HWT and anisotropic texture analysis

Transcription

1 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)

2 Introduction Anisotropic image = different characteristics according to the directions.

3 Introduction Anisotropic image = different characteristics according to the directions. Wide range of applications : medical imaging, hydrology, image processing...

4 Introduction Anisotropic image = different characteristics according to the directions. Wide range of applications : medical imaging, hydrology, image processing...

5 Introduction Some natural questions : What is anisotropy?

6 Introduction Some natural questions : What is anisotropy? Isotropic images vs anisotropic images?

7 Introduction Some natural questions : What is anisotropy? Isotropic images vs anisotropic images? Relevant parameters to describe anisotropy?

8 Introduction Some natural questions : What is anisotropy? Isotropic images vs anisotropic images? Relevant parameters to describe anisotropy? Efficient tools to analyze anisotropic images? Directional wavelets (J.P. Antoine et al, 1993), Hyperbolic wavelet analysis (De Vore et al. 1998), Ridgelets (Candès and Donoho 1999), Curvelets (Candès and Donoho 2002), Bandlets (Le Pennec and Mallat 2003), Shearlets (Kutyniok et al. 2010)...

9 Introduction Study of an anisotropic self similar model.

10 Introduction Study of an anisotropic self similar model. Classical notion of self similarity : {X(ax)} x R 2 L = {a H 0 X(x)} x R 2,H 0 (0,1).

11 Introduction Study of an anisotropic self similar model. Classical notion of self similarity : {X(ax)} x R 2 L = {a H 0 X(x)} x R 2,H 0 (0,1). Example : Fractional Brownian Motion (Kolmogorov, 1940, Mandelbrot Van Ness 1968).

12 Introduction Anisotropic and self similar random fields? (H.Biermé M.Meerschaert H.P.Scheffler, 2007) {X(a E 0 x)} x R 2 L = {a H 0 X(x)} x R 2 where a E 0 = exp(e 0 log(a)), E 0 : anisotropy matrix, H 0 : self similarity index.

13 Introduction Anisotropic and self similar random fields? (H.Biermé M.Meerschaert H.P.Scheffler, 2007) {X(a E 0 x)} x R 2 L = {a H 0 X(x)} x R 2 where a E 0 = exp(e 0 log(a)), E 0 : anisotropy matrix, H 0 : self similarity index. E 0 = Id : classical notion of self similarity. Isotropic field.

14 Introduction Usual scaling x λx replaced with linear scaling x λ E 0 x. Action of a linear scaling x λ E 0 x on an ellipse.

15 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Construction of anisotropic operator scaling random fields with stationary increments.

16 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Construction of anisotropic operator scaling random fields with stationary increments. Gaussian case : can be defined using an harmonizable representation ( ) X ρ (x 1,x 2 ) = Re (e i(x 1ξ 1 +x 2 ξ 2 ) (H 1)ρ(ξ) 0+1) dŵ(ξ). R 2

17 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Construction of anisotropic operator scaling random fields with stationary increments. Gaussian case : can be defined using an harmonizable representation ( ) X ρ (x 1,x 2 ) = Re (e i(x 1ξ 1 +x 2 ξ 2 ) (H 1)ρ(ξ) 0+1) dŵ(ξ). R 2 ρ (R 2,E t 0 ) pseudo-norm = positive, t E0 -homogeneous continuous function : ξ = (ξ 1,ξ 2 ) R 2, ρ(a te 0 ξ) = aρ(ξ).

18 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Example ( ) α0 0 E 0 = with α 0 2 α 0 (0,2), 0 H 0 (0,min(α 0,2 α 0 )).

19 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Example ( ) α0 0 E 0 = with α 0 2 α 0 (0,2), 0 H 0 (0,min(α 0,2 α 0 )). ρ(ξ 1,ξ 2 ) = ξ 1 1/α 0 + ξ 2 1/(2 α0) (R 2,E t 0 ) pseudo-norm.

