Béatrice Pesquet-Popescu and Jacques Lévy Véhel · 2016-01-22 · of f at t 0 translates intoα...

Béatrice Pesquet-Popescuand Jacques Lévy Véhel

re fractals everywhere? A very often cited ex-ample of a fractal curve is the Brittany coast inFrance. The view one has by looking at thishighly irregular coast at coarse scale is in-

deed quite similar to what can be observed by lookingmore precisely at a small part of the sea shore. In a fa-mous book [1], it was further claimed that fractals areeverywhere.

Actually, recently, image processing using fractalmodels has represented an active research area stimu-lated by a plethora of applications [2] (infographics [3];geophysics [4], [5]; turbulence phenomena [6], [7];satellite imagery, texture modeling, classification andsegmentation [8], [9]; compression, watermarking,etc.) where these concepts are potentially interesting.Fractals correspond to the general idea, which can beeasily understood from an intuitive point of view, that agiven object, especially a textured area, can be repre-

sented by similar characteristics “repeated” at differentscales. This versatile idea, however, can be translatedinto different forms from a mathematical point of view.In this tutorial, we are interested in a stochastic view offractals, which will focus on more or less sophisticatedmodels for describing image textures. As will be shown,these models rely on the notion of statistical “self-simi-larity.” This concept is at the origin of relevant modelsfor several natural phenomena. In the meantime, thenonstationary structure of self-similar processes putthem at the core of the most recent preoccupations inimage and signal processing.

Our trip in fractal landscapes will depart from the sim-plest but yet effective model of fractional Brownian mo-tion and explore its two-dimensional (2-D) extensions.We will focus on the ability to introduce anisotropy in thismodel, and we will also be interested in considering itsdiscrete-space counterparts. We will then move towards

48 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 20021053-5888/02/$17.00©2002IEEE

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recent multifractional and multifractal models providingmore degrees of freedom for fitting complex 2-D fields.

We note in this introduction that many of the modelsand processing described below are implemented inFracLab, a software MATLAB/Scilab toolbox for fractalprocessing of signals and images. FracLab is available athttp://www-rocq.inria.fr/fractales/.

A Simple Example of a Fractal Field:The Fractional Brownian Motion

Reminders About the1-D Fractional Brownian MotionFractional Brownian motion (fBm) is one of the mostpopular stochastic fractal models for images. It was intro-duced by Kolmogorov and studied by Mandelbrot andVan Ness in 1968 [4]. It is a nonstationary process, but ithas stationary increments. In one dimension, it is the onlyself-similar (i.e., “fractal”) Gaussian process with station-ary increments.

Let us briefly recap some of the most important prop-erties of the one-dimensional (1-D) fBm to compare andto extend it to two (or higher) dimensions. A process{ ( ), }B t t RH ∈ is a 1-D fBm if it is Gaussian, zero-mean,and its autocorrelation function is given by

[ ]R t s B t B s

t s t s

B H H

H H H

H( , ) { ( ) ( )}

,

=

= + − −

E

σ 22 2 2

2 (1)

where 0 1< <H is the fractal scaling parameter, also calledthe Hurst parameter. For H =1 2/ we obtain thewell-known Brownian motion (or Wiener process), de-noted by B t( ). One of the most important properties of theBrownian motion is the independence of its increments.

The fBm is a random process that is continuous in themean-square sense. It admits several integral representa-tions. The most popular one, as introduced in [4], is

[ ]B t t s s dB s

t s d

HH H

Ht

( ) ( ) ( ) ( )

( )

/ /

/

∝ − − −

+ −

−∞

− −

−

∫∫

0 1 2 1 2

1 2

0B s( ).

(2)

Equation (2) allows us to interpret the fBm as a deriva-tive of fractional order 1 2/ − H of Brownian motion.Another useful representation is the “spectral representa-tion” of the fBm [10].

The so-called “power law” or “T H law” of the fBmstates that the variance of the increments of B tH ( )is givenby E{[ ( ) ( )] } | |B t B tH H

H+ − =τ σ τ2 2 2 . We will call thisquantity the “structure function” of the fBm. As will bediscussed later, it also plays an important role in the studyof more general processes with stationary increments. Asthe fBm is nonstationary (and therefore one cannot de-fine its power spectrum), but it has stationary increments,it is more convenient to study the characteristics of thisprocess using the autocorrelation and the spectrum of itsincrements. This approach was used in [11] to define the“generalized power spectrum” of B tH ( ), which is propor-tional to| | ,( )ω − + < <2 1 0 1H H .

The first-order increment of the sampled fBm is thefractional Gaussian noise (fGn):G k B kH H( ) ( ; )= =∆ 1B k B k kH H( ) ( ), .− − ∈1 Z It is a stationary process, whoseautocorrelation function decays hyperbolically as| |k H2 2−

(for H ≠1 2/ ), and therefore the discrete-time process ex-hibits a short-range dependence for 0 1 2< <H / , inde-pendence for H =1 2/ (Brownian motion), and long-range dependence for 1 2 1/ < <H . Moreover, as all sec-ond-order self-similar processes with stationary incre-ments have the same second-order statistics as the fBm, itfollows that their increments have all the sameautocorrelation function as the fGn. The power spectrumdensity (psd) of the fGn is given by S( ) /| |ω ω∝ −1 2 1H .

Another important property of fBm is that its localregularity, as measured by the Holder exponent α [12], iswith probability one equal to H everywhere. Let us ex-plain the geometrical meaning of this statement. Roughlyspeaking, saying that a function f has exponent α at t 0means that, around t 0 , the graph of f “looks like” thecurve t f t c t ta ( ) | |0 0+ − α in the following sense: For anypositive ε, there exists a neighborhood of t 0 such that thepath of f inside this neighborhood is included in the en-velope defined by the two curves t f t c t ta ( ) | |0 0+ − −α ε

and t f t c t ta ( ) | |0 0− − −α ε , while this property is no lon-ger true for any negative ε (see Fig. 1). Thus, a “large” αmeans that f is smooth at t 0 , while an irregular behavior

SEPTEMBER 2002 IEEE SIGNAL PROCESSING MAGAZINE 49

� 1. Graphical interpretation of the Holder regularity of a func-tion f at a point t0.

