HYPERBOLIC WAVELET LEADERS FOR ANISOTROPIC MULTIFRACTAL TEXTURE ANALYSIS

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1 HYPERBOLIC WAVELET LEADERS FOR ANISOTROPIC MULTIFRACTAL TEXTURE ANALYSIS SG Roux, P Abry, B Vedel, S Jaffard, H Wendt Pysics Dept, CNRS (UMR 6), Ecole Normale Supérieure de Lyon, Université de Lyon, France LMBA, CNRS (UMR 6), Université de Bretagne, Vannes, France LAMA, CNRS (UMR 8), Université Paris-Est Créteil, France IRIT-ENSEEIHT, CNRS (UMR ), Université de Toulouse, France ABSTRACT Scale invariance as proven a crucial concept in texture modeling and analysis Isotropic and self-similar fractional Brownian fields (D-fBf) are often used as te natural reference process to model scale free textures Its analysis is standardly conducted using te D discrete wavelet transform Generalizations of D-fBf were considered independently in two respects: Anisotropy in te texture is allowed wile preserving exact self-similarity, analysis ten needs to be conducted using te D-Hyperbolic wavelet transform ; Multifractality enables more versatile scale free models but requires isotropy, analysis is ten acieved using wavelet leaders Te present paper proposes a first unifying extension, wic is enabled troug te following two key contributions: Te definition of D process tat incorporates jointly anisotropy and multifractality : Te definition of te corresponding analysis tool, te yperbolic wavelet leaders Teir relevance are studied by numerical simulations using syntetic scale free textures Index Terms Texture, Multifractality, Anisotropy, Hyperbolic wavelet leaders INTRODUCTION Context Scale free properties, or scale invariance, is often regarded as a central concept in real-world texture modeling (cf eg, [ 9]) Fractional Brownian fields (D-fBf), because of teir conceptual and practical simplicity and because of teir being natural isotropic D extension of D fractional Brownian motion [], are massively used in teory and applications as reference processes to model scale free textures D-fBf are isotropic, Gaussian exactly self-similar processes Teir dynamics are tus controlled by te unique so-called self-similarity parameter < H < For practical analysis, it is nowadays well-accepted tat wavelet transforms are ideal tools to evidence scale free properties and to estimate te corresponding scaling parameters [, ] Notably, it is well documented tat te moments Work supported by Frenc ANR BLANC AMATIS BS of te coefficients of a D-Discrete Wavelet Transform (D- DWT), d X (j, k), computed on a D-fBf, follow, across analysis scales a = j, a power-law beavior wit scaling exponent controlled by H, E d X (j, k) = C jh, tus easily permitting to reveal isotropic self-similarity and to estimate H (cf eg, []) However, as a modeling paradigm, D-fBf suffers from two major limitations: i) Scale free properties are governed by a single scaling exponent H, wic excludes te use of multifractality (wit a collection of scaling exponents) to better account for te ricness of scaling encountered in real-world textures [ 6] ; ii) Anisotropy is not permitted in te classical definition of D-fBf wile it is anoter natural and important property of texture [] Related works Tese two restrictions ave been addressed independently Multifractal processes [, ] are often used as versatile models permitting to enric scale free properties by allowing fluctuations along space of te local regularity of te process and tus requiring a collection of Hölder exponents to account for scaling in textures instead of one single parameter H, a ricness quantified by te so-called multifractal spectrum D() [,,, ] Tese ricer scaling properties are obtained toug at te price of maintaining isotropy Te relevant analysis of multifractality in textures requires te replacement of te D-DWT coefficients, by wavelet leaders, consisting of local ierarcical suprema of suc coefficients [] Independently, anisotropy as been been married wit scale free properties in Operator Scaling Gaussian Random Fields (OSGRF) processes [] Tese anisotropic Gaussian processes are exactly self-similar, wic precludes multifractal extensions It as recently been sown tat te use te D-yperbolic wavelet transform (D-HWT) [6] enables a teoretically sound and practically efficient joint estimation of bot self-similarity and anisotropy [] However, tere as been so far no attempt to propose models accommodating bot anisotropy and multifractality in textures, or to devise te corresponding analysis tools Goals, contributions and outline Te present contribution as tus a double aim: First, a candidate model for anisotropic