20 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Example ( ) α0 0 E 0 = with α 0 2 α 0 (0,2), 0 H 0 (0,min(α 0,2 α 0 )). ρ(ξ 1,ξ 2 ) = ξ 1 1/α 0 + ξ 2 1/(2 α0) (R 2,E t 0 ) pseudo-norm. Anisotropic operator scaling Gaussian field e i(x1ξ1+x2ξ2) 1 X(x 1,x 2 ) = ( ξ1 1/α 0 + ξ2 ) 1/(2 α H 0) 0 +1 dŵ(ξ). R 2

21 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Example ( ) α0 0 E 0 = with α 0 2 α 0 (0,2), 0 H 0 (0,min(α 0,2 α 0 )). ρ(ξ 1,ξ 2 ) = ξ 1 1/α 0 + ξ 2 1/(2 α0) (R 2,E t 0 ) pseudo-norm. Anisotropic operator scaling Gaussian field e i(x1ξ1+x2ξ2) 1 X(x 1,x 2 ) = ( ξ1 1/α 0 + ξ2 ) 1/(2 α H 0) 0 +1 dŵ(ξ). R 2 Operator scaling relationship satisfied by X : a > 0, {X(a α 0 x 1,a 2 α 0 x 2 )} (x1,x 2 ) R 2 L = {a H 0 X(x 1,x 2 )} (x1,x 2 ) R 2.

22 An anisotropic model (case H 0 = 0.6, α = 0.7) (H.Biermé M.Meerschaert H.P.Scheffler, 2007) X(x) = R 2 e i(x1ξ1+x2ξ2) 1 ( ξ1 1/α 0 + ξ2 1/(2 α 0) ) H 0 +1 dŵ(ξ).

23 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Studied model : operator scaling Gaussian fields driven by two parameters E 0 (anisotropy), H 0 (self similarity).

24 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Studied model : operator scaling Gaussian fields driven by two parameters E 0 (anisotropy), H 0 (self similarity). Estimation of these two parameters?

25 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Studied model : operator scaling Gaussian fields driven by two parameters E 0 (anisotropy), H 0 (self similarity). Estimation of these two parameters? Characteristic properties of E 0 et H 0?

26 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Studied model : operator scaling Gaussian fields driven by two parameters E 0 (anisotropy), H 0 (self similarity). Estimation of these two parameters? Characteristic properties of E 0 et H 0? Global smoothness of theses fields in well adapted anisotropic functional spaces Optimality results.

27 An anisotropic model (H.Biermé M.Meerschaert H.P.Scheffler, 2007) Studied model : operator scaling Gaussian fields driven by two parameters E 0 (anisotropy), H 0 (self similarity). Estimation of these two parameters? Characteristic properties of E 0 et H 0? Global smoothness of theses fields in well adapted anisotropic functional spaces Optimality results. Extension of results of Biermé Meerschaert Scheffler (2007), Xiao (2009) (Gaussian case), Biermé Lacaux (2009) (α stable operator scaling random fields).

28 Optimality results Anisotropic topology Let E be a (R 2,E) pseudo norm. Then for some C > 0 (x,y) (R 2 ) 2, x +y E C( x E + y E ).

29 Optimality results Anisotropic topology Let E be a (R 2,E) pseudo norm. Then for some C > 0 (x,y) (R 2 ) 2, x +y E C( x E + y E ). Two (R 2,E) pseudo norms are equivalent ie define the same topology.

30 Optimality results Anisotropic topology Isotropic and anisotropic balls.

31 Optimality results Anisotropic Hölder spaces E anisotropy, E (R 2,E) pseudo-norm, 0 < H < ρ min (E). f C H (R d,e) if C > 0, (x,y) R 2, f(y) f(x) C y x H E.

32 Optimality results Anisotropic Hölder spaces E anisotropy, E (R 2,E) pseudo-norm, 0 < H < ρ min (E). f C H (R d,e) if C > 0, (x,y) R 2, f(y) f(x) C y x H E. Critical exponent in anisotropic Hölder spaces s(f,e) = sup{h, f C H (R 2,E)}.

33 Optimality results Anisotropic Hölder spaces E anisotropy, E (R 2,E) pseudo-norm, 0 < H < ρ min (E). f C H (R d,e) if C > 0, (x,y) R 2, f(y) f(x) C y x H E. Critical exponent in anisotropic Hölder spaces s(f,e) = sup{h, f C H (R 2,E)}. Local version of this exponent s loc (f,e).

34 Optimality results Theorem (M.C., B. Vedel) E 0 anisotropy matrix, ρ a (R 2,E 0 ) pseudo norm. ( X ρ (x 1,x 2 ) = Re (e i(x 1ξ 1 +x 2 ξ 2 ) 1)ρ(ξ) R 2 Then a.s. (H 0+1) dŵ(ξ) ). H 0 = max{s loc (X ρe0,e),e anisotropy}, and E 0,D 0 Argmax({s loc (X ρe0,e),e anisotropy}). D 0 : diagonalizable real part of de E 0.

