The hyperbolic wavelet transform : an efficient tool for anisotropic texture analysis . M.Clausel (LJK–Grenoble) Joint work with P.Abry (ENS–Lyon), S.Roux (ENS–Lyon), B.Vedel (Vannes), S.Jaffard(Créteil)

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M.Clausel (LJK–Grenoble)

Joint work with P.Abry (ENS–Lyon), S.Roux

(ENS–Lyon), B.Vedel (Vannes), S.Jaffard(Cr´eteil)

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Introduction

Anisotropic image = different characteristics according to the directions.

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directions.

Wide range of applications: medical imaging, hydrology, image processing...

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Introduction

Anisotropic image = different characteristics according to the directions.

Wide range of applications: medical imaging, hydrology, image processing...

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Some natural questions : What is anisotropy ?

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Introduction

Isotropic images vs anisotropic images ?

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Isotropic images vs anisotropic images ? Relevant parameters to describe anisotropy ?

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Introduction

Isotropic images vs anisotropic images ? Relevant parameters to describe anisotropy ? Efficient tools to analyze anisotropic textures ?

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Several phenomenas combinig anisotropy and scale–invariance properties :

Rainfall and cloud fields (Schertzer et al. 1985,1987, Kumar et al 1993).

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Introduction

Roughness surface properties (Davies and Hall 1999).

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Solute transport in aquifer

(Schumer–Benson–Meerschaert–Baeumer 2003).

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Introduction

Solute transport in aquifer

(Schumer–Benson–Meerschaert–Baeumer 2003).

Experimental fracture surfaces (Ponson et al. 2006)...

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Different anisotropic and self-similarmodels :

Operator self–similar processes : Dobrushin 1978, Laha–Rohatgi 1982, Hudson–Mason 1982.

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Introduction

Fractional Brownian Sheet : Kamont 1996, L´eger 2000, Ayache–L´eger–Pontier 2002.

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Fractional Brownian Sheet : Kamont 1996, L´eger 2000, Ayache–L´eger–Pontier 2002.

Operator Scaling Random fields : Schertzer 1985, 1987, Bierm´e Meerschaert–Scheffler 2009, Li–Xiao 2011.

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Introduction

Study of the anisotropic self–similarmodel of Bierm´e–Meerschaert–Scheffler (2009).

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Classical notion of self–similarity : {X(ax)}_x∈R²

=L {a^H⁰X(x)}_x∈R² ,H₀ ∈(0,1).

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Introduction

Classical notion of self–similarity : {X(ax)}_x∈R²

=L {a^H⁰X(x)}_x∈R² ,H₀ ∈(0,1). Example : Fractional Brownian Motion (FBM).

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Sample paths of FBM for H₀ = 0.2 etH₀= 0.5

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Introduction

Anisotropic self–similarity (Schertzer–Lovejoy, 1985) {X(a^E⁰x)}_x=(x₁_,x₂₎∈R²

=L {a^H⁰X(x)}_x=(x₁_,x₂₎∈R², witha^E⁰ = exp(E₀log(a)).

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E₀ : anisotropy matrix,H₀ : self–similarity index.

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Introduction

Examples :

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Examples :

E0=Id : classical notion of self–similarity. Isotropic field.

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Introduction

Examples :

E0=Id : classical notion of self–similarity. Isotropic field.

E0=

α0 0

0 β0

,H0∈(0,min(α0, β0)) {X(a^α⁰x1,a^β⁰x2)}x=(x1,x2)∈R²

=L {a^H⁰X(x1,x2)}x=(x1,x2)∈R²

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Usual scalingx 7→ax replaced withlinear scaling x7→a^E⁰x.

Action of a linear scaling x7→a^E⁰x on an ellipse.

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An anisotropic model

Construction of anisotropic self–similar Gaussian field : Bierm´e–Meerschaert–Scheffler, 2009.

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Synthesis in the Fourier domain = harmonizable representation

Xρ(x1,x2) =Re Z

R²

(e^i(x¹^ξ¹^+x²^ξ²⁾−1)ρ(ξ)⁻^(H⁰⁺¹⁾dWc(ξ)

.