Fractals correspond to thegeneral idea that a given object,especially a textured area, canbe represented by similarcharacteristics “repeated” atdifferent scales.


of f at t 0 translates into α close to zero. For instance, in2-D, pixels having an exponent smaller than two are “ir-regular,” while an image is smooth in regions where allthe pixels have α >2. In the case of fBm, α( )t H0 = for allt 0 with probability one, and thus the larger H is, thesmoother the path of the process will look.

The classical 1-D fBm can be generalized to images ormultidimensional processes in several ways, dependingon the property one is looking for.

Multidimensional Extensions of the fBmLet us first keep in mind the Gaussianity assumption inthe original definition of the 1-D fBm and describe thecorresponding processes by their correlation function.

The extension to two or higher dimensions, however, isnot unique. For example, a possible generalization is themultiparameter Wiener process, or Brownian sheet, de-noted by { ( ), }W N

Hx x ∈R . It is an anisotropic extension,

which may have different Hurst parameters in each of theN directions, gathered in the vector parameterH= ( , , )H H N1 K . The idea to generalize the definition offBm in (1) is here to set the covariance between the pointsx = ≤ ≤( )xi i N1 and y = ≤ ≤( )y i i N1 inR N equal to the productof covariances between the components of the two vec-tors. This is similar to what is done for defining theBrownian sheet. We get the fractional Brownian sheet,which is thus defined to be the centered Gaussian processwith covariance function

( )E F F x y x yi

N

iH

iH

i iHi i i{ ( ) ( )} | | | | | |

H Hx y ∝ + − −

=∏ 1

2 2 2 .

This field also admits a harmonizable representation,which is a separable extension of the 1-D one.

Another generalization of an fBm for 2-D surfaces,called isotropic fractional Brownian field or Lévy frac-tional Brownian field, is the Gaussian zero-mean field BHwith autocorrelation function

( )R BH H H

H( )x,y x y x -y∝ + −2 2 2 ,

(3)

where0 1< <H and ⋅ is the usual Euclidean norm inR 2 .Note that the 1-D process obtained by “cutting” an iso-tropic 2-D fBm with a line passing through the origin ofthe plane is a 1-D fBm with the same self-similarity index.As said above, the interpretation of the parameter H is re-lated to the “roughness” of the fBm image: the closer Hto zero, the rougher the image (the more similar to awhite noise) and the closer H is to one, the smoother thecorresponding texture (cf. Fig. 2).

The two (or n)-dimensional isotropic fBm can also bedefined, as in the 1-D case, as a stochastic integral existing inthe mean-square sense [11]. The basic interpretation of sucha construction is that the increments of the fBm are gener-ated by passing a 2-D white noise through a bidimensionalfractional filter. For an isotropic field, the fractional filter isdefined by its frequency response G H N( ) /ω ω= − −1 2 whereN is the dimension of the space (N =2 for images). Usingthis construction, Reed et al. [11] defined a generalizedpower spectrum of an n-dimensional fBm asΦ B

H N

H( )ω ω= − −2 .

Alternatively, one can give an interpretation of theFourier transform of the correlation function (3) in termsof a psd defined as the Fourier transform of a tempereddistribution. Using this result, one can establish a link be-tween the so-obtained power spectrum of the fBm andthe expression of the averaged Wigner-Ville spectrum ofthe fBm introduced by Flandrin [13]. One shows [14]that the expected value of the Wigner-Ville distribution[15] of the 2-D fBm is the inverse Fourier transform ofthis power spectrum. Moreover, with the averagedWigner-Ville spectrum being defined as a spatial mean of

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(a)

(b)

� 2. Isotropic fBms with Hurst parameters (a) H = 0 2. and (b) 08. .


the average of the Wigner-Ville distribution, it followsthat the averaged power spectrum of the 2-D fBm is, forall nonzero frequencies, proportional to ω − −2 2H , whichis also formally consistent with the previous expression ofΦ BH

( )ω .Another possibility for building multidimensional ex-

tensions of the 1-D fBm is to define self-similar fields withstationary 2-D increments. The definition of the self-sim-ilarity in the multidimensional case is analogous to theone in 1-D: let F( )x , x ∈R 2 , be a continuous-space ran-dom field. It is called self-similar of parameter H >0 if forall a >0 we have F a a Fd H( ) ( )x x= , where =d means theequality of all its finite-dimensional probability distribu-tions. This implies that the statistical characteristics of theprocess are invariant (up to a multiplicative constant) un-der scale changes. The extension of the notion of station-ary increments is a little bit more tricky and, dependingon the definition, one can be led to fields with differentproperties. For example, in 1-D, one can see the (strictsense) stationarity of the increments as the invariance oftheir probability distribution to translations (which arethe only rigid body motions in a space with only one di-mension). In two dimensions, the extension of this prop-erty can be seen as the invariance of the probabilitydistribution of the increments F F( ) ( )x 0− with respect toall possible rigid body motions (translations, rotationsand compositions of these operations). In the sense ofthis definition, the isotropic fBm is the only self-similarGaussian field with stationary increments. If we relax theGaussianity constraint, there exist several self-similarfields with (strict-sense) stationary increments [10], [16].In particular, they may have stable distributions. Eventhough the stable fields discussed in [10] have not beenused in many image processing applications, the exten-sion of Gaussian models to heavy-tailed distributions forimages is a very promising research problem which has re-cently received increasing interest. Some hints on theon-going developments will be given in “Non-GaussianFractional Fields.”

Besides, if the stationarity is defined not with respectto all the rigid body motion but only with respect totranslations, we come up with another definition, whichwill be discussed in the next section.

Looking at AnisotropyThe extension from isotropic to anisotropic fractal imagemodels can be realized in several ways. A first method isto build anisotropic fields by linear spatial transforms ofisotropic fractal fields. This approach is discussed in “Fil-tering and Linear Spatial Transforms.” Another possibleconstruction is by spatial filtering of fields with desiredcharacteristics. Last, but not least, it is possible to adopt adirect approach by building intrinsically anisotropicfields, as was done for building the fractional Browniansheet. Another example of a direct construction will bediscussed. In all these constructions, the structure func-

tion of the fields will play an important role inintroducing and characterizing the anisotropy.