2 multifractal texture will be defined and described (Section ) ; Second, a tool for te joint analysis of anisotropy and multifractality, te Hyperbolic wavelet leaders, will be proposed (Section ) Te relevance of te proposed model and analysis tools will be illustrated and assessed using syntetic scalefree textures (Section ) ANISOTROPIC MULTIFRACTAL TEXTURES Definition Combining te definition of multidimensional isotropic multifractal random fields, introduced in [] as a natural extension of univariate multifractal random walks (cf eg, [8]), wit te definition of of OSGRF in [], an anisotropic extension of D-fBf, we propose te following original operational definition of anisotropic multifractal textures (e i x, ξ ) X,H,λ (x) = G R ξ (H+) λ (ξ)dξ (), were x = (x, x ) and ξ = (ξ, ξ ) denote respectively te space and Fourier domains coordinates Tis toy-model is constructed to incorporate te tree-properties targeted ere: Parameter < H < controls te global scale free property of te process via te fractional intergration kernel ξ (H+), ; Parameter λ >, referred to as te intermittency or multifractality parameter, controls te degree of variation of te local regularity along space, via te intermediate process G λ (ξ) defined as te Fourier transform of g(x) = e ω(x) dw (x), were dw (x) correspond to a D-Wiener measure and ω(x) to an independent Gaussian stationary process wose covariance is cosen of te form cov(ω(x), ω(y)) = λ log( x y ), () to ensure scale-free properties ; Parameter < < controls te global anisotropy via te use of an anisotropic pseudo-norm ξ, = ξ / + ξ /( ) in te fractional integration kernel Setting obviously permits to recover te isotropic case and tus corresponds to te isotropic multifractal random fields in [] Wile λ = is in principle excluded from te definition, we will by convention set ω(x) for λ = In tis case, one recovers te classical exactly self-similar Gaussian OSGRF defined in [] Setting jointly and λ obviously yields te simple Gaussian isotropic exactly self-similar D-fBf Anisotropic multifractal analysis Elaborating on isotropic multifractal analysis, we propose ere to caracterize scalefree properties by an anisotropic multifractal spectrum, defined mutatis mutandis from its isotropic counterpart [, 9] D () = d H ({x (x) = }), () were (x) is defined as an anisotropic local regularity (or Hölder) exponent as (x) = sup s {s; X,H,λ (y) P x (y) C y x s,} () wit P x (y) a polynomial of te form P (y, y ) = a β,β y β yβ (β,β ) N Scale-free properties Based on te teoretical results obtained for isotropic multifractal fields in [] and anisotropic self-similar processes in [], one can naturally conjecture tat te scale-free properties of X,H,λ (x) are wellcaracterized by its anisotropic multifractal spectrum taken in te nominal anisotropy direction (wit c = c ( ) = H λ and c = c ( ) = λ ): D () = ( c ) /c () HYPERBOLIC WAVELET LEADERS Hyperbolic wavelet transform Te definition of D-DWT coefficients, d X (j, k) = X(x) j ψ( j x k, relies on te use of one same dilation factor j for bot directions, wit te moter wavelet written as ψ(x, x ) = ψ (x )ψ (x ), cf eg, [] In contrast, te yperbolic wavelet transform (HWT) [6] is defined using two independent dilations factors (, j ) for te x and x coordinates d X (j, k) = X(x, x ) ψ( x k ) j ψ( j x k ) were j = (j, j ) It was sown in [,, ] tat it is precisely te use of tese two independent dilation factors in te analysis tat permits