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An anisotropic model

Synthesis in the Fourier domain = harmonizable representation

Xρ(x1,x2) =Re Z

R²

(e^i(x¹^ξ¹^+x²^ξ²⁾−1)ρ(ξ)⁻^(H⁰⁺¹⁾dWc(ξ)

.

ρ (R²,^tE₀) pseudo-norm= positive,^tE₀-homogeneous continuous function :

∀ξ= (ξ₁, ξ₂)∈R², ρ(a^t^E⁰ξ) =aρ(ξ).

⇒ X self–similar : anisotropy matrixE , self similarity index

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E0 =

α₀ 0 0 β₀

withα₀, β₀ >0, H0 ∈(0,min(α0, β₀)).

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An anisotropic model

Example

E0 =

α₀ 0 0 β₀

withα₀, β₀ >0, H0 ∈(0,min(α0, β₀)).

ρ(ξ₁, ξ₂) =|ξ₁|^1/α⁰+|ξ₂|^1/β⁰ (R²,^tE₀) pseudo–norm.

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E0 =

α₀ 0 0 β₀

withα₀, β₀ >0, H0 ∈(0,min(α0, β₀)).

Anisotropic self–similar Gaussian field X(x₁,x₂) =

Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

|ξ₁|^1/α⁰+|ξ₂|^1/β⁰H₀+1dWc(ξ).

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An anisotropic model

Example

E0 =

α₀ 0 0 β₀

withα₀, β₀ >0, H0 ∈(0,min(α0, β₀)).

Anisotropic self–similar Gaussian field X(x₁,x₂) =

Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

|ξ₁|^1/α⁰+|ξ₂|^1/β⁰H₀+1dWc(ξ). X satisfies :

∀a>0,{X(a^α⁰x₁,a^β⁰x₂)}_(x₁_,x₂₎∈R² (L)

= {a^H⁰X(x₁,x₂)}_(x₁_,x₂₎∈R² .

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X(x) = Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

|ξ₁|^1/α⁰+|ξ₂|^1/β⁰H0+1dWc(ξ).

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An anisotropic model

Hypothesis α₀+β₀= 2 : X_α₀_,H₀(x) =

Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

|ξ₁|^1/α⁰+|ξ₂|^1/(2⁻^α⁰⁾H0+1dWc(ξ) described by two parameters

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Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

AnisotropyE0=

α0 0

0 2−α0

forα0∈(0,2).

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An anisotropic model

Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

AnisotropyE0=

α0 0

0 2−α0

forα0∈(0,2).

H0(self–similarity).

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Z

R²

e^i(x¹^ξ¹^+x²^ξ²⁾−1

AnisotropyE0=

α0 0

0 2−α0

forα0∈(0,2).

H0(self–similarity).

Estimation of these two parameters ?

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An anisotropic model

FBM case : Hurst index H related to self–similarity properties of the random field B_H. For anyq >0

E(|B_H(x+h)−B_H(x)|^q) =C_H|h|^qH .

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E(|B_H(x+h)−B_H(x)|^q) =C_H|h|^qH . Classical approach for the estimation of the Hurst index : quadratic variations (Istas–Lang 1998)

Tj = 2⁻^j

2X^j−1

k=0

BH

k+ 1 2^j

−BH

k 2^j

q

∼2⁻^jqH .

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An anisotropic model

Tj = 2⁻^j

2X^j−1

k=0

BH

k+ 1 2^j

−BH

k 2^j

q

∼2⁻^jqH . Increments may be replaced by wavelets coefficients.

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Tj = 2⁻^j

2X^j−1

k=0

BH

k+ 1 2^j

−BH

k 2^j

q

∼2⁻^jqH . Increments may be replaced by wavelets coefficients.

Under suitable assumptions, CLT satisfied by estimator based on the regression log–log of this estimator.

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An anisotropic model

Our case : E₀ assumed to be diagonal with trace 2.

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Self–similarity implies

E(|X(x+a^E⁰h)−X(x)|^q) =C_X|a|^qH⁰ .

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An anisotropic model

E(|X(x+a^E⁰h)−X(x)|^q) =C_X|a|^qH⁰ . If E 6=E₀ andTr(E) = 2

E(|X(x+a^Eh)−X(x)|^p)≤C_X|a|^qH . withH ≤H₀ (M.C.–Vedel 2013).