Fields with Stationary Incrementsof Fractional OrderAs mentioned above, the manner of defining incrementsin R 2 is important, as it leads to fields with differentproperties. By taking different orders of differencingalong two orthogonal directions of space, we are empha-sizing the anisotropy of the fields under study. This en-ables the construction of self-similar Gaussian fields withstationary increments other than the isotropic fBm [17].

Consider an arbitrary spatial direction ( ),∆ ∆x y . Ba-sically, the increments ∆ ∆ ∆( , ) ( , ; , )D D

x yF x y′ of fractionalorder( , )D D′ ∈ +R 2 of a 2-D continuous (respectively, dis-crete) random field F x y( , ) are obtained by filteringF x y( , ) with a linear filter whose transfer function is( ) ( ) ,1 1− −− − ′e ep D s Dx y∆ ∆ ∀ ∈( , )p s C 2 . In this frame-work, a zero-mean field F x y( , ) has (wide-sense) station-ary increments of order ( , )D D′ if it is with finite variancesand, for all ( , , , )∆ ∆ ∆ ∆x y x y′ ′ , the cross-correlation of∆ ∆ ∆( , ) ( , ; , )D D

x yF x y′ and ∆ ∆ ∆( , ) ( , ; , )D Dx yF x y′ ′ ′ ′ ′ is a

function depending only on x x− ′, y y− ′, and∆ ∆ ∆ ∆x y x y, , ,′ ′ . More properties of such fields are dis-cussed in [17] and [18]. In particular, the isotropic 2-DfBm with parameter0 1< <H has stationary fractional in-crements of order ( , )D 0 /( , )0 D for all D D H∈ >R , .

A process F x y( , ) with stationary increments of or-der (1,0)/(0,1) is characterized by its structure functionϕ( , )∆ ∆x y , representing the variance ofF x y F x yx y( , ) ( , )− − −∆ ∆ for all ( , )∆ ∆x y . This functionis also used in geostatistics problems [19] and for charac-terizing the roughness of mechanical surfaces, which areby their nature anisotropic and nonstationary [20]. Thestructure function is a fundamental concept, as it can beused to express the autocorrelation function of the fieldswith stationary increments, which shows that zero-meanGaussian fields with stationary increments are entirelycharacterized by this function. Moreover, one can deter-mine explicitly the form of the structure function of such afield having self-similar properties. More precisely, its ex-press ion reads: ϕ ρ θ( , ) ( ),x y fH= 2 where0 1 2 2< < = + = +H x y x jy, , ( )ρ θ angle and f is a π-pe-riodic function. For example, the function f reduces to aconstant in the case of isotropic fBm.


Fractional Brownian motion isone of the most popularstochastic fractal models forimages. It is a nonstationaryprocess, but it has stationaryincrements.


Filtering and Linear Spatial TransformsIf we consider a second-order random field F( ),x x ∈R 2 ,with stationary increments of order (1,0)/(0,1) and suchthat F( )0 0= , we can build the fieldG h F

R( ) ( ) ( )x u x - u=∫ 2

du, where h is the impulse response of the considered realfilter. Provided that hdecays fast enough, the existence ofthis field is guaranteed [14]. It has also stationary incre-ments of order (1,0)/(0,1) and its structure function canbe easily deduced from the structure function of the fieldF.

Another way of introducing the anisotropy is to ob-serve that, if ϕ F ( ),∆ ∆ ∈R 2 is the structure function of afield F x y( , ) with stationary increments of order

(1,0)/(0,1), then, for all 2 2× real matrix M, F M( )x is afield with stationary increments of order (1,0)/(0,1) withstructure functionϕ F M( )∆ . Moreover, by using polar co-ordinates ∆ = ρ θ θ(cos ,sin ), the structure function of anisotropic 2-D fBm takes the form ϕ ρ ρB

HH

( )∝ 2 , whereasthe structure function of any self-similar field with sta-tionary increments of order (1,0)/(0,1) resulting from alinear spatial transformation of the 2-D fBm will be

ϕ ρ α θ θ( ) ( cos[ ( )])M H H∆ ∝ − −201 2 . (4)

The parameter α ∈[ , ]0 1 controls the degree of aniso-tropy, while θ π0 0∈[ , ] indicates the “privileged” directionof the field (see Fig. 3). More precisely, the incrementshave the smallest (respectively, largest) variation in the di-rection θ0 (respectively, θ π0 2+ / ). The two extremecases, α =0 and α =1, correspond, respectively, to an iso-tropic fBm and to a completely oriented field obtained bystacking 1-D fBm’s along each line of directionθ π0 2+ / .

Is Stationarity a Realistic Assumption?The stationary-increments property of fBm is useful be-cause it allows one to simplify the analysis. However,most real world images do not share this property. To ob-tain realistic models, one must consider more complexprocesses that have nonstationary increments of any or-der. One of the simplest fractal models that falls into thiscategory is the multifractional Brownian motion (mBm).The major difference between the two processes is that,unlike fBm, the Holder exponent of mBm is allowed tovary along the trajectory, a useful feature when one needsto model processes whose regularity varies in space, as isthe case for most images. While all the properties of fBmare governed by the unique number H, a functionH x y( , ) is available in the case of mBm. Let us give twoexamples that explain why this is important. As we haveseen, the long term correlations of the increments of fBmdecay as k H( )2 2− , where k is the lag, resulting in long rangedependence (LRD) when H >1 2/ and antipersistent be-havior when H <1 2/ . In this respect, fBm is “degener-ated”: since H rules both ends of the Fourier spectrum, i.e.,the high frequencies related to the Holder regularity and thelow frequencies related to the long term dependence struc-ture, it is not possible to have at the same time a very irregu-lar local behavior (H close to zero) and LRD (H >1 2/ ).fBm is thus not adapted to model processes which displayboth those features, such as Internet traffic or certainhighly textured images with strong global organization,as are, e.g., MR brain images. In contrast, mBm is per-fectly fitted in this case. Another example is image synthe-sis: fBm has frequently been used for generating artificialmountains [3]. Such a modeling assumes that the irregu-larity of the mountain is everywhere the same. This is notrealistic, since it does not take into account, e.g., erosion,which smooths some parts of the mountains more thanothers. To model these and other fine features of land-


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(a)

(b)

� 3. Anisotropic fields with structure function given by (3), hav-ing, respectively, the parameters (a) H = 0 2. , α = 0 9. , θ π0 2 3= /and (b) H = 08. , α = 0 9. , θ π0 3= / .


scapes, mBm is a much better candidate. We briefly pres-ent below the 1-D mBm and then move to its 2-Disotropic and nonisotropic extensions.

