point of view, thefact of using wavelet orthogonal basis provides the frame-

    work for some rigorous mathematical treatment. This isout of the scope of the present work but we hope to elab-orate about this point in a forthcoming publication. Froma practical point of view, the fact of using separable multi-resolution schemes provides a simple and very efficient al-gorithmic procedure for designing and synthesizing mul-tifractal rough surfaces. Let us point out that this multi-resolution synthesis algorithm has enough flexibility formaking tractable the implementation of anisotropic scaleinvariance. Since our second and main objective in thispaper is to test the reliability of the 2D WTMM methodintroduced in Section 4 of paper I [68], we will copi-ously use this multi-resolution synthesis algorithm to gen-

    erate collections of numerical images of multifractal roughsurfaces, each collection corresponding to a particular

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    cascading process (e.g., log-normal process, log-Poissonprocess, ...). Then, for each set of images, we will com-pute the (q) and D(h) multifractal spectra with the2D WTMM method and compare the numerical results

    with the theoretical predictions. Within the perspectiveof experimental application of the 2D WTMM methodto geophysical data in paper III [73], we will proceedto a comparative multifractal analysis of rough surfacesgenerated by the fractionally integrated singular cascademodel [18,20,69,70].

    But from the very analogy that links the multifractaldescription and statistical thermodynamics [2,49,7779],there still exists some fundamental insufficiency in the de-termination of the multifractal spectra. In particular, (q)and D(h) play respectively the role of thermodynamicalpotentials, namely free energy and entropy, which intrin-sically contain only some degenerate information about

    the Hamiltonian of the problem, i.e., the underlyingcascading process [80,81]. Therefore, it is not surprisingthat previous experimental determinations of the(q) andD(h) spectra of turbulent fields, have failed to providea selective test to discriminate between existing (deter-ministic or random) cascade models. In order to go be-yond the classical multifractal description, Castaing andco-workers [24,8288] have proposed some approach ofthe intermittency phenomenon which amounts to modelthe evolution of the shape of the turbulent velocity incre-ment probability density function (pdf), from Gaussian atlarge scales to more intermittent profiles with stretchedexponential-like tails at smaller scales [82, 8992], by a

    functional equation that relates two scales using a ker-nel G. This self-similarity kernel actually contains deepinformation on the underlying multiplicative process: itsshape is determined by the nature of the elementary stepin the cascade, while the way Gdepends on the coarse andfine scales reflects the scale invariance properties of thecascade. In their original work, Castaing et al. [24,8287]mainly focused on the estimate of the variance ofG andits scale behavior. A generalization of this approach tothe wavelet transform of the velocity field has been pro-posed in a previous work and shown to provide direct ac-cess to the entire shape of the kernel G [6163,81]. Thiswavelet-based method has been tested on synthetic 1Dturbulent signals and further applied to turbulence data.

    At the highest accessible Reynolds numbers, the computa-tion of the self-similarity kernel Gyields a very convincinglog-normal law on a well defined range of scales that canbe further used as an objective definition of the inertialrange [62,63,81]. Moreover, the number of cascade stepsis found to evolve as a power-law (and not logarithmi-cally) as a function of the scale, which is the signatureof the breaking of scale-invariance [62, 63,81,93]. How-ever, the investigation of data sets recorded at differentReynolds numbers suggests the asymptotic validity of thelog-normal multifractal description [63, 81,93].

    As emphasized in a recent work [94], one can andone must go even deeper in the multifractal analysis bystudying correlation functions in both space and scales.The wavelet transform skeleton defined by the wavelet

    transform modulus maxima actually contains the keyspace-scale information required for this two-point sta-tistical analysis. In the arborescent structure of thewavelet transform is somehow uncoded the underlying

    multiplicative cascade process [81,9496]. The computa-tion of the space-scale correlations functions from thewavelet transform skeleton has been proved to provideconclusive evidence for the existence of some internalultrametric structure [94]. We refer the readers to ref-erences [81,94, 97] for preliminary applications of thiswavelet based space-scale correlation method to fully de-veloped turbulence data and financial time-series. Ourthird goal in this paper, is to generalize the self-similaritykernel and space-scale correlation function approaches tothe statistical study of multifractal rough surfaces. Part ofthe work will consist in adapting the mathematical con-cepts and in turn enriching our algorithmic park from 1Dto 2D analysis. The final touch will be test applications of

    the corresponding softwares to numerical synthetic imagesgenerated by 2D randomW-cascade processes for whichthe self-similarity kernel as well as the space-correlationfunctions are known analytically.

    The paper is organized as follows. In Section 2, wedescribe the 2D versions of the fractionally integrated sin-gular cascade model and of the random W-cascade processon separable wavelet orthogonal basis. These models canbe used to synthesize rough surfaces that display multi-fractal scaling properties with prescribed (q) and D(h)spectra. In Section 3, we report the results of test applica-tions of the 2D WTMM method introduced in Section 4 of

    paper I [68], to previously synthesized multifractal roughsurfaces. Our main goal is to calibrate our numerical toolswith respect to finite-size effects and statistical conver-gence. Sections 4 and 5 are respectively devoted to thegeneralization in 2D of the self-similarity kernel and space-scale correlation function methods [81]. We illustrate ourpurpose with numerical applications which confirm the ne-cessity of investigating two-point statistics across scalesto get definite and conclusive evidence for a multiplicativehierarchical structure underlying the roughness fluctua-tions of multifractal surfaces. We conclude in Section 6 bydiscussing the wide range of potential applications of this2D wavelet-based statistical analysis in fundamental aswell as in applied sciences, from image processing to image

    synthesis, from numerical to experimental data analysis.

    2 Hierarchical models for multifractal roughsurface synthesis

    There are many well documented methods for gen-erating multifractal measures using multiplicativecascades [9, 1325,3234]. In contrast, the literature onspecific ways of synthesizing multifractal functions isrelatively small [18,47, 48,5364,98]. We present twoprocedures here: the first one, the Fractionally Inte-grated Singular Cascade (FISC) has been introducedand plentifully applied for multifractal geophysicalfield modeling [18,20,69,70,99101]; the second one is

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    Fig. 1.(a) Multifractal measure generated using the binomial cascade model with the parameter value p= 0.32. (b) Multifractalrough surface obtained after fractional integration with an exponent H = 0.638. In the top (1024 1024) panels, (x) andf(x) are coded using 32 grey levels from white (min or min f) to black (max or max f).

    quite original since it consists in generating multifractalrough surfaces from random cascade process on separable

    wavelet orthogonal basis.

    2.1 Fractionally integrated singular cascades

    A stochastic model with continuously variable intermit-tency was originally proposed by Schertzer and Lovejoy forrain (1D), clouds (2D), landscapes (2D) or any other com-pliant multifractal geophysical fields [18, 20,69, 70,72, 101].The idea is to start with a specific model of singular mul-tiplicative cascade and then to proceed to a straightfor-ward filtering in Fourier space (fractional integration) inorder to bring multifractal measures into the realm ofcontinuous multifractal functions. Let us note that this

    strategy is strongly inspired from the devils staircaseconcept which is nothing but a straightforward integra-tion of some Bernoulli measure distributed on a Cantorset [1,4749].

    A multiplicative cascade process consists in startingwith some 2D spatial domain, let say a square of charac-teristic size L, on which a measure = L is uniformlydistributed. At the first step of the construction, the ini-tial domain breaks into smaller domains, let say the initialsquare breaks into four smaller squares of characteristicsizeL/2, each receiving a fraction of the original measureas defined by a random variable Mwith a certain probabil-ity distribution P(M). By repeating the same procedurerecursively at smaller scales using independent realizationsof the random variableM, one generates a random singu-

    lar measure over the L L square:

    n(x; l) =L

    ni=1

    Mi, l/L= 2n

    0. (1)Since the cascading process is space-filling, all the informa-tion on the singular nature of this multiplicative process iscontained in the specific shape of the so-called measuremultiplier [4,9] probability distribution. In the presentstudy, we will mainly focus on the p-model (also calledbinomial model) originally proposed by Meneveau andSreenivasan [17] to simulate the highly intermittent fluc-tuations of the kinetic energy dissipation field in fully de-veloped turbulence. In this model, P(M) has the followingsimple form [4,17]:

    P(M) = 1/2

    M M(1)

    +

    M M(2)

    , (2)

    where

    M(1) =p/2, M(2) = (1 p)/2, 0 p 1/2, (3)independently of the cascade step. For the present dis-cussion, we additionally impose conservation of the mea-sure at each step; this means that one selects at randomamong the 4 sub-squares, the two which will receive afractionM(1) =p/2, the two others receiving the fractionM(2) = (1 p)/2 of the measure at the previous step. Astraightforward computation (see Refs. [4,17], for detailedcalculations) of the (q) spectrum defined in Section 4.1

    of paper I [68] (Eqs. (59, 60)) yields:(q) = (q+ 1) log2

    pq + (1 p)q. (4)

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    Fig. 2. 1D profiles obtained along some horizontal cut inFigures 1a and 1b respectively: (a) vs. x; (b) f vs.x.

