B. Vedel Joint work with P.Abry (ENS Lyon), S.Roux (ENS Lyon), M. Clausel LJK, Grenoble),

Hyperbolic wavelet analysis of textures : global regularity and multifractal formalism B. Vedel Joint work with P.Abry (ENS Lyon), S.Roux (ENS Lyon), M. Clausel LJK, Grenoble), S.Jaffard(Créteil) Harmonic Analysis, Probability and Applications, Conference in honor of Aline Bonami, Orléans, June, the 10th 2014

Introduction

Introduction Same maximal directional regularity s = 0.2 More or less roughness in other direction A classical wavelet analysis will give H 1 = 0.2, H 2 = 0.2 1.3 0.15, H 3 = 0.2 1.3, H 4 = 0.2 1.7 0.12 A multifractal analysis will not distinguish between these textures and a fbm of Hurst exponent H i.

Introduction

Introduction Numerous tools of harmonic analysis are adapted to isotropic situations (functional spaces, wavelets, multifractal analysis)

Introduction Numerous tools of harmonic analysis are adapted to isotropic situations (functional spaces, wavelets, multifractal analysis) Numerous models to describe rough textures : fractional Brownian motions, multifractal functions, multifractal fields...

Outline Anisotropic fields and spaces

Anisotropy Anisotropy is here characterized by The axes A parameter α (0, 2) or a basis of R 2 and a matrix D α = α = 1 Isotropic case ( α 0 ) 0 2 α

Anisotropy Notion of Operator scaling (anisotropic self-similarity) a > 0 f (a Dα x) = f (a α x 1, a 2 α x 2 ) = a H f (x 1, x 2 ) Quasi (or pseudo) norm : positive continuous function, D α homogeneous ρ α (a Dα x) = aρ α (x) Properties : ρ α (x + y) C(ρ α (x) + ρ α (y)) 2 D α pseudo-norms define the same topology. ( ) Example : ρ α (x) = x 1 2 2 1/2. α + x 2 2 α

Anisotropy Anisotropic unit ball - Action of the scaling x λ Dα x on an ellipse.

Operator scaling Gaussian Random Field Operator Scaling Gaussian Random Field (OSGRF and extensions to OSSRF) - Biermé et al. (07) Fix α and a pseudo-norm ρ α 0 < H< min(α, 2 α) Define X α,h,ρα (x) = (e i x, ξ 1) dŵ (ξ), R 2 (ρ α (ξ)) (H+1)

Some anisotropic models OSGRF (and extensions to OSSRF) X α,h,ρα is a gaussian random field has stationary increments is Operator Scaling, that is X α,h,ρα (a α x 1, a 2 α x 2 ) d = a H X α,h,ρα (x 1, x 2 ).

Typical OSGRF (H=0.6) E = ( ) 1 0 0 1 ( ) 0.7 0 E = 0 1.3 E = ( ) 1 0 1 1 E = ( 1 ) 1 1 1

Anisotropic functional spaces Basic example 1 : Anisotropic Sobolev space (α = 1/2) H 3,α (R 2 ) = {f L 2 ; (s 1 = 6 = 3 α, s 2 = 2 = 3 2 α ). 6 x 1 f L 2 and 2 x 2 f L 2 and...}

Anisotropic functional spaces Basic example 2 : Anisotropic Hölder spaces : C > 0, (x, y) R 2, f (y) f (x) C (ρ α (y x)) s ( C y 1 x 1 2 α + y2 x 2 ( (α = 1/2) C y 1 x 1 4 + y 2 x 2 4 3 ) 2 s 2 α ) s/2

Anisotropic functional spaces Anisotropic Littlewood-Paley analysis : let ϕ α 0 0, S(R2 ) such that ϕ α 0 (x) = 1 if ξ 1, For j N, we define and ϕ α 0 (x) = 0 if 2 Dα ξ 1. ϕ α j (x) = ϕ α 0 (2 jdα ξ) ϕ α 0 (2 (j 1)Dα ξ). Then + j=0 ϕ α j 1, and (ϕ α j ) j 0 is called an anisotropic resolution of the unity. supp (ϕ α 0 ) R1 α, supp ( ϕ α ) j R α j+1 \ Rj 1 α, where R α j = {ξ = (ξ 1, ξ 2 ) R 2 ; ξ 1 2 jα, and ξ 2 2 j(2 α) }.