One-Dimensional mBmThe name “multifractional Brownian motion” was intro-duced in [21] to designate the following generalization offBm: the mBm with functional parameter H t( ) is thezero-mean Gaussian process defined as

W t t s s dB s

t

H tH t H t

( )( ) / ( ) /( ) [( ) ( ) ] ( )

(

∝ − − −

+ −

− −

−∞∫ 1 2 1 20

s dB sH tt) ( ),( ) /−∫ 1 2

0 (5)

where H is a C1 function ranging in (0,1). A harmon-izable representation of mBm was introduced independ-ently in [22]. The increments of mBm are in general nei-ther independent nor stationary. When H t H t( )= ∀ ,mBm is an fBm of exponent H. The covariance of mBmreads [23]:

R t s t s t sWH t H s H t H s H t H s

H( , ) ( ) ( ) ( ) ( ) ( ) ( )∝ + − −+ + + .

One can show that, contrarily to fBm, the increments ofmBm display LRD for all admissible functions H t( ) (ofcourse, the notion of long range dependence must be re-defined carefully for nonstationary increments; see [23]).

The main feature of mBm is that its Holder exponentvaries in time: at each point t 0 , it equals H t( )0 with proba-bility one. This is in sharp contrast with fBm, where the al-most sure Holder exponent is constant: as announcedabove, mBm allows one to describe phenomena whoseregularity evolves in time/space. Equally with probabilityone, the Hausdorff and box dimensions [ , ]c d are bothequal to 2 − ∈min{ ( ), [ , ]}H t t c d . Another importantproperty of mBm is that it is asymptotically locallyself-similar. Basically, this means that, at each t, there existsan fBm with exponent H t( ) which is “tangent” to themBm: a path of an mBm is thus a “lumping” of infinitesi-mal portions of fBms with well-chosen exponents [22].

Let us finally mention that one can define a generalizedmBm where the function H is no longer restricted to beC1

but may be very irregular. This allows the modeling ofGaussian processes with arbitrary local regularity, and it is auseful tool for applications such as image segmentation[24].

Two-Dimensional Extensions of mBmOne can imagine various extensions of mBm in higher di-mensions. We describe two versions that are direct gener-alizations of the ones considered earlier for fBm.

An isotropic multifractional Brownian fieldW H( )xis a

centered Gaussian process, whose covariance functiondepends on a deterministic function H R( ),x x ∈ 2 andreads

E W WH H

H H H H H

{ ( ) ( )}( ) ( )

( ) ( ) ( ) ( ) ( )

x y

x y x y x

x y

x y x y

∝

+ − −+ + +{ }H( ) .y

It admits a moving average representation extendingin the obvious way (5) to several dimensions.

An anisotropic version called multifractionalBrownian sheet (mBs) is obtained by extending the defi-nition of the Brownian sheet. For H N N: ( , )R → 0 1 asmooth enough function, the mBs is the centered Gaussi-an field FH( )x

with autocorrelation function:

E F F

x y x

H H

iH H

iH H

ii i i i

{ ( ) ( )}( ) ( )

( ) ( ) ( ) ( )

x y

x y x y

x y ∝

+ −+ +{ }− +

=∏ y iH H

i

N i i( ) ( )x y

1

with x = ≤ ≤( )xi i N1 and y = ≤ ≤( )y i i N1 (N =2 for images).As in 1-D, one can study the regularity properties of thesetwo fields and show that their Holder exponents are con-trolled by the function H. Multifractional Brownianfields have been used in [25] for the modeling of X-rayimages of bones in view of early detection of osteoporo-sis. Another application is fine terrain modeling [26].

How to Analyze and SynthesizeFractal Surfaces

AnalysisOne of the most popular methods for estimating theHurst parameter of a fractal field is based on the waveletanalysis (WA). Being scaled and translated versions of asingle oscillating function, wavelets [27] perform amathematical zooming into signals. In the 2-D case,wavelets (and wavelet packets) also provide an appropri-ate tool for characterizing fields with fractal features.The most usual way to extend WA to two (or higher) di-mensions is to consider separable wavelet bases. The un-derlying notion of scale invariance in the WA naturallyrelates it to nonstationary self-similar processes, whosestatistical properties are invariant under scale changes.This special relation was first pointed out for the 1-DfBm [28]: wavelet coefficients of fBm form stationarysequences at a given scale and satisfy a “power-law”property. Applied to the 2-D isotropic fBm, this meansthat the variance of its wavelet coefficients at resolutionlevel j is proportional to 2 2 2j H( )+ [29]. Written in loga-rithmic scale, this leads to a simple method for estimat-ing the Hurst parameter of an isotropic fBm. In the caseof 2-D separable wavelets, we have three different re-gression lines allowing the estimation of H, correspond-ing to the horizontal, vertical, and diagonal details. Amore precise estimation of H can be obtained by makinga joint linear regression on all the subbands.

Under some assumptions on the number of vanishingmoments of the WA, the same property of station-



arization was shown to be consistent with the propertiesof more general 1-D random processes with stationary in-crements [30]-[32], and the results have also been ex-tended to fields with stationary increments of arbitraryfractional order [18].

Other classical methods for estimating the Hurst param-eter include the box-counting and the variation method[33], [34], maximum likelihood estimates [35]-[37], leastsquares regression in the spectral domain [38], methodsbased on the log-periodogram [39], or estimators based onthe moments of the increments [40], [41]. The fractal di-mension has also been measured by morphological covers[42] or by algorithms adapted to a specific application butderived from existing methods (e.g., the “reticular cellcounting” method [43], related to spectral algorithms orthe “fractal interpolation function models” [44]).

SynthesisSeveral methods have been proposed for synthesizing frac-tional Brownian fields. Synthesizing fBms is by no meansan easy process, especially if one needs to build large im-ages. The problem lies mainly in the non-Markovian na-ture of fBm. The Choleski method allows exact synthesis,but a plain implementation requires a large computationaltime and a large amount of memory. Under some minorrestrictions, it is possible to use fast and efficient algo-rithms for Choleski decomposition, but various approxi-mate methods have also been designed that allowreasonable computer time/memory requirements.