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E(|X(x+a^E⁰h)−X(x)|^q) =C_X|a|^qH⁰ . If E 6=E₀ andTr(E) = 2

E(|X(x+a^Eh)−X(x)|^p)≤C_X|a|^qH . withH ≤H₀ (M.C.–Vedel 2013).

Approximation of |X(x+a^Eh)−X(x)|^q by means of well–adapted wavelet coefficients ?

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Usual bidimensional wavelet transform

1D wavelet transform : ϕ(L filter),ψ (H filter).

2D wavelet transform : 1 scaling function (LL filter), 3 mother wavelets (LH,HL,HH filters).

L² basis :

ϕ_k₁_,k₂(x₁,x₂) = ϕ(x₁−k₁),

ψ_j⁽¹⁾_,k₁_,k₂(x₁,x₂) = ϕ(2^jx₁−k₁)ψ(2^jx₂−k₂), ψ_j⁽²⁾_,k₁_,k₂(x₁,x₂) = ψ(2^jx₁−k₁)ϕ(2^jx₂−k₂), ψ_j⁽³⁾_,k

1,k2(x₁,x₂) = ψ(2^jx₁−k₁)ψ(2^jx₂−k₂). Wavelet coefficients

Z

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ψ_j₁_,j₂_,k₁_,k₂(x₁,x₂) =ψ(2^j¹x₁−k₁)ψ(2^j²x₂−k₂) whereψ 1D wavelet.

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Our analyzing tool : hyperbolic wavelet transform

Hyperbolic wavelet basis associated to ψ

{ψ_j₁_,j₂_,k₁_,k₂,(j₁,j₂)∈Z²,(k₁,k₂)∈Z²}, (DeVore,Konyagin,Temlyakov, 1998).

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Hyperbolic wavelet basis associated to ψ

{ψ_j₁_,j₂_,k₁_,k₂,(j₁,j₂)∈Z²,(k₁,k₂)∈Z²}, (DeVore,Konyagin,Temlyakov, 1998).

Hyperbolic wavelet coefficients c_j₁_,j₂_,k₁_,k₂ = 2^j¹^+j²

Z

R²

f(x₁,x₂)ψ_j₁_,j₂_,k₁_,k₂(x₁,x₂)dx₁dx₂.

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Estimation

Scaling invariance properties of X_α₀_,H₀?

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Scaling invariance properties of X_α₀_,H₀? Structure functions

S(q,j₁,j₂) = 1 Nj₁,j2

X

(j1,j2)∈Z²

|c_j₁_,j₂_,k₁_,k₂|^q ,

p >1,N_j₁_,j₂ : number of available coefficients at scales (j₁,j₂).

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Estimation

Theoretical result

S(q,j₁,j₂)≈2^(j¹^+j²^)q/22⁻^q(H⁰^{+1) max(}

j1 α0,₂₋^j²_α

0)

.

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Theoretical result

S(q,j₁,j₂)≈2^(j¹^+j²^)q/22⁻^q(H⁰^{+1) max(}

j1 α0,₂₋^j²_α

0)

. For α∈(0,2), define

τ(q, α) = lim inf

j→∞

−log₂(S(q,[αj],[(2−α)j])) j

.

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Estimation

Theoretical result

S(q,j₁,j₂)≈2^(j¹^+j²^)q/22⁻^q(H⁰^{+1) max(}

j1 α0,₂₋^j²_α

0)

τ(q, α) = lim inf

j→∞

−log₂(S(q,[αj],[(2−α)j])) j

.

α7→τ(q, α) maximal forα=α0.

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Theoretical result

S(q,j₁,j₂)≈2^(j¹^+j²^)q/22⁻^q(H⁰^{+1) max(}

j1 α0,₂₋^j²_α

0)

τ(q, α) = lim inf

j→∞

−log₂(S(q,[αj],[(2−α)j])) j

.

α7→τ(q, α) maximal forα=α0. τ(q, α0) =qH0.

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Estimation

Estimation ofα₀ : ˆ

α(q) =Argmax_α∈(0,2)

τ(q, α) q

,

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Estimation ofα₀ : ˆ

α(q) =Argmax_α∈(0,2)

τ(q, α) q

,

Estimation ofH₀

H(q) = maxˆ

α∈(0,2)

τ(q, α) q

.