Anisotropic functional spaces Anisotropic Littlewood analysis : For f S (R 2 ) let α j (ϕ f = F 1 α f ) j. 80 60 40 20 0-20 - 40-60 - 80-80 - 60-40 - 20 0 20 40 60 80

Anisotropic functional spaces Anisotropic Besov spaces : ( α j f = F 1 ϕ α f ) j. Definition The Besov space Bp,q s,α (R 2 ), for 0 < p +, 0 < q +, s R, is defined by Bp,q s,α (R 2 ) = f S (R 2 ); 1/q 2 jsq α j f q p < + j 0.

Anisotropic wavelets Classical wavelets are clearly not adapted 40 80 30 60 20 40 10 20 0 0-10 - 20-20 - 40-30 - 60-40 - 40-30 - 20-10 0 10 20 30 40-80 - 80-60 - 40-20 0 20 40 60 80 It does not fit with the anisotropic Littlewood Paley analysis

Anisotropic wavelets : Triebel basis Hochmuth, Triebel : construction of a orthonormal anisotropic wavelet basis adapted to an anisotropy α : Looks like : ψ(2 jα x 1 k 1 )ψ(2 j(2 α) x 2 k 2 ), j 0, k 1, k 2 Z ϕ(2 jα x 1 k 1 )ψ(2 j(2 α) x 2 k 2 ), j 0, k 1, k 2 Z ψ(2 jα x 1 k 1 )ϕ(2 j(2 α) x 2 k 2 ), j 0, k 1, k 2 Z

Anisotropic wavelets : Triebel basis Hochmuth, Triebel : construction of a orthonormal anisotropic wavelet basis adapted to an anisotropy α : Looks like : ψ(2 [jα] x 1 k 1 )ψ(2 [j(2 α)] x 2 k 2 ), j 0, k 1, k 2 Z ϕ(2 [jα] x 1 k 1 )ψ(2 [j(2 α)] x 2 k 2 ), j 0, k 1, k 2 Z ψ(2 [jα] x 1 k 1 )ϕ(2 [j(2 α)] x 2 k 2 ), j 0, k 1, k 2 Z = provides characterizations of Hölder, Sobolev, Besov spaces with anisotropy α... = Bp,q s,α (R 2 ) Bp,q(R s 2 ) = Transference method... (Triebel, Theory of functions III)

What to do when α is not known?

Anisotropic wavelets Directional wavelets (J.P. Antoine et al, 93) Hyperbolic wavelet analysis (De Vore et al, 98) Curvelets (Candès - Donoho, 02) Bandlets (Le Pennec - Mallat, 03) Anisotropic wavelets (Triebel, 10) Shearlets (Kutyniok et al.,10)...

Hyperbolic wavelet analysis Hyperbolic wavelet basis : ψ j1,j 2,k 1,k 2 (x 1, x 2 ) = ψ(2 j 1 x 1 k 1 )ψ(2 j 2 x 2 k 2 ), j 1, j 2 0, k 1, k 2 Z (DeVore, Konyagin, Temlyakov, 98) Rectangular supports of size 2 j 1 2 j 2

Hyperbolic wavelet analysis Theorem (P. Abry, M. Clausel, S. Jaffard, S. Roux, B.V.) If f C s,α (R 2 ) then (j 1, j 2 ) sup c j1,j 2,k 1,k 2 (f ) C2 max( k 1,k 2 j 1α j, 2 2 α )s ( ) (starting with a 1D Meyer wavelet).

Hyperbolic wavelet analysis Theorem (P. Abry, M. Clausel, S. Jaffard, S. Roux, B.V.) If f C s,α (R 2 ) then (j 1, j 2 ) sup c j1,j 2,k 1,k 2 (f ) C2 max( k 1,k 2 j 1α j, 2 2 α )s ( ) Conversely, if ( ) holds, f C s,α log (R2 ). General version of this result for anisotropic Besov spaces : f B s,α p,q log (j 1,j 2 ) N 2 0 2 max( j 1 j α, 2 2 α )sq 2 (j 1 +j 2 )q p c j1,j 2,, q l p < +. (starting with a 1D Meyer wavelet).

Sketch of the proof 1st step : To obtain a characterization of B s,α p,q (R 2 ) in term of Hyperbolic wavelet characterization : θ j1,j 2 = τ j1 τ j2 with τ a 1D resolution of unity. j1,j 2 (f ) = F 1 ( j1,j 2 ( f )).