35 Optimality results Case E = E 0 well known : Biermé et al.(2007), Xiao (2009), Biermé Lacaux (2009).

36 Optimality results Case E = E 0 well known : Biermé et al.(2007), Xiao (2009), Biermé Lacaux (2009). L p version of this result in anisotropic Besov spaces.

37 Optimality results Optimality results Methodology to estimate the anisotropy and the self similarity index.

38 Optimality results Optimality results Methodology to estimate the anisotropy and the self similarity index. Our tool : hyperbolic wavelet analysis.

39 Optimality results Optimality results Methodology to estimate the anisotropy and the self similarity index. Our tool : hyperbolic wavelet analysis. Key point : hyperbolic wavelet characterization of functional spaces.

40 Our analyzing tool : hyperbolic wavelet transform 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 ) where ψ 1D wavelet.

41 Our analyzing tool : hyperbolic wavelet transform 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 ) where ψ 1D wavelet. Hyperbolic wavelet basis associated to ψ {ψ j1,j 2,k 1,k 2, (j 1,j 2 ) Z 2, (k 1,k 2 ) Z 2 }, (DeVore,Konyagin,Temlyakov, 1998).

42 Our analyzing tool : hyperbolic wavelet transform 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 ) where ψ 1D wavelet. Hyperbolic wavelet basis associated to ψ {ψ j1,j 2,k 1,k 2, (j 1,j 2 ) Z 2, (k 1,k 2 ) Z 2 }, (DeVore,Konyagin,Temlyakov, 1998). Hyperbolic wavelet coefficients of f c j1,j 2,k 1,k 2 = 2 j 1+j 2 R 2 f(x 1,x 2 )ψ j1,j 2,k 1,k 2 (x 1,x 2 )dx 1 dx 2.

43 Our analyzing tool : hyperbolic wavelet transform Theorem (P. Abry, M.C., S. Jaffard, B. Vedel) ( ) α 0 Let D = with α (0,2). 0 2 α (i) If f C s (R 2,D) then there exists some C > 0 such that : (j 1,j 2 ), sup (k 1,k 2 ) Z 2 c j1,j 2,k 1,k 2 C2 max( j 1α, j 2 2 α )s. (1) (ii) Conversely, assume that (1) holds then f C s log (R2,D). General version of this result for anisotropic Besov spaces.

44 Our analyzing tool : hyperbolic wavelet transform Partial reformulation of our optimality results using hyperbolic wavelet analysis.

45 Our analyzing tool : hyperbolic wavelet transform Partial reformulation of our optimality results using hyperbolic wavelet analysis. ( ) α0 0 Anisotropy of the analyzed field E 0 = diagonal. 0 2 α 0

46 Our analyzing tool : hyperbolic wavelet transform Partial reformulation of our optimality results using hyperbolic wavelet analysis. ( ) α0 0 Anisotropy of the analyzed field E 0 = diagonal. 0 2 α 0 Hyperbolic wavelet characterization of anisotropic Besov spaces with diagonal anisotropy

47 Our analyzing tool : hyperbolic wavelet transform { H 0 = max lim inf j and E 0 Argmax { lim inf j ( log2 ( c [αj],[(2 α)j], l p) j ( log2 ( c [αj],[(2 α)j], l p) j ) }, α (0,2) ) }, α (0,2),.

48 Our analyzing tool : hyperbolic wavelet transform { H 0 = max lim inf j and E 0 Argmax { lim inf j ( log2 ( c [αj],[(2 α)j], l p) j ( log2 ( c [αj],[(2 α)j], l p) j ) }, α (0,2) ) }, α (0,2),. Advantage of our method :requires only one analysis basis.

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

50 Beyond OSSRGF Analysis of complex textures taking into account local regularity and anisotropy? In the isotropic case, a well known tool : multifractal analysis.

51 Beyond OSSRGF 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?

52 Beyond OSSRGF 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? Definition of an anisotropic multifractal formalism for fixed anisotropy using Triebel bases (Ben Braiek Benslimane, 2011).

53 Beyond OSSRGF Pointwise smoothness : isotropic case f locally bounded on R 2, s > 0. f C s log β (x 0 ) if C,δ > 0, P x0 with deg(p x0 ) s s.t. x x 0 δ f(x) P x0 (x) x x 0 s log( x x 0 ) β. Case β = 0 : C s log 0 (x 0 ) = C s (x 0 ).