    When p= 1/2 (p = 1/2 corresponds to a uniform distri-bution of the measure over the original square), one getsa nonlinear(q) spectrum which is the hallmark of multi-

    fractal scaling properties. Figure 1a shows a realisation ofsuch a random measure for the parameter value p= 0.32after 10 cascade steps. The intense spikiness is witness tothe singularity of this binomial cascade model. The inter-mittent character of this measure is clearly seen on 1Dcuts as illustrated in Figure 2a. In the limit of an infinitenumber of steps, we are clearly not dealing with a functionof (x, y) [0, L]2: the product ni=1 Mi in equation (1) isgenerally zero but infinite just often enough to keep thedomain average at unity.

    As a mean of introducing continuity, the form of frac-tional integration we use here, is a low-pass power-lawfiltering in Fourier space [18,20]. The measure n(x) is

    smoothed into a function using:

    fn(x) =n(x) |x|(1H) , 0< H

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    Fig. 4. Fourier transforms of the 3 separable wavelets calculated from a 1D Daubechies 8 wavelet [74]: (a) 1(kx, ky); (b)

    2(kx, ky); (c) 3(kx, ky).

    and

    hmax= limq

    f(q)

    q =H

    1 log2(p), (13)where we recall that 0 p 1/2. For p = 1/2, one re-covers monofractal scaling with a unique Holder exponenth= H. For any other value of the parameter p, one hasmultifractal scaling with a Holder exponenth which fluc-tuates from point to point, taking value in the intervalh [hmin, hmax].

    2.2 Random cascades on separable wavelet orthogonalbasis

    As mentioned in the introduction, aW-cascade [64] isbuilt recursively on the two-dimensional square grid ofseparable wavelet orthogonal basis, involving only scalesthat range between a given large scale L and the scale 0(excluded). Thus the corresponding fractal functionf(x)will not involve scales greater than L. For that purpose,we will use compactly supported wavelets [74].

    2.2.1 Two-dimensional wavelet orthogonal basis [7476]

    As in one dimension, the notion of resolution is formalizedwith orthogonal projections in spaces of various sizes. Theapproximation of an image f(x) =f(x, y), at the resolu-tion 2j, is defined as the orthogonal projection off ona space V2j that is included in L

    2(R2). The space V2j is

    the set of all approximations at the resolution 2j. Whenthe resolution decreases, the size ofV2j decreases as well.Here we consider the particular case of separable multi-resolution. Let {Vj}jZ be a multi-resolution ofL2(R). Aseparable 2D multi-resolution is composed of the tensorproduct spaces:

    V2j =VjVj . (14)

    Let W2j be the detail space equal to the orthogonal com-plement of the lower resolution approximation space V2j

    in V2j1:

    V

    2

    j1=V

    2

    jW2

    j . (15)One can rewrite this equation in the following way:

    Vj1 Vj1= (Vj Vj) W2j . (16)Then, by inserting Vj1 = Vj Wj in equation (16),simple algebra yields:

    W2j = (VjWj) (WjVj) (Wj Wj). (17)

    As in the 1D case, the overall space L2(R2) can be de-composed as an orthogonal sum of the detail spaces at allresolutions [7476]:

    L2(R2) = +j=W2j . (18)

    A separable wavelet orthonormal basis ofL2(R2) canthus be constructed with separable products of a scal-ing function and a wavelet generating a wavelet or-thogonal basis ofL2(R). Let us define the following threewavelets [7476]:

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

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

    3(x, y) = (x)(y), (19)

    which take the following values on the 2D square grid:

    kj,m,n(x, y) =

    1

    2jk

    x 2j(m + 1/2)2j

    ,y 2j(n + 1/2)

    2j

    , (20)

    where 1 k 3. According to equation (17), the waveletfamily

    1j,m,n(x, y), 2j,m,n(x, y),

    3j,m,n(x, y)

    (m,n)Z2

    (21)

    is an orthonormal basis ofW2j . From equation (18), onededuces that

    1j,m,n(x, y), 2j,m,n(x, y),

    3j,m,n(x, y)

    (j,m,n)Z3

    (22)

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    is an orthonormal basis ofL2(R2).

    The Fourier transform of the three separablewavelets [76] calculated from a 1D Daubechies 8 waveletare shown in Figure 4. Wavelet coefficients calculated with

    1 and 2 are large along edges which are respectively hor-izontal and vertical. The wavelet3 produces large coeffi-cients at the corners. Note that the separable wavelet ex-pressions (Eq. (19)) imply that 1(kx, ky) = (kx)(ky),

    2(kx, ky) = (ky)(kx) and 3(kx, ky) = (kx)(ky).Let us point out that in order to approach spatial isotropy,the respective weights on (1, 2, 3) have to be in the ra-tios (1, 1, 1/2(2)/4+1), at least on average.

    2.2.2 RandomW-cascades

    Let us consider the set{1j,m,n,

    2j,m,n,

    3j,m,n}of 2D sep-

    arable compactly supported wavelets that form an or-thonormal basis of L-periodic functions of L2per([0, L]

    2),

    whereL = 2N. Thusf L2per([0, L]2),fcan be writtenunder the form:

    f(x) =Nj=0

    2Nj1m,n=0

    3k=0

    ckj,m,nkj,m,n(x), (23)

    where the set of coefficients {ckj,m,n = kj,m,n|f} providesa complete characterization of the function f. The notionof cascade is then rather natural on the 2D square grid.The construction rule is very similar to the one used in

    1D on wavelet dyadic trees [6064]. We build a randomfunction f(x) by specifying its wavelet coefficients {ckj,m,n}in a recursive way. Actually, it is the modulus

    dj,m,n =

    c1j,m,n

    2+

    c2j,m,n2

    +

    c3j,m,n21/2

    , (24)

    that one generates at successive scales, by iterating thefollowing system:

    dj1,2m,2n = M(r1)j,m,ndj,m,n ,

    dj1,2m+1,2n = M(r2)j,m,ndj,m,n ,

    dj1,2m,2n+1= M

    (r3)

    j,m,ndj,m,n ,dj1,2m+1,2n+1= M

    (r4)j,m,ndj,m,n, (25)

    for all j (1 j N), m (0 m < 2Nj) and n(0 n < 2Nj) and where the M(ri)j,m,n are independentidentically distributed (i.i.d.) positive valued random vari-ables with prescribed law P(M). To go from the dj,m,nmodulus coefficients to the ckj,m,n(1 k 3) wavelet co-efficients, one has to specify the value of the angles (, )involved in the following expressions:

    c1j,m,n= cos()cos()dj,m,n ,

    c

    2

    j,m,n= cos()sin()dj,m,n,c3j,m,n= sin()dj,m,n. (26)

    where [, ] and [/2, /2].To generate a given realization of theW-cascade, one

    starts, at large scale, from arbitrarily chosen values of thecoefficients ckN,0,0 (1

    k

    3), i.e., from an arbitrarily

    chosen value ofdN,0,0. Then one generates the coefficientsdj,m,n at successive scales, by iterating equation (25). Ateach step and for each realization of the random vari-able M, the angle and in equation (26) are indepen-dently and randomly chosen with a white distribution on[min, max] and [min, max] respectively. Let us pointout that the so-obtained random function f(x) (assuming

    that the sumN

    j=0in equation (23), which actually should

    be an infinite sumN

    j=, converges) is self-similar inthe sense that the law of a wavelet coefficient dj,m,n atthe scale 2j can be linked to the law of another waveletcoefficient dj,m,n at the scale 2

    j > 2j using a multi-plicative random variable depending only on the ratio ofthe two scales:

    dj,m,nldj,m,nXjj , (27)wherel stands for the equality in law and where Xj =Nj

    i=1 Mi (theMis are i.i.d. positive valued random vari-ables with the same law as M). Thus, from a statisticalpoint of view, the details of the function f at a scale aare the same as the details at a scale a up to a rescalingfactor that depends only on a/a.

    Remark

    Let us note that equation (25) can be rewritten as

    ln dj1,2m,2n = ln dj,m,n+ ln M(r1)j,m,n,

    ln dj1,2m+1,2n = ln dj,m,n+ ln M(r2)j,m,n,

    ln dj1,2m,2n+1= ln dj,m,n+ ln M(r3)j,m,n,

    ln dj1,2m+1,2n+1= ln dj,m,n+ ln M(r4)j,m,n. (28)

    If M is log-normal, these equations correspond to whatone could call a tree-autoregressive process. This processis of order 1 in the sense that the regression involves onlyone term. We refer the reader to the work of Basseville andco-workers [102] for some introduction to autoregressive

    models lying on a tree (including the orthonormal waveletdyadic tree). As previously emphasized in Ref. [64], ourapproach ofW-cascades in 1D as well as in 2D, is signif-icantly different from theirs since we concentrate on theanalysis of the fractal function f itself and not on theproperties of the tree-process.

    2.2.3 Numerical simulations of 2DW-cascades

    As inspired from the modeling of the energy cascading pro-cess in fully developed turbulence by log-infinitely divisi-ble multiplicative processes [2125,6163], we will mainlyconcentrate here on the synthesis of multifractal roughsurfaces using log-normal and log-PoissonW-cascades.