Sketch of the proof Isotropic case : we can replace j (f ) with j 1,j 2 j1,j 2 (f ) with max(j 1, j 2 ) = j. About 3 j terms

Sketch of the proof Anisotropic case

Sketch of the proof Anisotropic case we can replace α j (f ) with j 1,j 2 j1,j 2 (f ) with max( j 1 j α, 2 2 α ) j. About C α j terms

Sketch of the proof f B s,α p,q (R 2 )? 1/q 2 jsq α j f q p j 0 2 jsq j 0 2 jsq j 0 < + j1,j2 j 1,j 2 ; max( j 1 j α, 2 2 α ) j j1,j2 j 1,j 2 ; max( j 1 j α, 2 2 α ) j (f ) (f ) q p q p 1/q 1/q < + < +

Sketch of the proof f B s,α p,q (R 2 )? 1/q 2 jsq α j f q p j 0 2 jsq j 0 2 jsq j 0 < + j1,j2 j 1,j 2 ; max( j 1 j α, 2 2 α ) j j1,j2 j 1,j 2 ; max( j 1 j α, 2 2 α ) j (f ) (f ) q p q p 1/q 1/q < + < + OK if p = q = 2

Sketch of the proof f B s,α p,q (R 2 )? 1/q 2 jsq α j f q p j 0 2 jsq j 0 2 jsq j 0 < + j1,j2 j 1,j 2 ; max( j 1 j α, 2 2 α ) j j1,j2 j 1,j 2 ; max( j 1 j α, 2 2 α ) j (f ) (f ) q p q p 1/q 1/q < + < + OK but with a logarithmic correction in the other cases.

Sketch of the proof 2nd step : The wavelet coefficients are obtained by the Fourier analysis of j1,j 2 (f ).

Hyperbolic wavelet analysis Benefits : No a priori knowledge of α (Universal basis... but axis are known) Orthonormal basis (in particular, no redundancy useful also for synthesis)

Hyperbolic wavelet analysis Benefits : No a priori knowledge of α (Universal basis... but axis are known) Orthonormal basis (in particular, no redundancy useful also for synthesis) Applications : Texture analysis : OSGRF (cf. last part)

Beyond global regularity Analysis of complex textures taking into account local regularity and anisotropy?

Beyond global regularity Analysis of complex textures taking into account local regularity and anisotropy? In the isotropic case, a well-known tool : Multifractal analysis

Beyond global regularity Analysis of complex textures taking into account local regularity and anisotropy? In the isotropic case, a well-known tool : Multifractal analysis Definition of an extended multifractal analysis

Beyond global regularity Pointwise regularity : isotropic case f locally bounded on R 2, s > 0. f C s (x 0 ) if C > 0, δ and P x0 with deg(p x0 ) s such that x x 0 δ = f (x) P x0 (x) C x x 0 s where = ρ is the euclidian norm isotropic pointwise exponent : h f (x 0 ) = sup{s, f C s (x 0 )} Characterization of h f (x 0 ) via the classical wavelet basis.

Beyond global regularity Pointwise regularity : Anisotropic case f locally bounded on R 2, s > 0. f C s,α (x 0 ) if C > 0, δ and P x0 with deg α (P x0 ) s such that x x 0 δ = f (x) P x0 (x) C x x 0 s α where α = ρ α is an anisotropic pseudo-norm. Anisotropic pointwise exponent : h f,α (x 0 ) = sup{s, f C s,α (x 0 )} Characterization of h f,α (x 0 ) via the Triebel wavelet basis.

Wavelet leaders dyadic rectangle : λ = λ(j 1, j 2, k 1, k 2 ) = [ k 1 2 j, k 1 + 1 1 2 j [ [ k 2 1 2 j, k 2 + 1 2 2 j [, 2 Hyperbolic wavelet leaders For any x 0 = (a, b) R 2, where d j1,j 2 (x 0 ) = 3λ j1,j 2 (x 0 ) = [ [2j 1 a] 1 2 j 1 d λ = sup c λ. λ λ sup c λ. λ 3λ j1,j 2 (x 0 ), [2j 1 a] + 2 2 j 1 [ [ [2j 2 b] 1 2 j 2, [2j 2 b] + 2 2 j [, 2

Local regularity Hyperbolic wavelet characterizations of C s,α (x 0 ) f C s,α (x 0 ) = C > 0 such that for any j 1, j 2 one has d j1,j 2 (x 0 ) C2 max( j 1 α, 2 α )s. (1) j 2 f C ε 0 (R 2 ) + (1) = f C s,α log 2 (x 0 ).