We first describe the steps involved in the Choleskimethod. We wish to generate samples of an index-HfBm BH at N equidistant points of [0,1]. The discreteincrements ∆B k N B k N B k NH H H( / ) ( / ) (( ) / )= − −1form a stationary Gaussian sequence with zero mean, andthe statistical properties of the vector ∆ ∆B BH H= (( / ), ( / ),..., ( ))1 2 1N B N BH H∆ ∆ are determined by itsautocovariance matrix AN. Since AN is positive definite, itmay be written using its Choleski decomposition asA L L LN N N

TN= where is an invertible lower triangular

matrix. Let ∆ ΝY be an N-sample realization of a unit vari-ance centered white Gaussian noise. Then it is clear thatthe autocovariance matrix of the random vector L YN N∆is exactly AN. Setting ∆ ∆B L YH N N= , we thus generate arealization of BH as B k N B p NH Hp

k( / ) ( / )==∑ ∆1

. The

synthesis of an fBm is now reduced to the computation ofLN from AN. Note that the procedure above may be ap-plied for synthesizing any discrete Gaussian process. A di-

rect method for general Choleski factorization has com-plexity O N( )3 and cannot be used for building largetraces. Fortunately, when the process is stationary (this iswhy we work with ∆BH rather than BH) and when thesamples are equi-spaced, the considered matrix is Toeplitzand one can use fast algorithms, such as the Schur orLevinson ones, which have complexity O N( )2 and needO(N) memory. It is possible to do even better if one forcesN to be a power of two. Such a requirement is common tomany methods (e.g., FFT or dyadic wavelet-based algo-rithms). Then, the doubling Schur algorithm [45] allowsthe complexity to be reduced to O N N( (log ( ))2

2 . Thismethod was used in [46] to synthesize 1-D fBms with131,072 sample points and 2-D fBms with 2048 × 2048sample points.

Let us now briefly comment on some approximatemethods. The oldest one is the midpoint displacement al-gorithm. In the case of Brownian motion, this is the origi-nal construction by Lévy of the Wiener process. Thismethod has a linear complexity [47]. When H ≠1 2/ ,however, the resulting process has second-order propertiesthat differ significantly from those of fBm. In [48] and[37], an improvement of this scheme is proposed, whichallows one to recover approximately the right covariancefunction with a low computational burden. The algorithmis based on the notion of a multiscale process and improveson the classical method by using statistical descriptions ofthe interpolation and displacement steps.

Wavelet-based methods [49] rely mainly on the factthat the wavelet transform acts as a “whitening filter” onfBm. This allows easy synthesis of the wavelet coefficientsof fBm. However, the problem of building the low fre-quency nonstationary approximation of the signal re-mains. Other wavelet methods [50], [29] assume that thedetail coefficients are uncorrelated, thus leading to“1/ f -type” fields rather than 2-D fBm.

A spectral method for generating an approximation ofa 2-D N N× fBm is the incremental Fourier synthesis de-scribed in [51]. The idea is to create first a periodic ran-dom field of size2 2N N× with statistics close to those ofthe increments of the 2-D fBm over half the spatial pe-riod. Such random fields are easy to generate becausetheir Karhunen-Loeve transform is simply the 2-D dis-crete Fourier transform. The approximate 2-D fBm isthen obtained by adding up the increments. The com-plexity of this method is O N N( log ( ))2

2 . This methodcan also be used to generate anisotropic fields with sta-tionary increments [52]. Other approximate Fouriermethods include the ones described in [3] and [53].

Finally, while methods based on differential models areinteresting because they have a physical meaning, they donot yield correct approximations to fBm. In fact, thesemethods are used to study generic “1 / f ” noises [54].

Let us now discuss the synthesis of mBm. The usualtechnique is based on the property that a path of an mBmwith function H t( )is “tangent” at each t 0 to the one of anfBm with exponent H H t0 0= ( ) (see [21]). This allows


One of the most popularmethods for estimating the Hurstparameter of a fractal field isbased on the wavelet analysis.


the generation of a sample path of an mBm B tH t( ) ( )at theN points( )t i by generating first N plain fBm-s Bi with ex-ponents H H ti i= ( ). One then lumps together the appro-priate points B ti i( ). The complexity of the procedure isO N N( (log ( )) )2

22 . An alternate method is to make use

of the Choleski decomposition to generate directly thesamples of an mBm. The complexity is now O N( )3 . Thissecond method is thus more complex than the first one,but it is exact, whereas the previous one is approximate.

Discrete Fractional Modelsfor a Digitized WorldContinuous fields, and in particular fractal fields, have tobe sampled to be processed. Another approach, whichmay appear much easier to apply in practice, is to definedirectly discrete-space models. This idea is also supportedby the fact that the increments of continuous-spaceself-similar fields present long-range dependence (LRD)properties. They can be modeled by 2-D extensions offractionally integrated auto-regressive moving average(FARIMA) processes, which have been successfully usedin finance, hydrology, and other applications. The 2-DFARIMA fields can be built in an isotropic or anisotropicmanner, the latter being much more flexible in taking intoaccount the characteristics of natural scenes. These mod-els also constitute a discrete alternative to the 2-D fGn.Another argument in favor of discrete-space models isthat the parameter estimation methods will be directly de-rived from the classical approaches existing in the litera-ture for time-series analysis.

Long-Range Dependence and Fractal ProcessesFor a stationary second-order random process,( ( ))X k k∈Z ,the LRD is defined by the nonsummability of itsautocorrelation functionΓ( )k . In particular, a condition ofLRD is the existence of a parameter d ∈( , / )0 1 2 such thatlim ( ) , *

kdk k K K→ ∞

−+= ∈1 2 Γ γ γwith R or, equivalently,

the psd is divergent at the origin lim | | ( )ω ω ω→ =02 d S

K KS Swith ∈ +R . Defining H d= +1 2/ , the LRD ap-pears for H ∈( / , )1 2 1 . A typical example of an LRD pro-cess is the (discrete-time) fractional Guassian noise (fGn)described earlier.

Another example of LRD processes are FARIMA pro-cesses, also called fractionally differenced noises, whichhave been introduced by Hosking [55]. The psd of aFARIMA(0,d,0) process is S( ) |sin( / )|ω σ ω= −2 22 d . Itcan be remarked that, for ω → 0, it is equivalent to that ofthe fGn. The constraint d >0 leads to LRD, whereasd <1 2/ is necessary for finite variance.