54 Beyond OSSRGF Anisotropic concept of degree (Ben Braiek Benslimane, 2011) P polynomial of the form : P(t 1,t 2 ) = a β1,β 2 t β1 1 tβ 2 2, (β 1,β 2 ) N 2 and α (0,2). (α,2 α) homogeneous degree of P deg α (P) = sup{αβ 1 +(2 α)β 2,a β1,β 2 0};.

55 Beyond OSSRGF f locally bounded on R 2, s > 0, E = anisotropy. ( ) α 0 diagonal 0 2 α Pointwise smoothness and anisotropy (Ben Braiek Benslimane, 2011) f C s,(α,2 α) (x log β 0 ) if C,δ > 0, P x0 with deg α (P x0 ) s f(x) P x0 (x) C x x 0 s E log( x x 0 E ) β, with E (R 2,E) pseudo norm. Case β = 0 : C s,(α,2 α) (x log 0 0 ) = C s,(α,2 α) (x 0 ). Anisotropic pointwise exponent of f at x 0 : h f,α (x 0 ) = sup{s, f C s,(α,2 α) (x 0 )}. (2)

56 Beyond OSSRGF Hyperbolic dyadic cube λ = λ(j 1,j 2,k 1,k 2 ) = [ k 1 2 j, k j [ [ k j, k j [. 2

57 Beyond OSSRGF Hyperbolic dyadic cube λ = λ(j 1,j 2,k 1,k 2 ) = [ k 1 2 j, k j [ [ k j, k j [. 2 Hyperbolic wavelet leaders : and 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

58 Beyond OSSRGF Hyperbolic wavelet characterization of spaces C s,(α,2 α) (x 0 ) 1 f C s,(α,2 α) (x 0 ) C > 0, j 1,j 2 N { 1}, d j1,j 2 (x 0 ) C2 max(j 1 α, 2 f uniformly Hölder + (3) f C s,(α,2 α) log 2 (x 0 ). j 2 2 α )s. (3)

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

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

61 Beyond OSSRGF Hyperbolic partition functions of a locally bounded function as : S(j,p,α) = 2 2j d p j 1,j 2,k 1,k 2. (j 1,j 2 ) Γ j (α)(k 1,k 2 ) Z 2

62 Beyond OSSRGF Hyperbolic partition functions of a locally bounded function as : S(j,p,α) = 2 2j d p j 1,j 2,k 1,k 2. (j 1,j 2 ) Γ j (α)(k 1,k 2 ) Z 2 Anisotropic scaling function ω f (p,α) = liminf j logs(j,p,α) log2 j.

63 Beyond OSSRGF Hyperbolic partition functions of a locally bounded function as : S(j,p,α) = 2 2j d p j 1,j 2,k 1,k 2. (j 1,j 2 ) Γ j (α)(k 1,k 2 ) Z 2 Anisotropic scaling function ω f (p,α) = liminf j Legendre hyperbolic spectrum : logs(j,p,α) log2 j. L f (H,α) = inf p R {Hp ω f(p,(α,2 α))+2}.

64 Beyond OSSRGF Extended multifractal formalism f uniform Hölder function : (H,a) (R +) (0,2), d f (H,a) L f (H,a).

65 Bibliography 1 H. Biermé, M.M. Meerschaert, and H.P. Scheffler, Operator scaling stable random fields., Stoch. Proc. Appl. 117 (2009), no. 3, H. Biermé and C. Lacaux Holder regularity for operator scaling stable random fields. Stoch.Proc.Appl. 119(8) (2009) H. Ben Braiek and M.BenslimaneDirectional and anisotropic regularity and irregularity criteria in Triebel wavelet bases Submitted (2001). 4 R. A. DeVore, S. V. Konyagin, and V. N. Temlyakov, Hyperbolic wavelet approximation, Constr. Approx. 14 (1998), 1 26.

66 Bibliography 1 M.Clausel and B.Vedel Two optimality results about sample paths properties of Operator Scaling Gaussian Random Fields Submitted. 2 S.G. Roux, M.Clausel, B.Vedel, S.Jaffard, P.Abry, Transformée hyperbolique en ondelettes 2D pour la caractérisation des images autosimilaires anisotropes, XXIII GRETSI Bordeaux S.G. Roux, M.Clausel, B.Vedel, S.Jaffard, P.Abry, The Hyperbolic Wavelet Transform for self similar anisotropic texture analysis, Submitted P.Abry, M.Clausel, S.Jaffard, B.Vedel, S.G. Roux, The Hyperbolic Wavelet Transform : an efficient tool for anisotropic texture analysis, In preparation.