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Estimation

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Estimation

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Estimation

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Estimation

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First results

Estimation performance of Hb andαbperformed on 100 realizations of (α,H₀).

.

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100 realizations of (α,H) of sizeN×N .

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Beyond OSSRGF : unknown axes

What happens if the axes are unknown ?

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What happens if the axes are unknown ? Model

X_α₀_,H₀_,θ₀ = Z

R²

(e^ixξ−1)f(ξ)⁻^H⁰⁻¹dWc(ξ), withf(ξ) =|ζ₁|^1/α⁰+|ζ₂|^1/(2⁻^α⁰⁾ whereζ =R_θ₀ξ.

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Beyond OSSRGF : unknown axes

What happens if the axes are unknown ? Model

X_α₀_,H₀_,θ₀ = Z

R²

(e^ixξ−1)f(ξ)⁻^H⁰⁻¹dWc(ξ), withf(ξ) =|ζ₁|^1/α⁰+|ζ₂|^1/(2⁻^α⁰⁾ whereζ =R_θ₀ξ.

Self–similarity properties and anisotropy

{X_α₀_,H₀_,θ₀(a^E⁰x)}⁽=^L⁾{a^H⁰X(x)}, withE₀ =R_θ₀

α₀ 0 0 2−α

R_θ⁻¹

0 .

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Method

Rotation of the image I :I_θ=R_θI. Estimation ofH,b αb onI_θ ⇒ H(θ),b α(θ).b Empirical resultH maximal ifθ=θ₀,α=α₀.

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Hyperbolic wavelet analysis and anisotropy : what else ?

Statistical properties of the estimators (work in progress).

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Beyond anisotropic self–similarity : anisotropic multifractal analysis (P.Abry–M.C.–S.Jaffard–S.G.Roux–B.Vedel, 2015).

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Hyperbolic wavelet analysis and anisotropy : what else ?

Another point of view for anisotropy : oriented textures (G.

Peyr´e 2007).

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Peyr´e 2007).

Decomposition of multicomponent images into several modes and estimatio of the orientation at any point using

bidimensional synchrosqueezing (M.C.–T.Oberlin, V. Perrier, 2015).

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Hyperbolic wavelet analysis and anisotropy : what else ?

Peyr´e 2007).

Synthesis of textures with prescribed orientation (K.

Polisano–M.C.–V. Perrier–L. Condat, 2014)

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Peyr´e 2007).

Synthesis of textures with prescribed orientation (K.

Polisano–M.C.–V. Perrier–L. Condat, 2014)

Denoising of anisotropic ultrasound images (Y. Farouj, J.M.

Freyermuth, L.Navarro, M.C., P. Delachartre 2015, to be submitted).

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Bibliography

1 D. Schertzer and S. Lovejoy, Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades, J. Geophys. Res. 92 (1987), 9693–9714.

2 H. Bierm´e, M.M. Meerschaert, and H.P.

Scheffler, Operator scaling stable random fields., Stoch.

Proc. Appl. 117 (2009), no. 3, 312–332.

3 R. A. DeVore, S. V. Konyagin, and V. N.

Temlyakov, Hyperbolic wavelet approximation, Constr.

Approx. 14 (1998), 1–26.

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sample paths properties of Operator Scaling Gaussian Random Fields Ann. Univ. Buch. (2013).

2 S.G. Roux, M.Clausel, B.Vedel, S.Jaffard,

P.Abry, The Hyperbolic Wavelet Transform for self–similar anisotropic texture analysis,IEEE TIP 2013.

3 P.Abry, M.Clausel, S.Jaffard, B.Vedel, S.G.

Roux, The Hyperbolic Wavelet Transform : an efficient tool for anisotropic texture analysis,Rev. Mat. Iber. 2015.

4 M.Clausel, T.Oberlin, V. Perrier, The Monogenic Synchrosqueezed Wavelet Transform : A tool for the Decomposition/Demodulation of AM-FM images. ACHA 2015.

5 K. Polisano, M. Clausel, V. Perrier, L. Condat, Texture modeling by Gaussian fields with prescribed local