Gaussian FARIMA Modelsand Related Anisotropic FieldsA possible bidimensional extension of the fGn [11] is anisotropic field with 2-D psdS( , ) ( ) ,ω ω ω ωx y x y

d∝ + −2 2 2

0 1 2< <d / . The link between d and the Hurst parameter

is then: 2d H= . Its discrete-space equivalent is an isotro-pic 2-D FARIMA. This is a zero-mean Gaussian fieldwith psd

S( , ) sin sinω ωω ω

x yx y

d

∝ +

−2 2

2

2 2.

When ω x → 0 and ω y → 0, the psd of the 2-D FARIMAprocess tends towards that of the isotropic fGn.

As already pointed out, it is often useful to dispose ofan anisotropic model for LRD fields. It can be built in thesame way as the isotropic 2-D fractionally differenced


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� 4. Level sets of the psd of the anisotropic fGn, for: (a)ϕ π α= − = =/ , . , .5 0 97 03d and (b) ϕ π α= = =/ , . , .5 017 03d . Theangle ϕ / 2 is measured clockwise with respect to the vertical axis.


Gaussian noise, taking into account jointly thelong-range dependence properties, the anisotropy, andthe directionality in the image. A form for the spectraldensity answering to these requirements is

S( , ) ( , ) ( , )( ),ω ω σ ω ω ω ωα ϕx y x y

dx yI A= − −2 2 1 2

where

A x y x

y

α ϕ ω ω α ϕ ω

α ϕ ω, ( , ) ( cos )sin ( / )

( cos )sin ( /

= −

+ +

1 2

1 2

2

2 )( / ) sin sin sin .− 1 2 α ϕ ω ωx y

This expression is able, on the one hand, to model an iso-tropic part: I x y x y( , ) sin ( / ) sin ( / )ω ω ω ω= +2 22 2 , de-pending only on the fractional coefficient d and capturingthe LRD behavior, and, on the other hand, an anisotropicpart A x yα ϕ ω ω, ( , ) depending only on α ≥0 and ϕ ∈[ , )0 1 .A polar coordinates change ( , ) (cos ,sin )ω ω ω ω ωθ θx y r=in the previous expression helps in studying the psd in theneighborhood of the origin of the spatial frequency plane.Indeed, for ω r → 0, we get

S( cos , sin )~

[ cos ( ( / ))] .

ω ω ω ω

ω α α ω ϕθ θ

θ

r r

rd− −+ − −4 2 11 2 2

For an LRD field, this spectral density diverges at the fre-quency ( , ) ( , )ω ωx y = 0 0 , for d >0. Another constraint co-mes from the fact that, in order to define the correlationfunction of the field, the integral of its psd has to be finite,and this leads to d <1 2/ . The form of the psd near the ori-gin also suggest the interpretation of the parameters char-acterizing the model (see Fig. 4). The parameterϕ π/ [ , )2 0∈ determines the orientation of the resultingfield: the psd is maximum in the neighborhood of the ori-gin forω ϕθ = / 2 and therefore the privileged direction inthe field is ϕ π/ /2 2+ . The parameter α ∈[ , )0 1 character-izes the dispersion of the psd of the field around this privi-leged direction and is called the anisotropy coefficient. Inparticular, we remark that, for α =0, we find an isotropicmodel. The fractional parameter d determines the degreeof “roughness” of the field. The closer d to 1 2/ , the“smoother” the field. Note also that the LRD and the ani-sotropy terms being separated when ω r → 0, the orienta-tion and the roughness of the field can be tunedindependently. The influence of these parameters on thegenerated texture is illustrated in Fig. 5.

The four parameters of the model, σ α2 , ,d , ϕ, can beeasily estimated in the noise-free case, either by a leastmean squares approach or by maximum likelihood[14]. The first method relies on the fact that an asymp-totically unbiased estimator of log ( , )S ω ωx y is, for( , ) ( , )ω ωx y ≠ 0 0 , log ( , )PN x yω ω γ+ (γ ≅057721. ), wherePN x y( , )ω ω is the periodogram of the observed image. Inthe second case, one can use the Whittle spectral approxi-mation to derive an explicit form for the log-likelihood[56], [57]. When the realizations of the anisotropicFARIMA are corrupted by an additive Gaussian noise,the direct estimation of parameters from the log-likelihoodexpression leads to a complex nonlinear optimization. Thiscan be alleviated by an expectation-maximization (EM)approach [58]. At each iteration of the algorithm, an op-timization of the same complexity as that of the noiselessmodel is performed.

Nonstationary ExtensionsIt is also important to note that it is possible to proposenonstationary extensions of the FARIMA 2-D fields[14]. A construction of such fields has been detailed in


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� 5. (a) Discrete-space fractional noise withϕ π α= = =/ , . , .3 0 97 03d . (b) Nonstationary field with stationaryincrements, having the above realization as first-order increment.


[59]: starting from a stationary field ( ( , ))( , )

U n mn m ∈Z2 ,

we first filter it by the filters with frequency responses

[ ]G ex x yj x x y( , ) ( ) sin ( ) sin ( )

/ω ω ω ω ω= − +−

−1 2

22

2

1 2

and G Gy x y x y x( , ) ( , )ω ω ω ω= . The resulting fields Bx

and By are stationary and can be considered as the incre-ments of a nonstationary field F n m( , )with stationary in-crements of order (1,0)/(0,1). This one can beconstructed by summing up in an appropriate way Bx

and By . Moreover, if the original field U n m( , ) is theanisotropic FARIMA field defined earlier, then( ( , ))

( , )F n m

n m ∈Z2 is a field with stationary increments oforder ( , )α 0 /( , )0 α , for all α α∈ >R d, 2 . Therefore, thesefields are the discrete-space equivalent of the continu-ous-space models with stationary increments of fractionalorder discussed in “Fields with Stationary Increments ofFractional Order.” In particular, for isotropic fields, theyrepresent discrete-space counterparts of the isotropic 2-DfBm (see Fig. 5(b)).