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    Log-normal W-cascadesLet us first start with Mbeing a log-normal random vari-able. Ifm and 2 are, respectively, the mean and the vari-ance of ln M, then a straightforward computation leads tothe following (q) spectrum:

    (q) = log2Mq 2,q R

    = 2

    2 l n 2q2 m

    ln 2q 2, (29)

    where. . . means ensemble average. The correspondingD(h) singularity spectrum is obtained by Legendre trans-forming equation (29):

    D(h) = (h + m/ ln2)2

    22/ ln 2 + 2 . (30)

    According to the convergence criteria established in

    1D [64], we will only consider parameter values that satisfythe conditions:

    m < 0 and |m|

    2 >2

    ln 2. (31)

    Moreover, by solving D(h) = 0, one gets hmin andhmax:

    hmin = mln 2

    2ln 2

    ,

    hmax= mln 2

    + 2

    ln 2 (32)

    In Figure 5 are illustrated 3 realizations of a log-normal

    W-cascade corresponding to 3 different ways of distribut-ing the weights on 1,2 and3 during the construction.The Fourier transforms of these 3 images are also shownin order to evidence some underlying anisotropy inducedby the intrinsic anisotropy of the separable wavelets k

    (1 k 3). The model parameters are fixed to the val-ues m =0.38ln2 and 2 = 0.03 ln 2; 10 cascade stepshave been used to generate these images.

    (i) In Figure 5a, we start from the following wavelet coef-ficients at the largest scale:

    c1N,0,0= 1, c2N,0,0= 1, c

    3N,0,0= 0. (33)

    At each step of the cascade, the multiplier M israndomly chosen with the previously defined log-normal distribution. is randomly chosen between[, ] while = 0 is kept fixed. This means thatthe so-generatedW-cascade develops on the wavelets1 and 2 only. The corresponding Fourier transformin Figure 5b clearly displays anisotropic scaling withenhanced values along thekx andky axis. This squarelattice anisotropy results from the intrinsic anisotropicshape of 1 and 2.

    (ii) In Figure 5c, theW-cascade is generated on 3 only.One starts at the largest scale from the following coef-

    ficients:

    c1N,0,0= 0, c2N,0,0= 0, c

    3N,0,0= 1, (34)

    Fig. 5. Three realizations of a log-normal W-cascade (n =10) with parameter values m= 0.38ln2 and 2 = 0.03ln2,on separable orthonormal wavelets, using 1D Daubechies 8wavelet. (a) W-cascade on1 and2 (case (i) in the text). (c)W-cascade on 3 (case (ii)). (e) W-cascade on 1, 2 and 3

    (case (iii)). The corresponding Fourier transforms are shown in(b), (d), and (f) respectively. The fuzzy curves made of smallblack dots are level curves to guide the eyes. The amplitudesoff and fare coded using 32 grey levels from white (min f or

    minf) to black (max for maxf).

    and during the cascading process, = /2 is chosenat random. As seen in the image of f(x) itself, theparticular shape of 3(x, y) = (x)(y) inducesthe presence of a texture with main axis along thediagonals. This anisotropy is clearly patent on the

    corresponding Fourier transform shown in Figure 5d.

    (iii) In order to approach isotropy, one needs to adapt ourstrategy to the specificity of theW-cascade one in-tends to generate. The specific shape of the analyzingwavelets 1, 2 and 3 in Fourier space (Fig. 4), re-quires to adjust in a clever way the weights we put ateach step on each of these wavelet modes. One startsat the largest scale from the following coefficients:

    c1N,0,0= 1, c2N,0,0= 1, c

    3N,0,0= 2

    ((2)/4+1) ,(35)

    where(2) =

    4 is chosen according to the power-law decay of the energy spectrum one wants to impose.At each step, is randomly chosen between [, ]

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    Fig. 6. Multifractal (1024 1024) rough surfaces generated using random W-cascades (n= 10 steps). (a) Log-normal cascadewith parameter values m = 0.38ln2 and 2 = 0.03 ln 2. (b) Log-Poisson cascade with the parameter values = 1/9ln2,= (2/3)1/3 and = 2 ln 2. Same coding as in Figure 5.

    while the domain for needs to be adapted so that [, ] (white distribution), where satisfiesthe following equation:

    sin2

    4 =

    2(2)/2+3

    1 + 2(2)/2+3 1

    2 , >0 . (36)

    The image shown in Figure 5e as well as its Fouriertransform in Figure 5f, no longer display significantdeparture from isotropy. Even though each of thewavelets k (1 k 3) are clearly anisotropic, thisdoes not prevent the synthesis of multifractal functionsfrom approaching isotropic scaling properties (Notethat there still remains some anisotropy coming fromthe underlying fragmentation process).

    Figure 6a illustrates another realization of an isotropiclog-normalW-cascade together with the correspondingmultifractal rough surface. The intermittent nature off(x) is enlightened on 1D cuts as shown in Figures 7aand 7b. From the power-spectrum power-law behavior inFigures 8a and 8b, one extracts a rather accurate estimateof the power-spectral exponent = (2) + 4 = 2.70, forthe set of considered parameter values.

    Log-Poisson W-cascades

    Let be the mean and the variance of the Poisson distri-bution Y. We consider that the law of ln M is the sameasY ln + . A straightforward computation leads to the

    Fig. 7. 1D profiles obtained along some horizontal and ver-tical cuts in Figures 6a and 6b respectively. Log-normal W-cascade: (a) horizontal profile; (b) vertical profile. Log-PoissonW-cascade: (c) horizontal profile; (d) vertical profile.

    following(q) spectrum:

    (q) = (1 q) qln 2

    2. (37)

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    Fig. 8. Power-spectrum analysis of the (1024 1024) im-ages shown in Figures 6a and 6b respectively. Log-normalW-cascade: (a) ln |fn=10(k)|; (b) S(|k|) vs. |k| in logarithmic

    representation. Log-Poisson W-cascade: (c) ln |fn=10(k)|; (d)S(|k|) vs. |k| in logarithmic representation. The solid lines in(b) and (d) correspond to the theoretical power-law predictionwith exponent = (2) + 4 = 2.70.

    The corresponding D(h) singularity spectrum obtained by

    Legendre transforming(q) writes:

    D(h) =

    h

    ln +

    ln2ln

    ln

    h + / ln 2

    (/ ln2)ln

    1

    ln 2

    + 2 . (38)

    Very much like the spectra obtained for log-normalW-cascades, (q) is a nonlinear function ofqwhileD(h) has asingle humped shape characteristic of multifractal scalingproperties.

    In Figure 6b is shown a multifractal rough surface gen-

    erated with the log-Poisson W-cascade model with param-eter values =1/9ln2, = (2/3)1/3 and = 2 ln2.1D profiles obtained along horizontal and vertical 1D cutsare illustrated in Figures 7c and 7d. From a qualitativepoint of view, this rough surface looks intermittent verymuch like the one obtained with a log-normal W-cascade.Actually, as seen on the power-spectrum analysis inFigures 8c and 8d, the numerical data are in good agree-ment with the theoretical prediction for the spectral ex-ponent= (2)+4 = 2.70. Let us remark that because ofour specific choices for the model parameter values, boththe log-normal and log-Poisson W-cascades generate mul-tifractal rough surfaces with the same (2) exponent andin turn the same spectral exponent . From a quantita-tive point of view, the data in Figures 8c and 8d fail todistinguish log-Poisson from log-normal W-cascades. This

    observation is not so surprising since we already knowthat power spectrum analysis fails to discriminate betweenrough surfaces that display multifractal scaling proper-ties from those which are homogeneous and monofractal.

    In particular, a fractional Brownian surface BH(x

    ) withthe index H = ((2) + 2)/2 = 0.35, diplays the samepower-law decay than the power spectra in Figures 8band 8d (see Sect. 5 in paper I [68]).

    Remark

    As previously discussed for 1D randomW-cascades inreference [64], there is no reason, a priori, that all therealizations of the same stochastic multifractal functionscorrespond to a unique D(h)-curve. Each realization hasits own unique distribution of singularities and the cru-cial issue is to relate the theoretical singularity spectrum

    defined by equation (58) in paper I [68] to the statisticalD(h) spectrum given by equations (30) and (38) respec-tively. From the mathematical results proved for 1D ran-domW-cascades [64], the statistical D(h) spectrum ob-tained with the multifractal WTMM formalism, is likelyto be an upper bound for the theoretical singularity spec-trum (at least for its left increasing branch).

    3 WTMM analysis of synthetic multifractalrough surfaces

    This section is devoted to the application of the 2DWTMM method to synthetic multifractal rough surfacesgenerated with the two classes of models described inSection 2. We systematically follow the numerical im-plementation procedure described in Section 4.3 of pa-per I [68]. For each model, we first wavelet transform 32(1024 1024) images of the stochastic multifractal roughsurfaces with an isotropic first-order analyzing wavelet. Tomaster edge effects, we then restrain our analysis to the(512 512) central part of the wavelet transform of eachimage. From the wavelet transform skeleton defined by theWTMMM, we compute partition functions from which weextract the (q) and D(h) multifractal spectra (Sect. 4.2

    of paper I [68]). We systematically test the robustness ofour estimates with respect to some change in the shape ofthe analyzing wavelet, in particular when increasing thenumber of zero moments.