to disentangle self-similarity from anisotropy HWT coefficients can be efficiently computed using alternate iterations of te classical D and D pyramidal algoritms underlying te DWT Hyperbolic wavelet leaders To extend te anisotropy resolving property of HWT to multifractal analysis, we define yperbolic wavelet leaders [] in analogy wit te D-DWT leaders for isotropic multifractality proposed in [] Let β > and let λ j (k) denote yperbolic dyadic rectangles [] λ j (k) =] k, k + ] ] k j, k + j ], (6) Let 9λ j (k) denote te union of suc rectangles: 9λ j (k) =] k, k + ] ] j k j, j k + j ] Hyperbolic wavelet leaders at scales j and location k are defined as local suprema of te HWT coefficients in a spatial neigborood across finer scales j j et j j : L (β) X (j, k) = sup (j +j )β d X (j, k ) () j,k 9λ j (k)

3 Te interest of using leaders stems from te fact tat it can be sown tat wen X as an anisotropic local regularity exponent (x ) > at location x, ten [, ]: L X (j, k) ( +β)(x) max( j ( ) ), (8) wic paves te way towards te estimation of D Here, te parameter β permits to ensure tat te process X as enoug regularity for te corner stone Eq (8) to old, cf, [, ] Anistropic multifractal spectrum estimation To empasize te targeted anisotropic nature of te analysis, we make use of relabelled scales (j) = (j, ( )j), wit < < te analysis anisotropy angle Elaborating by analogy on calculations acieved independently for anisotropic Gaussian self-similar process [] and isotropic multifractal processes [], Eq (8) above enables to conjecture tat E( L X (j,( )j, k) q ) C ( ) = n j, ( ) q (τ (q)+(+β)q) max(, ) were τ (q) denotes te Legendre transform of D [] Te quantity τ(q, ) = (q (τ (q)+(+β)q) max(, )) is furter maximal wen =, wic motivates te proposed estimation procedure Te estimation of te multifractal spectrum in isotropic multifractal analysis can be based on several procedures (cf eg, [] for reviews) We follow ere a parametric strategy targeting te direct estimation of c and c in Eq () [], based on cumulants of te log yperbolic wavelet leaders: C ( ) = [ ] log L (β) X n (, k), () j k Z [ log L (β) X (, k)] [C ( )], () k Z were n j denotes to te number of yperbolic wavelet leaders available at scales For te anisotropic multifractal texture defined in Eq (), Eq (9) leads to j, (9) C (j, ( )j) c () ln j () C (j, ( )j) c () ln j () and estimates ĉ p () can tus be obtained by linear regressions of C p (j, ( )j) versus j, cf [, ] Te parameters (, ĉ, ĉ ) are ten obtained as cp = argmax c p (), ĉ p = c p ( cp ) () and are straigtforward to translate into estimates for H and λ Note tat te use of te analysis angle in tis procedure essentially amounts to recovering te classical D- DWT leaders proposed in [] PERFORMANCE ASSESSMENT Syntetic textures and performance assessment To assess te relevance and performance of te proposed model and estimation procedure, tree different settings for te parameters C (j, j ) j C (j, j ) j C ( ) c () ĉ C ( ) c () 6 ĉ Fig Anisotropic multifractal textures Plots of C p and c p for p = (top row) and p = (bottom row) Left column: C p (j, j ) surface wit te dased black and te blue solid lines indicating te isotropic and te estimated anisotropic directions and, respectively Middle column: C p ( (j)) as functions of j, for (black line wit ) and for (blue line wit o ) Rigt column: c p () as functions of (average and standard deviations across realizations) Te location and value at te maximum of c p () permits to estimate and c p Te teoretical values for (, c ) are sown wit red lines (, H, λ ) are studied: a multifractal anisotropic texture wit (8,, ); a multifractal isotropic texture wit (,, ); a self-similar (non multifractal) anisotropic texture (OSGRF) wit (8,, ) For eac parameter setting, N = independent copies of size are generated wit MATLAB routines designed by ourselves (and available upon request) Samples for te tree cases are sown in Fig Daubecies least asymmetric moter wavelet wit vanising moments [] are used to compute te yperbolic wavelet