Non-Gaussian Fractional FieldsThe driving noise in the discrete anisotropic model dis-cussed so far has a Gaussian distribution. A large class ofdistributions can be approximated by Gaussian mixturesand texture models combining such distributions withLRD characteristics have been proposed in [60]. How-ever, in some applications (e.g., SAR, ultrasound, or as-tronomical imaging), the analyzed fields may be highlyvariable and it may be interesting to use “heavy” tail prob-ability distributions. In particular, stable processes haveturned out to be good models for many impulsive signalsand noises. Alpha-stable distributions have infinite vari-ance, undefined higher-order moments, and in general,there does not exist an explicit form for their probabilitydensities [61], [10]. They are interesting, however, in lin-ear modeling, as linear transforms preserve the distribu-tion of any linear combination of i.i.d. α-stable randomvariables. Alpha-stable 2-D discrete-space processes hav-ing LRD properties have been studied in [62].

Multifractal SurfacesMultifractal analysis is concerned with the study of theregularity structure of functions or processes, both from alocal and global point of view. More precisely, one startsby measuring in some way the pointwise regularity, usu-ally with some kind of Holder exponents (see [12]). Thesecond step is to give a global description of this regular-ity. This can be done in a geometric fashion, usingHausdorff dimension, or in a statistical one through alarge deviation analysis. We describe below a simplifiedversion of multifractal analysis in the framework of imageprocessing. Let X t( ), t T∈ = [ , ]0 1 2 denote the image.

The Hausdorff spectrum f h ( )α describes the structureof the function t taα( ) (where α( )t is the Holder expo-nent at t) by evaluating the size, as measured by the

Hausdorff dimension, of its level sets. In other words,one sets f t T th H( ) { , ( ) }α α α= ∈ =dim , wheredim H E( )de-notes the Hausdorff dimension [63] of the set E. Sinceeach Eα is included in T , f h takes values in [ , ] { }0 2 ∪ −∞(the value −∞ occurs when Eα is empty, while f h ( )α 0 2=indicates that a whole region in the image has exponentα 0 ). From a heuristic point of view, the Hausdorffmultifractal spectrum thus describes the “size” of the setof pixels in the image which have a given regularity. Forinstance, if f h ( )α 0 2= for someα 0 2> and f h ( )α <2 for allα α≠ 0 , then we know that almost all points in the imagehave regularity α 0 and thus that the image is almost ev-erywhere smooth, because α 0 2> .

Another global description of the local regularity isprovided by the large deviation multifractal spectrum,f g ( )α . A heuristic way to introduce f g is as follows. Fixan integer n and partition the image into n 2 boxes of size1 2/ n . Now pick a box at random. One defines f g ( )α bywriting that the probability that the chosen box has a reg-ularity α behaves as n f g− −( ( ))2 α , when n is large. Thus,roughly speaking, f g ( )α measures the rate of decay, whenn tends to infinity, of the probability that a randomlypicked region of size 1 2/ n has regularity α. In particular,if f g ( )α is strictly smaller than two, then the probability ofobserving a regularity α goes to zero exponentially fast,with exponential rate2 − f g ( )α : when n is large, most pix-els have an α such that f g ( )α =2. We mention that it isnatural to interpret f g as a rate function in a large devia-tion principle. The theory of large deviations providesconditions under which such rate functions may be calcu-lated as Legendre transforms of moment generating func-tions. When applicable, this procedure yields a morerobust estimation than a direct computation. In general,this allows the definition of a new spectrum, theLegendre spectrum, not considered here.

Multifractal analysis has been the subject of numerousstudies both in the deterministic and random cases. Avery partial list of references is [64]-[68]. Many workshave been devoted to the comparison of the spectra andtheir computation in various cases. In particular, it isshown, for instance in [69], that the inequality f fh g≤holds in full generality. The spectra have been determinedmost notably in the case of multiplicative cascades [70],for which one has equality between f h and f g (one thensays that the multifractal formalism holds). TheHausdorff spectrum of Lévy processes was computed in[71], and the f h and f g spectra of a large class of Gaussi-


Multifractal analysis isconcerned with the study of theregularity structure of functionsor processes, both from a localand global point of view.


an processes are given in [72]. Multifractalanalysis has a number of important applica-tions in image processing, some of which aredescribed in the next section.

ApplicationsSome application of the fractal modeling ofsurfaces have already been mentioned, liketexture analysis and synthesis, and others willbe discussed in this section.

Geophysical Fields SimulationA popular way to synthesize various geo-physical phenomena is to use multifractals.The starting point of the method is based on

the remark that the simplest models leading to processeswith nontrivial spectra (i.e., not reduced to a point) arethe so-called multiplicative cascades and that these kindsof cascades are believed to occur commonly in many nat-ural phenomena. To construct the crudest cascade,choose a real number m0 in ( , )0 1 and set m m1 01= −(these are called the weights). Split the unit interval intotwo halves, and assign measure m0 to [ , / ]0 1 2 and mea-sure m1 to [1/2,1]. Iterate this process so that at step nwe are dealing with dyadic intervals [ ,( ) ]k kn n2 1 2− −+ ,each having a given measure µ n k, . Then split each ofthese interval into two halves and put measure m n k0µ ,on the left one and measure m n k1µ , on the right one.One shows that this procedure allows the definition of alimiting measure when n tends to infinity, called the bi-nomial measure. The binomial measure has amultifractal spectrum looking like the mathematicalsymbol ∩. More complex versions of these multiplica-tive processes are expected to be good models in a widevariety of fields including turbulence [68], [67], DLA[73], financial data modeling [74], Internet traffic [75],[76], geophysics [77], etc. In particular, random multi-plicative cascades with a continuous scale parameter(i.e., defined on a continuum of scales rather than dyadicones) and a Lévy stable distribution for the weights wereused in a series of works [78] to model various multidi-mensional geophysical phenomena, including cloudsand oceans.

Let us finally mention another method for synthesiz-ing clouds based on a wavelet model called TAON [79].Images obtained with this model are displayed in Fig. 6.

Medical ApplicationsFractal Brownian fields have been early used to model tex-tures and the Hurst parameter has been estimated in viewof segmentation and classification [80]-[82]. Recently,isotropic and nonisotropic 2-D fBm and multifractionalBrownian fields have been used for early detection of os-teoporosis from X-ray images [25], [83].


(a) (b)

� 6. Cloud images obtained using TAON.

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� 7. (a) Original image and edges obtained through (b)multifractal segmentation.