    3.1 Multifractal rough surfaces generatedby the fractionally integrated singular cascade model

    In Figure 9 is illustrated the computation of the max-ima chains and the WTMMM for an individual image ofa random multifractal rough surface generated with thefractionally integrated singular cascade model describedin Section 2.1 [72,99101], for the following parameter val-ues: p = 0.32 and H = 0.638 (Fig. 1b). In Figure 9b is

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    Fig. 9. 2D wavelet transform analysis of a multifractal rough surface generated with the fractionally integrated singular cascademodel (p= 0.32, H = 0.638). is a third-order radially symmetric analyzing wavelet (see Fig. 1 of paper I [68]). (a) 32 grey-

    scale coding of the original (10241024) image. In (b) a = 22.9W, (c)a = 21.9W and (d)a = 23.9W (whereW= 13 pixels),are shown the maxima chains; the local maxima ofM

    along these chains are indicated by () from which originates an arrowwhose length is proportional to M

    and its direction (with respect to thex-axis) is given by A

    ; only the (512 512) centralpart delimited by a dashed square in the original image in (a) is taken into account to define the WT skeleton. In (b), thesmoothed image b,a f is shown as a grey-scale coded background from white (min) to black (max).

    shown the convolution of the original image (Fig. 9a) withthe isotropic mexican hat smoothing filter . Accordingto the definition of the WTMM, the maxima chains corre-spond to well defined edge curves of the smoothed image.The local maxima ofM along these curves are indi-cated by (

    ) from which originates an arrow whose length

    is proportional toM and its direction (with respect tothex-axis) is given by A. After linking these WTMMMacross scales, one constructs the WT skeleton from whichone computes the partition functionsZ(q, a) (Eq. (59) ofpaper I [68]). As reported in Figure 10a, the annealedaverage partition functions display a well defined scalingbehavior over about 4 octaves when plotted versus a ina logarithmic representation. When processing to a linearregression fit of the data over the first four octaves, onegets the(q) spectrum shown in Figure 10c. In contrast tofractional Brownian rough surfaces studied in Section 5 ofpaper I [68], this (q) spectrum unambiguously devi-ates from a straight line. Moreover, for4 q 9,the data are in remarkable agreement with the theo-retical spectrum given in equation (8). The values of

    h = /q range in the interval [0.18, 1.30], whichmeans that hmax > 1, i.e., the multifractal function un-der study has singularities in its first derivative also.This result is corroborated by the scaling behavior ofthe expectation value h(q, a), defined in equation (66)in paper I [68], which clearly depends on q in Fig-

    ure 10b. When Legendre transforming the nonlinear(q) spectrum obtained in Figure 10c, one gets theD(h) singularity spectrum reported in Figure 10d. Itscharacteristic single humped shape over a finite range ofHolder exponents is a clear signature of the multifrac-tal nature of the synthetic rough surfaces generated bythe fractionally integrated singular cascade model. Forq = 0, the largest dimension is attained for singulari-ties of Holder exponent h(q = 0) = 0.74 0.02, a valuewhich is in good agreement with the theoretical predictionh(q= 0) = 0.738. Moreover, the corresponding maximumof the D(h) curve, D(h(q = 0)) =(0) = 2.00 0.02does not deviate substantially from the theoretical value

    DF= 2, which confirms that the considered random mul-tifractal functions are singular everywhere.

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    Fig. 10. Determination of the (q) and D(h) spectra of mul-tifractal rough surfaces generated with the fractionally inte-

    grated singular cascade model (p= 0.32,H

    = 0.638). The 2DWTMM method is used with a third-order radially isotropicanalyzing wavelet (). (a) log2Z(q, a)vs. log2 a; the solid linescorrespond to linear regression fit of the data over the first 4octaves. (b)h(q, a) vs. log2 a; the solid lines correspond to lin-ear regression fit estimates ofh(q). (c) (q) vs. q as obtainedfrom linear regression fit of the data in (a) over the first fouroctaves. (d) D(h) vs. h, after Legendre transforming the (q)curve in (c). In (c) and (d), the symbols () represent theresults obtained when using a first-order radially symmetricanalyzing wavelet. The solid lines correspond to the theoreti-cal (q) (Eq. (8)) and D(h) spectra. These results come fromannealed averaging over 32 (10241024) images.ais expressedinW units.

    In Figures 10c and 10d are also shown for compari-son the results obtained when applying the 2D WTMMmethod with a first-order analyzing wavelet (the smooth-ing function being a Gaussian). With our statistical sam-ple of 32 (10241024) images, it is clear that the estimatesof the (q) and D(h) spectra are poorer than the resultspreviously obtained with a third-order isotropic analyzingwavelet. This observation is a direct consequence of thefact that first-order analyzing wavelets are not adaptedto characterize the fluctuations of multifractal functionswhose singularity spectrum support extends beyond 1(hmax > 1, i.e., the weakest singularities are located inthe first derivative of f(x)) on one side (q < 0), mean-while it reaches almost 0 on the other side (q > 0). Ad-

    Fig. 11. Pdfs of the WTMMM coefficients of synthetic mul-tifractal rough surfaces generated with the fractionally inte-grated singular cascade model (p = 0.32, H = 0.638). (a)Pa(M) vs. M. (b) Pa(A) vs. A. is a third-order radiallysymmetric analyzing wavelet. Four different scales a = 1, 2, 4,8 (in Wunits) are shown. These results correspond to aver-aging over 32 (1024 1024) images.

    Fig. 12. Pdfs ofM when conditioned by A. The different

    curves correspond to fixing A (mod ) to 0/8,/4/8,/2 /8 and 3/4 /8. (a) a = 20.1; (b) a = 21.1 (inWunits). Same 2D WTMM computations as in Figure 11.

    ditional computations on 32 newly generated images im-prove the agreement with the theoretical (q) spectrum.For the class of multifractal rough surfaces studied in thissection, it is clear that the use of a third-order analyzingwavelet allows us to master in a more efficient way theproblem of the fractional integration to the benefit of abetter characterization of the multiplicative nature of theunderlying singular cascade.

    In Figure 11 are shown the corresponding pdfsPa(M) =

    dA Pa(M, A) andPa(A) =

    dM Pa(M, A),

    for four different values of the scale parametera.

    As seen in Figure 11a, whatever the value of a,Pa(M) displays some oscillatory decreasing tail asthe signature of the underlying discrete multiplica-tive process (prior to the fractional integration). Notethat the main peaks observed in these distributionsoccur for values of M within a proportional ratiovery close to (1 p)/p = 0.68/0.32 = 2.125. InFigure 11b, Pa(A) clearly does not evolve across scales,which means that the scale invariance properties of theconsidered synthetic rough surfaces are contained in thescale dependence of theM-pdf. However, some oscilla-tory departure from a flat distribution is observed with

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    Fig. 13. Distribution of the WTMMM in the plane (T1 , T2)for the following values of the scale parameter: a = 1 (a), 2 (b),4 (c) and 8 (d) in W units. Same 2D WTMM computationsas in Figure 11.

    well defined maxima forA = 0, /2, and 3/2. Thisanisotropy in the reconstructed images (Fig. 1b) is the di-rect consequence of the privileged role played by the x- andy-axis in the binomial cascade process prior to the frac-tional integration. Hopefully, this anisotropy does not dis-turb the scaling properties ofM since, as reported in Fig-ure 12 for two different scales, the pdf ofM, when condi-tioned by the argument A, is shown to be shape invariant.This independence ofM andA is further illustrated inFigure 13 where the WTMMM are plotted in the(T1 , T2) plane (see Sect. 2 of paper I [68]). For the4 scales represented, the distributions so obtained areclearly square symmetric as the signature of the factor-ization of the joint probability distribution: Pa(M, A) =Pa(F(A)M)Pa(A), where F(A) is a function that does notdepend on the scale parameter and that allows us to super-

    impose perfectly the curves in Figures 12a and 12b (F(A)accounts simply for this square anisotropy but does notaffect the scaling properties of the partition functions);Pa(A) is the scale independent distribution found in Fig-ure 11b.

    All the multifractal properties of the rough surfacesgenerated by the fractionally integrated singular cascademodel are thus contained in the way the shape of the pdf ofM evolves when one decreases the scale parametera. Thisevolution is illustrated in Figure 14a when using a semi-logarithmic representation. As experienced in paper I [68]for fractional Brownian surfaces, a test of monofractal self-similar scaling is the existence of a unique exponent hwhich allows us to rescale all the pdfs Pa(M) computedat different scales, onto a universal function P independent

    Fig. 14. Pdfs ofM as computed at different scales a = 1, 2,4 and 8 (in W units). (a) lnPa(M) vs. M. (b) lnPa(M) vs.M/ah(q=0) with h(q= 0) = 0.738. Same 2D WTMM compu-tations as in Figure 11.

    ofa, and which satisfies:

    PM

    L(a)= PML(a)/ah. (39)In Figure 14b, we have tried to find such an exponentwithout any success. The results reported in this fig-ure correspond to rescaling the WTMMM by ah(q=0),where h(q= 0) = lima0+h(q = 0, a)/ ln a (Eq. (68) inpaper I [68]) = /q|q=0, is the Holder exponent whichis the most frequently encountered in the rough surfacesunder consideration. As compared to the remarkable col-lapse obtained in Figure 27 of paper I for B1/3(x), thepdfs obtained after rescaling in Figure 14b are clearlydifferent. Equation (39) is thus absolutely not relevant forthose synthetic rough surfaces whose intermittent fluctu-ations display multifractal scaling properties as character-ized by a singularity spectrum which involves a continuumof (Holder) scaling exponents (Fig. 10d).