leaders for j and j ranging from to Eqs () and ()) are used to estimate c () and c () by linear regressions Estimation of c () and c () and of, c and c is ten performed on eac realization independently Average results for te tree cases are plotted in Figs -, respectively, and furter quantified in Tab Performance assessment Fig sows tat for anisotropic multifractal textures, bot functions c () and c () display clear maxima located at te same, tus yielding consistant estimates 9 of Because it is not granted by construction, suc consistency in te estimation of is a satisfactory outcome validating te relevance of te procedure Te values for c p are also satisfactorily estimated, wit small biases and variances For isotropic multifractal textures (cf, Fig ), consistently for bot p = and p =, indicating te absence of anisotropy For tat case, te yperbolic wavelet leaders essentially amounts to te classical D-DWT leaders, yet witout aving been selected a priori Moreover,

4 C (j, j ) C ( ) ĉ c () j C (j, j ) C ( ) c () ĉ = 6 j Fig Isotropic multifractal texture Legend as in Fig C (j, j ) j C (j, j ) j C ( ) ĉ c () C ( ) c () Fig Anisotropic non multifractal texture Legend as in Fig D() Image Iso MultiF Aniso MultiF ĉ Aniso Non MultiF Fig Hyperbolic versus classical wavelet leaders multifractal textures Top row: sample textures Bottom row: multifractal spectra estimated using D-DWT classical leaders, ie, (black) and yperbolic wavelet leaders, wit (blue) compared to te teoretical spectrum (red) Aniso MultiF Iso MultiF Aniso Non MultiF ĉ 99 (9) () 9 () c (9) 8 (8) () ĉ 6 (9) () () c (98) () (8) Table Bias and standard deviation for estimates Fig indicates tat for anisotropic self-similar (non multifractal) textures, c () also displays a clear maximum wose location 8 nicely estimates and wose value ĉ = c ( ) nicely matces H Conversely, te function c () remains essentially flat at, sowing no multifractality ĉ, wit a sligt practical maximum wit same location as tat of c (), a satisfactory practical outcome of te procedure Te performance for all tree parameter settings are furter quantified in Tab Te results indicate tat te proposed multifractal anisotropic analysis procedure consistently yields satisfactory estimates wit small biases and variances for all textures, be tey anisotropic or not, multifractal or not Finally, Fig compares te multifractal spectra obtained using yperbolic wavelet leaders to tose obtained using classical D-DWT leaders As expected, for isotropic textures, bot metods yield equivalent estimates tat matc well te teoretical multifractal spectrum Yet, for te anisotropic multifractal and te anisotropic non multifractal textures, D-DWT leaders fail to provide relevant estimates of te multifractal spectra and significantly overestimate te multifractality of te texture In addition, note tat it would not be possible for practitioners to detect tat teir D-DWT leaders based analysis is fooled by anisotropy In contrast, estimates obtained wit te proposed yperbolic wavelet leaders remain excellent, wic furter underlines teir benefits CONCLUSION AND FUTURE WORKS A double contribution as been proposed ere: An operational anisotropic multifractal texture as been defined, wic, to te best of our knowledge, was so far not available in te literature ; A procedure for te anisotropic multifractal analysis as been devised, wic again is seen as an original contribution Elaborating on te present work, te statistical properties of te proposed anisotropic multifractal texture model will be furter assessed in a more teoretical framework Accordingly, te properties of its yperbolic wavelet leaders will be furter consolidated teoretically At te practical level, we believe tat te proposed analysis tool is now ready and sound enoug for application to real-world textures, were bot anisotropy and scale free properties are likely to be relevant togeter, and will tus permit an estimation of multifractality not fooled by anisotropy

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