Satellite ImagingAn isotropic FARIMA model with a non-Gaussian(one-sided exponential) white noise driving sequencewas used in radar [84] to simulate the texture in syntheticaperture radar (SAR) imagery for ocean surveillance. Theisotropy of the model was established based on the sym-metry of the periodogram for the RADARSAT clutter.This model, combining an LRD part with an ARMAshort-range dependence part, was shown to better fit theexperimental clutter power spectral density than both aMA model and a simple fractional differencing field.Other applications in SAR imagery concern radar imagedata classification [85], [86] and the characterization andsegmentation of hydrological basins [87], both based onthe fractal dimension.

Image SegmentationIn many applications, the information contained in thelocal regularity of images is more important than the ac-tual values of the grey levels. A typical example is edgedetection: edges are not modified by an affine transfor-mation of the grey levels, while they always correspond tolow regularity pixels. It thus seems reasonable to expectthat estimating the Holder exponents of the image willyield relevant information for segmentation. In the spe-cific case of edge detection, one needs additional, global,information. Indeed, the property of being an edge pointis not only local: by definition, the set of the edges in theimage has the geometry of a set of lines (as opposed to2-D regions or isolated points) and thus must be of di-mension one. From a different, statistical, point of view,one also has that the probability that a randomly chosenpixel in an image of size n 2 pixels is an edge point shouldbe of the order of 1 / n. We may then characterize edges aspoints that have specific, low regularities (this is a localcriterion, measured through the Holder exponentα), andsuch that the associated spectra values are f h ( )α =1 (be-cause edges are 1-D objects) and f g ( )α =1 (because theyhave a given, resolution-dependent, probability to oc-cur). Multifractal edge detection thus consists of first esti-mating f g and then classifying as edge points those pixelswhose Holder exponent α is such that f g ( )α =1 (one as-sumes that the multifractal formalism is valid, i.e.,f fh g= ). An example of a multifractal segmentation isdisplayed in Fig. 7. See [8] for more on this topic.

Image DenoisingThe problem of image denoising may also be treated withmultifractal methods. Intuitively, it seems clear that mostpoints in a scene whose overall appearance is noisy (as, forinstance, SAR images) will have a low regularity. To the con-trary, “smooth” images contain mostly points with high val-ues ofα. In terms of the multifractal spectrum, noisy imageshave a “large” f g ( )α for “small” values of α and havef g ( )α <2 for α ≥2. For “clean” images, sup α α< <2 2f g ( ) ,and the maximum of f g is reached for exponents larger than

two. To denoise an image, a natural idea is then to modify itso that its multifractal spectrum is translated towards largevalues of α: in this way, the regularity of each point is in-creased, but the shape of the spectrum is left unchanged. Asa consequence, the image becomes more readable while therespective strength of each singularity remains the same(i.e., a noisy point on a contour will still be, after processing,more irregular than a noisy point in a smooth zone). From apractical point of view, the shift in Holder regularity is ob-tained through a nonlinear manipulation of the wavelet co-efficients of the image (see [88] for a detailed explanation).This method allows in particular the efficient processing of


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� 8. (a) Original SAR image and (b) its multifractal denoising:the multifractal spectrum of the image is shifted through anonlinear manipulation of its wavelet coefficients.


certain SAR images which resist most other techniques, asshown in Fig. 8.

Interpolation of Fractal SurfacesIn missing data problems, one can be interested in realiz-ing a linear interpolation of a field exhibiting fractal fea-tures. The statistical interpolation of nonstationary fields(as those we have seen so far) does not enter the classicalframework of mean-square prediction problems for sta-tionary processes [89]. However, if we restrict our analy-sis to fields having stationary increments, it is possible toextend the existing methods, by exploiting the propertiesof the structure function [90]. Indeed, let F( )n be thevalue of the field to be estimated andS be a finite subset ofZ n \{ }0 which defines a f inite neighborhood{ , }n p p− ∈S of the point n. Remark that we do not im-pose any constraint on the neighborhood, which can besymmetric or not. The filter is shift-invariant (i.e., its co-efficients do not depend on the position n of the esti-mated sample) if hnp

p∈∑ =S

( ) 1. The interpolationcoefficients are estimated by minimizing the mean-squareestimation error, and the problem reduces to a linearmean square estimation. As the increments of F are sta-tionary, we can express the normal equations, using thestructure function. Together with the shift invarianceconstraint, we obtain a set of linear equations allowingthe determination of the impulse response of the interpo-lation filter. This method has been applied to the interpo-lation of underwater terrain maps in [90]. Note that thedescribed approach is strongly related to kriging methods[19].

ConclusionsResulting from more than a century of theoretical worksrealized in mathematics and physics, the idea of fractalsemerged in the early 1970s. As we have shown in this arti-cle, this concept provides new sophisticated analysis andsynthesis tools for image processing. An important ques-tion that could be raised at this point is whether fractalsconstitute adequate models for real scenes. Recent ad-vances [91] have shown that even though few natural im-ages are actually fractal, methods inherited from thefractal and multifractal formalisms can be successfully ap-plied to a wide range of textures and images.

Béatrice Pesquet-Popescu received the engineering degreein telecommunications from the “Politehnica” Institutein Bucharest in 1995 and the Ph.D.degree from EcoleNormale Supérieure de Cachan in 1998. In 1998 she wasa Research and Teaching Assistant at Université Paris XI,and in 1999 she joined Philips Research France. Since2000 she has been an Associate Professor at EcoleNationale Supérieure des Télécommunications (ENST).EURASIP gave her a Best Student Paper Award in theIEEE Signal Processing Workshop on Higher-Order Sta-tistics in 1997, and in 1998 she received a Young Investi-

gator Award granted by the French Physical Society. Hercurrent research interests are in statistical image analysisand multimedia applications.

Jacques Lévy Véhel received the engineering degrees fromEcole Polytechnique, France, in 1983 and EcoleNationale Superieure des Telecommunications, France,in 1985. He holds a Ph.D. degree in applied mathematicsfrom Universite de Paris XI, France. He is now a researchdirector at INRIA, where he leads the Fractales Lab. Hiswork is mainly concerned with the development of fractaland multifractal methods for signal/image processing.He has written two books and more than 20 journal pa-pers in this area. He has also supervised the developmentof the FracLab toolbox for signal/image processing.

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