    3.2 Multifractal rough surfaces generated by randomcascades on separable wavelet orthogonal basis

    This section is devoted to the application of the 2DWTMM method to multifractal functions synthetizedfrom W-cascades on separable wavelet orthogonal basis asdefined in Section 2.2. We mainly report results obtainedwith the first-order radially symmetric analyzing waveletsshown in Figure 1 of paper I [68]. Possibly because ofthe range of Holder exponent values which is restricted toh [0, 1], but more probably because of the underlyingmultiplicative structure of the multifractal surface itself(without any need of some power-law filtering), a first-order analyzing wavelets leads to numerical multifractalspectra which are in remarkable agreement with the theo-retical predictions. Let us point out that quite robust re-sults are obtained with the third-order analyzing waveletused in the previous sub-section.

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    Fig. 15. 2D wavelet transform analysis of a multifractal rough surface generated with the log-normal W-cascade model. Samemodel parameters as in Figure 6a. is the first-order radially symmetric analyzing wavelet shown in Figure 1 of paper I [68].

    (a) 32 grey-scale coding of the original (1024 1024) image. In (b) a = 22.9

    W, (c) a = 21.9

    W and (d) a = 23.9

    W, are shownthe maxima chains and the WTMMM for the central (512 512) part of the original image (dashed square in (a)). In (b), thesmoothed image b,a f is shown as a grey-scale coded background from white (min) to black (max).

    3.2.1 Log-normalW-cascades

    In Figure 15 is illustrated the computation of the max-ima chains and the WTMMM for an individual imageof a multifractal rough surface generated with the log-normalW-cascade model described in Section 2.2. Themodel parameters are the same as the ones used in

    Figure 6a, namely m = 0.38ln2 and 2

    = 0.03ln2.Equations (35, 36) are implemented in order to approachas much as possible isotropic scaling. Again Figure 15billustrates perfectly the fact that the maxima chains cor-respond to edge curves of the original image after smooth-ing by a Gaussian filter. From the WTMMM defined onthese maxima chains, one constructs the WT skeleton ac-cording to the procedure described in Section 4 of paperI [68]. From the WT skeletons of 32 (1024 1024) imageslike the one in Figure 15a, one computes the annealed av-erage of the partition functionsZ(q, a). As shown in Fig-ure 16a, when plotted versusthe scale parameter a in alogarithmic representation, these annealed average parti-tion functions display a rather impressive scaling behaviorover a range of scales of about 4 octaves (i.e., W a 16W, whereW= 13 pixels). Let us point out that scal-

    ing of quite good quality is found for a rather wide rangeof values ofq: 6 q 8. When processing to a linear re-gression fit of the data over the first four octaves, one getsthe(q) spectrum () shown in Figure 16c. For the rangeofqvalues where scaling is operating, the numerical dataare in remarkable agreement with the theoretical nonlin-ear(q) spectrum given by equation (29). Similar quanti-tative agreement is observed on theD(h) singularity spec-

    trum in Figure 16d. Let us note that consistant parabolicshapes are obtained when using either the Legendre trans-form of the (q) data or the formula (68) and (69) ofpaper I [68] to compute h(q) and D(q). In Figure 16b arereported the results for the expectation valuesh(q, a) (Eq.(66) of paper I [68])vs.log2 a; it is clear on this figure thatthe slopeh(q) depends upon q, the hallmark of multifrac-tal scaling. Note that again, the theoretical predictionsh(q) = /q =2q/ ln 2 m/ ln2 provide very satis-factory fits of the numerical data. From equation (32), themultifractal rough surfaces under study, display intermit-tent fluctuations corresponding to Holder exponent valuesranging from hmin = 0.034 to hmax = 0.726. Unfortu-nately, to capture the strongest and weakest singularities,

    one needs to compute the (q) spectrum for very largevalues of|q|. This requires the processing of many more

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    Fig. 16. Determination of the (q) and D(h) spectra of mul-tifractal rough surfaces generated with the log-normal ()and log-Poisson () randomW-cascade models, using the 2D

    WTMM method. is the first-order radially symmetric ana-lyzing wavelet used in Section 3.1. (a) log2Z(q, a) vs. log2 a;the solid lines correspond to linear regression fit of the dataover the first four octaves. (b) h(q, a) vs. log2 a; the solid linescorrespond to linear regression fit estimates ofh(q). (c)(q)vs.qas obtained from linear regression fit of the data in (a) overthe first four octaves. (d) D(h) vs. h, after Legendre trans-forming the (q) curve in (c). In (c) and (d), the solid linesrepresent the theoretical log-normal spectra (Eqs. (29, 30));the log-Poisson predictions (Eqs. (37, 38)) are represented bythe dashed lines.

    images of much larger size, which is out of our current

    computer capabilities. Note that with the statistical sam-ple studied here, one has D(h(q= 0) = 0.38) = 2.000.02,which allows us to conclude that the rough surfaces underconsideration are singular everywhere.

    From the construction rule of these synthetic log-normal rough surfaces (Sect. 2.2.2), the multifractal na-ture of these random functions is expected to be containedin the way the shape of the WT modulus pdf Pa(M)evolves when varying the scale parameter a, as shownin Figure 17a. Indeed the joint probability distributionPa(M, A) is expected to factorize, as the signature of theimplicit decoupling ofM andA in the construction pro-cess. This decoupling is numerically retrieved in Figure 18where, for two different scales, the pdf ofM, when con-ditioned by the argument A, is shown to be shape invari-

    Fig. 17. Pdfs of the WTMMM coefficients of synthetic mul-tifractal rough surfaces generated with the log-normal W-cascade model (m= 0.38ln 2 and2 = 0.03 ln2). (a)Pa(M)vs M. (b) Pa(A) vs. A. is a first-order radially symmetricanalyzing wavelet. Four different scales a = 1, 2, 4, 8 (in Wunits) are shown. These results correspond to averaging over32 (1024 1024) images.

    Fig. 18. Pdfs ofM when conditioned by A. The differentcurves correspond to fixing A (mod ) to 0/8,/4/8,/2 /8 and 3/4 /8. (a) a = 20.1; (b) a = 21.1 (in Wunits). Same 2D WTMM computations as in Figure 17.

    ant. For further evidence of this statistical independencebetween M and A, we refer the reader to Figure 19 wherethe WTMMM are plotted in the (T1 , T2) plane. Whenvarying the scale parameter a, no significant angular de-pendent evolution is observed in the distribution of theWTMMM. As seen in Figure 17b, Pa(A) does not exhibitany significant change when increasing a, except some lossin regularity at large scales due to the rarefaction of the

    maxima lines. Let us point out that, even though Pa(A)looks globally rather flat, one can notice some small am-plitude almost periodic oscillations at the smallest scaleswhich reflects the existence of privileged directions in thewavelet cascading process. These oscillations are maxi-mum forA = 0, /2, and 3/2, as the witness to thesquare lattice anisotropy underlying the 2D wavelet treedecomposition.

    Another way to evidence multifractality is to reportthe failure of the self-similarity relationship (39). As ex-perienced in Figure 14b for the fractionally integrated sin-gular cascades, we have tried to find a value ofhsuch that,when rescaling the WTMMM by ah, all the pdf Pa(

    M)

    shown in Figure 20a in a semi-logarithmic representation,collapse onto a single curve. Clearly, we have not been able

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    Fig. 19. Distribution of the WTMMM in the plane (T1 , T2)for the following values of the scale parameter: a = 1 (a), 2 (b),4 (c) and 8 (d) in W units. Same 2D WTMM computationsas in Figure 17.

    Fig. 20. Pdfs ofM as computed at different scales a = 1, 2,4 and 8 (in W units). (a) lnPa(M) vs. M. (b) lnPa(M) vs.M/ah(q=0) with h(q= 0) = 0.38. Same 2D WTMM computa-tions as in Figure 17.

    to find such a remarkable exponent. As reported in Fig-ure 20b, when using the expected most frequent Holderexponent h(q= 0) = /q

    |q=0 =

    m/ ln2 = 0.38, the

    right tail of Pa(M) shrinks to smaller values ofM/ahwhena is increased. Although this evolution does not lookvery spectacular, its clearly deviates from the perfect col-lapse found for fBm surfaces in Figure 27b of paper I, thehallmark of monofractal scaling properties.