Beyond global regularity Iso anisotropic Hölder sets : E f (H, α) = {x R 2, h f,α (x) = H}

Beyond global regularity Iso anisotropic Hölder sets : E f (H, α) = {x R 2, h f,α (x) = H} Hyperbolic spectrum of singularities of f on (R + { }) (0, 2) d(h, α) = dim H (E f (H, α)).

Beyond global regularity For a locally bounded function Hyperbolic partition functions S(j, p, α) = 2 2j (j 1,j 2 ); max( j 1 α, j 2 α )=j (k 1,k 2 ) Z 2 d p j 1,j 2,k 1,k 2,

Beyond global regularity For a locally bounded function Hyperbolic partition functions S(j, p, α) = 2 2j Anisotropic scaling function (j 1,j 2 ); max( j 1 α, j 2 α )=j ω f (p, α) = lim inf j (k 1,k 2 ) Z 2 d p j 1,j 2,k 1,k 2, log S(j, p, α) log 2 j,

Beyond global regularity For a locally bounded function Hyperbolic partition functions S(j, p, α) = 2 2j Anisotropic scaling function (j 1,j 2 ); max( j 1 α, j 2 α )=j ω f (p, α) = lim inf j Legendre Hyperbolic Spectrum : (k 1,k 2 ) Z 2 d p j 1,j 2,k 1,k 2, log S(j, p, α) log 2 j, L f (H, α) = inf p R {Hp ω f (p, (α, 2 α)) + 2}.

Beyond global regularity Extended multifractal formalism For f C ε 0 (R 2 ), one has (H, α) (R +) (0, 2), d f (H, α) L f (H, α). If (H, α) (R +) (0, 2), d(h, α) = L f (H, α), then f is said to satisfy the hyperbolic multifractal formalism.

Analysis of OSGRF (θ 0 = 0, α 0 = 1, H 0 = 0.2) (θ 0 = 0, α 0 = 0.7,H 0 = 0.2) (θ 0 = 0, α 0 = 0.3, H 0 = 0.2) (θ 0 = π/6, α 0 = 0.7, H 0 = 0.2)(θ 0 = π/4, α 0 = 0.7, H 0 = 0.2)(θ 0 = π/3, α 0 = 0.7, H 0 = 0.2)

Directional Regularity : Anisotropic spaces Theorem Let D α = ( ) α 0 and 0 < α 0 2 α 0 < 2. One has 1. (Biermé, Lacaux, 09) 2. (Clausel, V.) sup{s, X Dα0,H 0,ρ C s,α 0 loc } = H 0 sup{s, X Dα0,H 0,ρ C s,α } < H 0 Proof : expansion on anisotropic wavelet bases.

Estimation procedure Wavelet coefficients d j1,j 2,k 1,k 2 Structure function Z(q, j 1, j 2 ) = 1 N j1,j 2 j 1,j 2 d j1,j 2,k 1,k 2 q N j1,j 2 : number of coefficients at scale (j 1, j 2 ). ζ(q, α) = lim inf j α log 2 (Z(q, [αj], [(2 α)j])) j Estimators α q = argmax α ζ(q, α) Ĥ q = ζ(q, α)/q.

Estimation procedure Image log 2 (Z(q = 2, j 1, j 2 )) 7 5 j1 3 1 1 3 5 7 j 2

Estimation procedure Image log 2 (Z(q = 2, j 1, j 2 )) 7 α = 1.5 ; j1 5 3 αj 1 = (2 α)j 2 ; α log 2 (Z(q, αj, [(2 α)j])) 2 2.5 3 3.5 4 4.5 Interpolation 1 3 5 7 j 1 1 3 5 7 j 2

Estimation procedure Image log 2 (Z(q = 2, j 1, j 2 )) 7 α = 1.5 ; j1 5 3 αj 1 = (2 α)j 2 ; α log 2 (Z(q, αj, [(2 α)j])) 2 2.5 3 3.5 4 4.5 Regression ζ(q, α) 1 3 j 5 7 ζ(q, α) 1 1 0.5 0 0.5 1 3 5 7 j 2 0 1. 2 α

Estimation procedure Image log 2 (Z(q = 2, j 1, j 2 )) 7 α = 1 ; j1 5 3 αj 1 = (2 α)j 2 ; α log 2 (Z(q, αj, [(2 α)j])) 3 2 1 0 1 ζ(q, α) 1 3 5 7 j ζ(q, α) 1 1 0.5 0 0.5 1 3 5 7 j 2 0 1. 2 α