    3.2.2 Log-Poisson W-cascades

    We have reproduced our 2D WTMM statistical analysis on32 (10241024) images of multifractal rough surfaces gen-erated with the log-PoissonW-cascade model describedin Section 2.2.2, with the parameter values = 1/9ln2, = (2/3)1/3 and = 2 ln 2. The results so obtained are

    of the same good quality as those reported before for log-normal W-cascades. As shown in Figures 16c and 16d, the(q) and D(h) spectra obtained with the log-PoissonW-cascades () are found in remarkable agreement with thecorresponding theoretical spectra (- - -) for values ofq inthe range6 q 8, where well-defined scaling is ob-served. Let us remark that in this range ofqvalues, for theconsidered sets of parameter values (which for both log-normal and log-PoissonW-cascades are directly inspiredfrom the multifractal 1D WTMM analysis of the longi-tudinal velocity fluctuations in fully developed turbulentflows [25,63,81]), the theoretical (q) spectrum given byequation (37) is almost undistinguishable from the theo-retical(q) spectrum for the log-normal W-cascades (29).It is therefore not surprising that our numerical analy-sis fails to distinguish log-Poisson from log-normal mul-tifractal rough surfaces with the available statisticalsample of synthetic (1024

    1024) images. As seen in

    Figures 16c and 16d, the log-normal and log-Poisson the-oretical multifractal spectra start departing one from theother at large (q 6) and small (q4) q values. Ournumerical data also start separating one from each otherat those rather large values of|q| in a way which is quiteconsistent with the theoretical predictions. However, letus emphasize that, as previously discussed in paper I [68],finite-size effects as well as some lack of statistical con-vergence start affecting our estimates of the (q) expo-nents (and in turn the D(h) singularity spectrum) whichare no longer as reliable as they should be. This obser-vation must not be considered as some limitation of the2D WTMM method to discriminating between log-normal

    and log-Poisson statistic since, with other choices of pa-rameter values, this discrimination can be shown to bequite operational. Because of the specific choice of themodel parameters, the above observation simply indicatesthat in order to distinguish log-Poisson multifractal spec-tra from their log-normal approximations (see the discus-sion in Sect. 4), one needs to be able to investigate valuesof|q| 6 which requires many more images of much largersize than those investigated in the present study.

    In Figures 21, 22 and 23, we report the results ofthe computation of the joint probability distributionPa(M, A) = Pa(M)Pa(A), which is found to factorizeas previously observed for log-normal

    W-cascades. These

    figures have to be compared with Figures 17, 18 and19 discussed just above. The similarities between thosetwo sets of data confirm the conclusions derived fromthe multifractal spectra computations. In Figure 24, onewitnesses the impossibility of finding a value of the ex-ponent h such that, when rescalingM by ah, all thepdfs computed at different scales collapse onto a singlecurve. When using the most frequent Holder exponenth = h(q= 0) =(/ ln2 + ln / ln2) = 0.381 (a valuewhich, up to 103, is the same as for the log-normalW-cascades in Fig. 20), the right tails of these pdfs fail tosuperimpose one on the top of each other. This discrep-ancy is the signature of multiscaling, i.e., the existence

    of a continuum of scaling exponent values which accountfor the roughness fluctuations of the multifractal surfaces

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    Fig. 21. Pdfs of the WTMMM coefficients of synthetic mul-tifractal rough surfaces generated with the log-Poisson W-cascade model ( = 1/9ln2, = (2/3)1/3 and = 2 ln 2).(a) Pa(M) vs. M. (b) Pa(A) vs. A. is a first-order radi-ally symmetric analyzing wavelet. Four different scales a= 1,2, 4, 8 (in W units) are shown. These results correspond toaveraging over 32 (1024 1024) images.

    Fig. 22. Pdfs ofM when conditioned by A. The different

    curves correspond to fixing A (mod ) to 0/8,/4/8,/2 /8 and 3/4 /8. (a) a = 20.1; (b) a = 21.1 (inWunits). Same 2D WTMM computations as in Figure 21.

    Fig. 23. Distribution of the WTMMM in the plane (T1 , T2)for the following values of the scale parameter: a = 1 (a), 2 (b),

    4 (c) and 8 (d) in W units. Same 2D WTMM computationsas in Figure 21.

    Fig. 24. Pdfs ofM as computed at different scales a = 1, 2,4 and 8 (in W units). (a) lnPa(M) vs. M. (b) lnPa(M) vs.M/ah(q=0) with h(q= 0) = 0.381. Same 2D WTMM compu-tations as in Figure 21.

    under study. The single-humped D(h) singularity spec-

    trum shown in Figure 16d quantifies statistically the rel-ative contributions of the corresponding singularities ofdifferent Holder exponents.

    4 WTMM computation of the self-similaritykernel of multifractal rough surfaces

    This section is devoted to the generalization in 2D of themethod for determining the self-similarity kernelG origi-

    nally introduced by Castaing and co-workers [24,8288] inthe context of the analysis of the velocity increment pdfin fully developed turbulent flows. Note that this methodhas been extended in previous works to the correspondingWTMM pdf [6163,81,93]. Our goal here is to pave theway from 1D to 2D studies, under some specific assump-tions that make this generalization rather straightforward.Our working hypothesis will be the factorization of thejoint probability distribution:

    Pa(M, A) = Pa(M)Pa(A). (40)

    This hypothesis of statistical independence between themodulus and the argument of the WTMMM, is satisfiedby construction when investigating the WT skeleton ofthe multifractal rough surfaces generated with the ran-domW-cascade models discussed in Section 2.2. We willsee in paper III [73], that this hypothesis is also quite re-alistic when analyzing high resolution satellite images ofstratocumulus radiance fields [103]. Our strategy will thusconsists in computing a kernel G which will account forthe evolution of the shape ofPa(M). As far as Pa(A) isconcerned, either this distribution is scale independent, aspreviously observed in Figures 17b and 21b, or it is foundto evolve as a function of a (or more likely of ln a) andsome important issue will be to explicit the underlyingevolution equation (or the underlying dynamical systemwhen dealing with discrete cascading process).

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    4.1 A method to determine the self-similarity kernel

    Under the hypothesis stated in equation (40), Castainget al.ansatz [24,82] can be revisited along the lines drawn

    in references [6163], i.e., by taking advantage of thespace-scale decomposition provided by the WT skeleton.Because of the implicit assumption of the existence of anunderlying multiplicative structure, the WTMMM pdf ata given scale a can be expressed as a weighted sum ofdilated pdfs at a larger scale a > a:

    Pa(M) =

    Gaa(u)Pa(euM)eudu , fora > a .

    (41)

    For any decreasing sequence of scales (a1,...,an), the ker-nelG satisfies the decomposition law:

    Gana1 =Ganan1 ... Ga2a1 , (42)where denotes the convolution product. According toCastaing et al. [24,82], the cascade is self-similarif thereexists a decreasing sequence of scales{an} such thatGanan1 = G is independent of n. The cascade is saidcontinuously self-similar [24,82] if there exists a posi-tive, decreasing function s(a) such that Gaa dependsupon a and a only through s(a, a) = s(a) s(a), i.e.,Gaa(u) = G(u, s(a, a)).s(a, a) actually accounts for thenumber of elementary cascade steps from scale a to scalea. (s(a) can be seen as the number of cascade steps fromthe integral scale L down to the considered scale a.) InFourier space, the convolution property (Eq. (42)) turns

    into a multiplicative property for G, the Fourier transformofG:

    Gaa(p) = G(p)s(a,a) , fora > a . (43)

    From this equation, one deduces that G has to be thecharacteristic function of an infinitely divisible pdf. Sucha cascade is referred to be a log-infinitely divisible cas-cade [21,24]. According to Novikovs definition [21], thecascade is scale-similar(or scale-invariant) if:

    s(a, a

    ) = lna

    a

    , (44)

    i.e., s(a) = ln(L/a).

    Remark

    In their original work, Castaing et al.[82] have developeda formalism, based on an extremum principle, which turnsout to be consistent with the Kolmogorov-Obukhov (1962)general ideas of log-normality [13,14], but which predictsan anomalous power-law behavior of the depth of the cas-cade. This departure from scale invariance has been con-firmed experimentally [6163,81,93] from the computa-tion of the kernelGaa of the WTMM pdf which corrobo-rates that s(a) =

    (L/a) 1 /, whereis a Reynolds

    number dependent exponent which quantifies the devi-ation from scale-similarity (scale-invariance being ulti-mately restored for 0, in the limit of infinite Reynoldsnumber).

    Our numerical estimation ofG [62,63] is based on thecomputation of the characteristic function M(p, a) of theWTMMM logarithms at scale a [103]:

    M(p, a) =

    eip lnMPa(M) dM . (45)

    From equation (41), it is easy to show that G satisfies:

    M(p, a) = Gaa(p)M(p, a). (46)

    After the WT calculation and the WTMMM detection,the real and imaginary parts of M(p, a) are computed

    separately as < cos(p ln M) > and < sin(p ln M) > re-spectively. The use of the WT skeleton instead of the con-tinuous wavelet transform prevents M(p, a) from gettingtoo small, as compared to numerical noise, over a reason-able range of values ofp, so that Gaa(p) can be computedfrom the ratio:

    Gaa(p) = M(p, a)

    M(p, a) (47)

    4.2 Computing the self-similarity kernel of multifractalrough surfaces generated with randomW-cascades

    The aim of this sub-section is to perform test applica-tions of our WTMMM methodology to compute the self-similarity kernel using as guinea pigs, the synthetic multi-fractal rough surfaces generated by the random W-cascademodels introduced in Section 2.2.

    4.2.1 Uncovering a continuously self-similar cascade

    In order to test the validity of equation (43), one first has

    to focus on the scale dependence ofGaa as calculated

    with equation (47). Figures 25a and 25c respectively showthe modulus logarithm ln |Gaa | and the phase aa , ofGaa , for various pairs of scales a < a

    , as computed fromthe WT skeletons of 32 (1024 1024) images of syntheticlog-normal rough surfaces like the one illustrated in Fig-ure 6a. In Figures 25b and 25d respectively, we succeed incollapsing all different curves in Figures 25a and 25c onto

    a single kernel G(p) = G1/s(a,a)aa , withs(a, a

    ) = ln(a/a)in good agreement with equations (43, 44) and the con-tinuously scale-invariant self-similar cascade picture. Letus point out that this collapse deteriorates for|p| 5, asthe consequence of finite-size effects as well as of a lack ofstatistics. As illustrated in Figure 27, similar observationsalso apply for synthetic log-Poisson multifractal rough sur-faces.

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    Fig. 25. Estimation ofGaa(p) for 32 (1024 1024) imagesof multifractal rough surfaces generated with the log-normalW-cascade model for the parameter values m = 0.38ln2and 2 = 0.03 ln 2. The analyzing wavelet is the first-orderradially symmetric wavelet used in Figure 15. (a) ln |Gaa(p)|

    vs. p; (b) ln |Gaa(p)|/ ln(a/a) vs. p; (c) aa(p) vs. p; (d)

    aa(p)/ ln(a/a) vs. p. The symbols correspond to the follow-

    ing pairs of scale (in W units): a = 20.5, a = 23.5 (), a = 2,

    a = 23.5 ( ), a = 2, a = 23 ( ), a = 21.5, a = 23 (). In (a)and (c), the solid lines correspond to the theoretical predictions

    for Gaa(p) (Eq. (55)) for a = 2, a = 23 ( ). In (b) and (d),

    the solid lines correspond to the theoretical self-similar kernelG(p) = exp

    (imp

    2

    p

    2

    /2)/ ln 2

    .

    4.2.2 Discriminating between log-normal and log-Poissoncascades

    The relevance of equation (43) being established, let usturn to the precise analysis of the nature ofG. Using theTaylor series expansion of ln G(p) [62, 63]:

    G(p) = exp

    k=1

    ck(ip)k

    k!

    , (48)

    equation (43) can be rewritten as:

    Gaa(p) = exp

    k=1

    s(a, a)ck(ip)k

    k!

    , (49)

    where the real valued coefficients ck are the cumulantsofG.

    Log-normal W-cascade process

    Let us come back to the construction rule of the 2DW

    -cascades defined in Section 2.2. Let Pj the pdf of thewavelet coefficient modulus dj,m,n (Eqs. (24, 25)). Let

    P(log)j (u) be the pdf of ln dj :

    P(log)j (u) = e

    uPj(eu). (50)

    Ifj1> j2, then from equation (28):ln dj2 = ln dj1 + ln Mj11+ .. + ln Mj2 . (51)

    This equation can be rewritten as

    P(log)j2 (u) = P(log)j1

    Gj2j1(u), (52)whereGj2j1(u) = G G...G,G(u) being the pdf of ln M.In the Fourier space, one gets

    P(log)j2

    (p) = P(log)j1

    (p)Gs(j2,j1)(p), (53)

    where

    s(j2, j1) = j1 j2 (54)represents the number of steps of the cascade from thescale 2j1 to the scale 2j2 . These rather straightforwardcomputations demonstrate the scale-invariance and theself-similarity of the random W-cascades.

    In the particular case of a log-normalW-cascade,the self-similarity kernel G(u) is expected to be Gaus-sian [6164] which implies the following behavior of theself-similarity kernel in Fourier space:

    Gaa(p) = exp imp

    ln 2

    2p2

    2 l n 2 lna

    a . (55)From equation (49), this Gaussian kernel corresponds tothe following set of cumulants:

    c1= m/ ln 2, c2= 2/ ln 2, ck= 0 fork 3. (56)

    As shown in Figure 26, those cumulants can be extractedfrom the behavior of:

    C2n+1(a, a) = (1)n2n+1aa/p2n+1|p=0, (57)

    and

    C2n+2(a, a) = 2n+2 ln

    |Gaa

    |/p2n+2

    |p=0, (58)

    for n 0, as functions of ln(a/a). When plottingC1(a, a)vs. ln(a/a), one can see in Figure 26a that for differ-ent values of the reference scale a, all the points ob-tained when varying the scale a fall on a unique straightline which matches perfectly the theoretical predictionC1(a, a

    ) = m ln(a/a)/ ln2 =0.38 ln(a/a). Very goodagreement is also found forC2(a, a

    ) = 2 ln(a/a)/ ln2 inFigure 26b, where the theoretical value 2/ ln 2 = 0.03provides a nice fit of the slope of the numerical data.In Figure 26c are shown the results for C3(a, a

    ) whichtheoretically should be equal to zero (Eq. (56)). Despitesome fluctuations resulting from finite-size effects and thelack of statistical convergence, all the data points fall in arather narrow neighborhood of the horizontal line corre-sponding toc3= 0.

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    Fig. 26. Computation of the cumulants of the self-similarity kernel G of multifractal rough surfaces generated with the log-normalW-cascade model. Same computations as in Figure 25. (a) C1(a, a

    ) vs. ln(a/a); (b)C2(a,a) vs. ln(a/a); (c) C3(a, a

    )vs.ln(a/a). The symbols correspond to the following values of the reference scale a = 20.5 (), 2 (), 21.5 (), 22 ( ), 22.5 ( ), 23

    () and 23.5 (N ), in W units. The solid lines correspond to the theoretical slopes c1 = m/ ln 2 = 0.38 (a),c2= 2/ ln2 = 0.03

    (b) and c3= 0 (c).

    Log-Poisson W-cascade process

    A log-Poisson W-cascade is characterized by the followingkernel shape [6164]:

    Gaa(p) = exp

    cos(p ln ) 1ln 2

    + i

    p + sin(p ln )

    ln 2

    ln

    a

    a

    , (59)

    where , and are parameters (see Sect. 2.2). Thislog-Poisson kernel corresponds to the following set of cu-mulants:

    c1 =+ ln

    ln 2 andck =

    (ln )k

    ln 2 fork 2. (60)

    Note that the log-Poisson process reduces to a log-normalcascade for|p ln | 1, i.e., in the limit 1 and(ln )2 2, where the atomic nature of the quantizedlog-Poisson process vanishes. In Figure 28 are reportedthe results of the computation ofC1(a, a

    ), C2(a, a) and

    C3

    (a, a) from the numerical log-Poisson self-similarity

    kernel Gaa(p) shown in Figure 27, for different valuesof the reference scale a. In Figures 28a and 28b, all thedata points obtained for C1(a, a

    ) and C2(a, a), remark-

    ably fall on a straight line when plotted versus ln(a/a).This is a clear numerical evidence for the scale-similarityof the underlying multiplicative process. Moreover, theslopes of these straight lines are found in very good agree-ment with the theoretical predictions for the first twocumulants (Eq. (60)): c1 = (+ ln )/ l n 2 =0.381and c2 = (ln )2/ ln 2 = 0.036. Although we are againfaced with finite-size effects and some lack of statisti-cal convergence, when comparing the results obtainedfor C3(a, a

    ) in Figure 28c to those reported in Fig-ure 26c for the log-normalW-cascades, one observes adefinite tendency of the numerical data to fall about

    Fig. 27. Estimation ofGaa(p) for 32 (1024 1024) imagesof multifractal rough surfaces generated with the log-PoissonW-cascade model for the parameter values = 1/9ln2, = (2/3)1/3 and = 2 ln2. Same representation as inFigure 25. In (a) and (c), the solid lines correspond to the

    theoretical predictions for Gaa(p) (Eq. (59)) for a = 2,a = 23 ( ) in W units; the dashed lines correspond to the

    log-normal approximation ofGaa(p). In (b) and (d), the solid

    lines correspond to the theoretical self-similar kernel G(p) =exp

    ,

    (cos(p ln) 1) + i(p + sin(p ln))

    / ln 2

    ; thedashed lines correspond to the log-normal approximationG(p) = exp

    (i(+ ln )p (ln)2p2/2)/ ln 2

    .

    a straight line of negative slope when plotted versusln(a/a). This is a strong indication that the third cu-mulant c3 is different from zero and that the cascadingprocess is no longer log-normal. Furthermore, as shown inFigure 28c, the theoretical value c3 = (ln )3/ l n 2 =0.0049 provides a reasonable fit of the data. The rele-vance of the theoretical self-similarity kernel (Eq. (59)) is

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    Fig. 28. Computation of the cumulants of the self-similarity kernel G of multifractal rough surfaces generated with the log-Poisson W-cascade model. Same computations as in Figure 27. (a) C1(a, a

    )vs. ln(a/a); (b)C2(a, a) vs. ln(a/a); (c) C3(a, a

    )vs. ln(a/a). The symbols correspond to the following values of the reference scale a = 20.5 (), 2 (), 21.5 (), 22 ( ), 22.5

    ( ), 23 () and 23.5 (N ), in W units. The solid lines correspond to the theoretical slopes c1 = (+ ln)/ ln2 = 0.381 (a),

    c2= (ln)2

    / ln2 = 0.036 (b) and c3 = (ln)3

    / ln 2 = 0.0049 (c).