Estimation procedure Image log 2 (Z(q = 2, j 1, j 2 )) 7 5 j1 3 1 1 1 3 5 7 j 2 0.5 ζ(q, α) 0 0.5 0 1. 2 α

Estimation procedure Image log 2 (Z(q = 2, j 1, j 2 )) 7 j1 5 3 ˆα ˆαj 1 = (2 ˆα)j 2 ; 1 1 3 5 7 j 2 α log 2 (Z(q, αj, [(2 α)j])) 3 2 1 0 1 2 ˆζ(q) 1 3 5 7 j ζ(q, α) 1 ˆζ(q) 0.5 0 0.5 0 ˆα 1. α 2 ˆα ζ(q) = argmax α ζ(q, α) ; ˆζ(q) = ζ(q, ˆα ζ(q) ) ;

Estimation : θ 0 unknown Isotropic Anisotropic ˆα Ĥ (α 0 = 0.7, H 0 = 0.6) (α 0 = 0.3, H 0 = 0.2) (α 0 = 0.7, H 0 = 0.2) 1.3 1.8 1.3 1.4 1.1 1.1 1 0.9 0.6 0.9 0.7 1 0.9 0.8 0.7 0.6 0.5 0 π 4 π 2 θ 3π 4 π 0.2 0.7 0.6 0.5 0.4 0.3 0.2 0 π 4 Efficient and fast estimation procedures. Joint estimation of parameters (α 0, H 0, θ 0 ) π 2 θ 3π 4 π 0.7 0.5 0.4 0.3 0.2 0 π 4 Bootstrap : test for isotropy (and multifractality). π 2 θ 3π 4 π

Extended Fractional Brownian Fields (EFBF) Bonami, Estrade (03) X (a,β) = (e i x, ξ 1)f (ξ)dŵ (ξ), R 2 f (ξ) = ξ 2h(arg(ξ)) 2, H 1 = max h(arg(ξ)), H 2 = min h(arg(ξ)). H 2 = H 1 isotropic. 1 Ĥ(α) 0.5 0 100 real. (2 10, 2 10 ) 0.5 0.6 0.4 0.8 1 1.2 1 0 0.5 1 1.5 2 H 1 = 0.8 et H 2 = 0.2, H 2 = 0.4, H 2 = 0.6, H 2 = 0.8. α.

Fractional Brownian Sheet (FBS) (e i<x1,ξ1> 1)(e i<x2,ξ2> 1) B H1,H 2 (x) = R 2 ξ 1 H 1+ 1 2 ξ 2 H 2+ 1 dŵξ 1,ξ 2, 2. 1.5 Ĥ(α) 1 100 real. (2 10, 2 10 ) 0.5 0 0 0.5 1 1.5 2 H 1 = 0.8 et H 2 = 0.2, H 2 = 0.4, H 2 = 0.6, H 2 = 0.8. α.

Multifractal case : (α 0 = 0.8, H 0 = 0.7,λ 0 = 0) (α 0 = 0.8,H 0 = 0.7,λ 0 = 0.1) (α 0 = 0.8, H 0 = 0.7,λ 0 = 0.2) (α 0 = 0.8, H 0 = 0.3,λ 0 = 0) (α 0 = 0.8, H 0 = 0.3,λ 0 = 0.1) (α 0 = 0.8, H 0 = 0.3,λ 0 = 0.2)

Estimation : Isotropic Anisotropic 2 1.5 1 0.5 0 2 1.5 1 0.5 ˆα ˆζ(q) 0 4 ζ(q) D(q) h(q) c(q) 4 2 0 2 4 q 4 2 0 2 4 4 2 0 2 2 1.5 1 0.5 0 2 1.5 1 0.5 ˆD(h) 0 0.2 0.6 1 h.

Conclusion A lot has been done for analysis at prescribed anisotropy

Conclusion A lot has been done for analysis at prescribed anisotropy Hyperbolic wavelet analysis is an efficient tool for unknown anisotropy

Conclusion A lot has been done for analysis at prescribed anisotropy Hyperbolic wavelet analysis is an efficient tool for unknown anisotropy What about axes (global, local)?

Conclusion A lot has been done for analysis at prescribed anisotropy Hyperbolic wavelet analysis is an efficient tool for unknown anisotropy What about axes (global, local)? Anisotropic multifractal models

Thank you for your attention!