    Fig. 29. Pdfs ofM as computed at different scales a = 1, 2, 4 and 8 (in W units). (a) ln(Pa(M)) vs. M; same data forlog-normal W-cascades as in Figure 20a, after being transformed according to equation (41) with the Gaussian kernel estimatedin Figures 25 and 26. (b) ln(Pa(M)) vs. M; same data for log-Poisson W-cascades as in Figure 24a, after being transformedwith a Gaussian kernel build from the first two cumulants c1 and c2 estimated in Figure 28. (c) Same pdfs as in (b) but afterbeing transformed with a kernel involving the first three cumulants c1, c2 and c3 estimated in Figure 28.

    confirmed in Figures 27b and 27d, where the correspond-ing solid lines are found in good agreement with the datafor 5 p 5. Note that the presence of a nonzero third-order cumulant is hardly perceptible in Figure 27d, wherethe dashed line corresponding to the log-normal linear ap-

    proximation ofaa cannot be discarded by the data ex-cept for values of|p| 4. Unfortunately, for these ratherlarge values of p, one can no longer trust quantitativelyour numerical data because of insufficient sampling.

    As a check of the reliability of our numerical methodto compute the self-similarity kernel Gaa(u), we have suc-ceeded in Figure 29a, in collapsing all the WTMMM pdfscomputed at different scales in Figure 20a for log-normalW-cascades, onto a single curve when using the convo-lution equation (41) with a scale-invariant Gaussian ker-nel (Eq. (55)).We have tried to do the same remarkableoperation for the WTMMM pdfs obtained for the log-Poisson

    W-cascades in Figure 24a. When using the log-

    normal approximation of the log-Poisson kernel, takinginto account c1 and c2 only, one clearly fails in Figure

    29b to collapse the right tails of theM pdfs onto a sin-gle curve. When putting into the game the value of thethird cumulantc3 estimated from a linear regression fit ofthe data in Figure 28c, the collapse is improved but stillnot perfect because of the statistical uncertainty in our

    estimate ofc3 when considering a statistical sample of 32(1024 1024) images only (note that in paper III [73] wewill have at our disposal 32 (10241024) images of cloudyscenes collected with Landsat satellite).

    Remark

    From the definition of the characteristic functionM(p, a) (Eq. (45)), one gets the following relationship withthe partition functionZ(q, a) involved in the multifractalWTMM description (Eq. (70) of paper I [68]) [63]:

    S(q, a) =Z(q, a)

    Z(0, a)=

    Mq

    (a) =M(

    iq, a),

    a(q)+2 , (61)

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    whereZ(0, a) a2, as previously discussed for multi-fractal rough surfaces that are singular everywhere. Fromthe expression (47) of the Fourier transform of the kernelGand equation (61), one deduces:

    S(q, a)

    S(q, a) = Gaa(iq). (62)

    When further using equation (49), this last equation be-comes

    S(q, a)

    S(q, a) =

    a

    a

    P k=1

    ckqk

    k!

    , (63)

    which is consistent with the scaling behavior ofS(q, a) inequation (61) provided

    (q) = k=1

    ck q

    k

    k! 2. (64)

    This general expression for the (q) spectrum reduces tothe specific spectra found in equations (29,37), when re-placing the cumulants ck by their respective analyticalexpressions found in equations (56, 60) for log-normaland log-PoissonW-cascades. Let us note that with thechoices of parameter values made in our numerical simula-tions, the first two cumulants of log-normal (c1 = 0.380,c2 = 0.030) and log-Poisson (c1 =0.381, c2 = 0.036)W-cascades are so similar that the respective (q) spec-tra mainly differ by the contribution of the cumulants of

    order k 3. This explains, a posteriori, the difficultiesencountered with the 2D WTMM method in Section 3 todistinguish the multifractal spectra of the correspondinglog-normal and log-Poisson rough surfaces.

    5 Space-scale correlation functions fromwavelet analysis

    5.1 Space-scale correlation functions

    Correlations in multifractals have already been experi-enced in the literature [104106]. However, all these stud-ies rely upon the computation of the scaling behavior ofsome partition functions involving different points; theythus mainly concentrate on spatial correlations of the lo-cal singularity exponents. The approach recently devel-oped in references [81,94,97] is different since it does notfocus on (nor suppose) any scaling property but ratherconsists in studying the correlations of the logarithms ofthe amplitude of a space-scale decomposition of the sig-nal. More specifically, if(x) is a bump function such that||||1= 1, then by taking

    2

    (x

    , a) =a

    4

    (x

    y

    )/aT

    [f](y

    , a)2

    d

    2y,(65)

    one has

    ||f||22=

    2(x, a)d2xda . (66)

    Thus,2(x, a) can be interpreted as the local space-scaleenergy density of the considered multifractal functionf(x). Since 2(x, a) is a positive quantity, we can definethe magnitudeof the function fat the point x and scalea as:

    (x, a) = 1

    2ln 2(x, a). (67)

    Our aim in this section, is to show that a multiplicativeprocess can be revealed and characterized through the cor-relations of its space-scale magnitudes:

    C(x1

    ,x2

    ; a1

    , a2

    ) =(x

    1, a

    1)(x

    2, a

    2), (68)

    where ...stands for ensemble average and for the cen-tered process .

    Remark

    Note that instead of working with the continuous wavelettransform, one can use its skeleton defined by theWTMMM. Within this alternative, the magnitude is sim-ply (x, a) = ln M(x, a), where the point (x, a) is by def-inition a WTMMM.

    5.2 Analysis of randomW-cascades using space-scalecorrelation functions

    The tree structure of aW-cascade defined on separablewavelet orthogonal basis as explained in Section 2.2, in-duces correlations between different details of the cor-responding function f(x) [9497]. These correlations canbe characterized by computing the correlation betweentwo wavelet coefficients at an arbitrary scale aj = 2j

    and at a distance |x| = x = 2jk, where for thesake of simplicity, we will focus on spatial separations inthe x-direction only (we will see later on, that our the-oretical predictions are independent of the chosen direc-tion provided the function f(x) displays isotropic scalingproperties). Let us fix k = (kx, ky) and let k1 = k andk2 = k+k, where k = (k, 0). Let us suppose thatthe last common ancestor (on the quaternary tree of theW-cascade) of the wavelet coefficient modulus dj,k1 anddj,k2 (Eqs. (24, 25)) is at a scale 2

    Nl(j,k1,k2), where in thefollowingl(j,k1,k2) will be referred as the ultrametric W-distance between two wavelet coefficients [64]. Then, fromequation (25), one can write:

    dj,k1 =M(N1)...MNl(j,k1,k2)

    MNl(j,k1,k2)1

    [1]

    ...M(j)[1]

    , (69)

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    and

    dj,k2 =M(N1)...M

    Nl(j,k1,k2)

    MNl(j,k1,k2)1[2] ...M(j)[2] , (70)where all the M(i), M(

    i)[1]

    and M(i)

    [2]are i.i.d. random vari-

    ables with the same law P(M) as M. Then their covari-ance is

    Cov(ln dj,k1, ln dj,k2) =

    l(j,k1,k2)i=1

    Cov

    ln M(Ni), ln M(Ni)

    =2l(j,k1,k2), (71)

    where 2 is the variance of ln M. But the ultrametricstructure of the

    W-cascades shows that such a process

    is not stationary (nor ergodic). Moreover, we will gener-ally consider uncorrelated realizations of size L = 2N ofthe same cascade process, so that, in good approximation,the correlation function depends only on the spatial dis-tancex = |x|. Thus one can replace ensemble averageby space average in the definition of the correlation func-tion [64]:

    C(xj,2p ; aj) = 4jN2

    l=Nj1l=0

    Nj,2p(l)l , (72)

    where Nj,2p(l) is the number of wavelet coefficients dj,k(0

    kx

    2j

    2p) such that d

    j,k and d

    j,k , where k =

    (kx, ky) and k = (kx+ 2p, ky), are at theW-distancel. Itis clear thatNj,2p(l) = 0 forl Nj p. Moreover, onecan easily show that:

    l < Nj p , Nj,2p(l) = 4pNjp,1(l). (73)

    Since Nj,1(l) = 2Nj+l, equation (72) becomes ( p sup(a1, a2), the mag-nitudes ofBH=1/3(x) are found uncorrelated.

    Remark

    We have reproduced our space-scale correlation analysisfor multifractal rough surfaces generated with the log-Poisson W-cascade model. When using the model param-eters defined in Figure 6b, one gets numerical data thatcannot be distinguished from those shown in Figure 30afor log-normalW-cascades. This can be explained fromthe fact that the variance parameter 2 = (ln )2 =0.025 is very close to the value 2 = 0.0 3 l n 2 = 0.021used for generating the log-normal rough surfaces. This isalso an indication that in order to discriminate betweenlog-Poisson and log-normal multiplicative processes, oneneeds to investigate multi-point space-scale correlationfunctions in order to extract information about the higher

    moments of the multiplier probability law that governs theW-cascading process.

    6 Perspectives

    To summarize, we have reported the results of numericalapplications of the 2D WTMM method to synthetic mul-tifractal rough surfaces. We have tested the reliability ofthis method to measure the (q) and D(h) spectra whichare at the heart of the multifractal description of multi-affine rough surfaces. We have further proposed a methodto compute the self-similarity kernel which provides addi-tional information on the nature of the underlying multi-plicative process (e.g., about the multiplier pdf) as wellas some subsequent tests of the scale-similarity propertiesof this process. Finally, we