A Wavelet-based Multifractal Analysis for scalar and vector fields: Application to developped turbulence

Transcription

1 UCLA/IPAM Multiscale Geometry and Analysis in High Dimensions MGA Workshop IV, nov. 8-12, 2004 A Wavelet-based Multifractal Analysis for scalar and vector fields: Application to developped turbulence Pierre Kestener, Alain Arneodo Laboratoire de Physique, ENS Lyon, France present address: CEA Saclay, DSM/DAPNIA/SEDI, Gif-Sur-Yvette, FRANCE

2 Contents Introduction Characterization of multifractal images with the 2D WTMM method Generalizations of the WTMM method from 2D to 3D from scalar to vector Application to developped turbulence (direct numerical simulations) 3D scalar case : dissipation field and enstrophy field 3D vector case : velocity field and vorticity field Application to galaxy distribution (simulation data)

3 Fractal Objects : self-similarity exact self-similarity : statistical self-similarity : f(x 0 + λu) f(x 0 ) λ h(x 0) ( f(x 0 + u) f(x 0 ) )

4 Self-similarity and Wavelet Transform Wavelet Transform : a mathematical microscope to study fractal objects self-similarity properties T ψ [f](a, b) =< f, ψ a,b >= 1 a + dx f(x)ψ ( x b a ) local self-similarity (Hölder exponent) can be seen in the wavelet transform coefficients under scaling laws (Jaffard, Mallat et coll., Holschneider and Tchamitchian) : T ψ [f](a, x 0 ) a h(x 0)+ 1 2, a 0 + statistical description of self-similarity : WTMM method (Wavelet Transform Modulus Maxima) by Muzy, Bacry, Arnéodo (1993) for multifractal signals. many applications in bioinformatics (DNA Sequences), in turbulence, in finance, in geophysics,...

5 classical multifractal formalism comes from turbulence The Navier-Stokes equations: t u + u. u = 1 p + f + ν u, ρ +.u = 0 + BC + CI turbulent regime : u. u >> ν u signal highly disorganized and structures at all scales, unpredictable as for details Statistical tools required

6 Multifractal description of intermittency in turbulence -

7 Multifractal description of intermittency in turbulence classical multifractal formalism comes from turbulence multifractal model for velocity : longitudinal structure functions based on (1D) velocity increments : S p (l) = < (e.δv(r, le)) p > l ζ p, p > 0 multifractal model for (1D surrogate) dissipation : RSH hypothesis S p (l) < ε p/3 l > l p/3 l τ ε(p/3)+p/3. Measure of the spectra τ ε (p) and f(α) with the box-counting method. multifractal formalism based on WT (WTMM method) for regularity analysis of functions, measures and distributions. generalization of the WTMM method 1D/2D 3D generalization to multidimensional vector fields. First application to velocity and vorticity fields from numerical turbulent flows.

8 2D WTMM Methodology : PhD work of N. Decoster 2D data : I a Tψ(r, a) = ( I φ a x (r) I φ a y (r) ) Tψ(r, a) = ( I φa ) (r) = ( M ψ(r, a), Aψ(r, a) )

9 WTMM Methodology : Skeleton WTMM Chains WTMM Chains at 3 different scales WTMMM WTMMM linking : WT skeleton

10 Local roughness characterization : Hölder exponent f(x 0 + λu) f(x 0 ) λ h(x 0) ( f(x 0 + u) f(x 0 ) ) Monofractal Image Multifractal Image M a h, single h M a h, a h, a h, h [h min, h max ]

11 WTMM Method : multifractal formalism Singularity spectrum : D D(h) D D(h) D(h) = d H { r R d, h(r) = h } D D h h h h h Analogy statistical physics : compute partition functions h Z(q, a) = L(a) ( Mψ (r, a) ) q a τ (q) Legendre transform : D(h) = min q ( qh τ (q) ) H(q, a) = L(a) ln M ψ (r, a) W ψ (r, a) a h(q) D(q, a) = L(a) ln W ψ (r, a) W ψ (r, a) a D(q)

12 WTMM Method : multifractal formalism Singularity spectrum : D D(h) D D(h) D(h) = d H { r R d, h(r) = h } D D h h h h h Analogy statistical physics : compute partition functions h Z(q, a) = L(a) ( Mψ (r, a) ) q a τ (q) Legendre transform : D(h) = min q ( qh τ (q) ) H(q, a) = L(a) ln M ψ (r, a) W ψ (r, a) a h(q) D(q, a) = L(a) ln W ψ (r, a) W ψ (r, a) a D(q)

13 fractional Brownian surfaces : B H (r) H < 0.5 : anti-correlated increments H = 0.5 : non-correlated increments H > 0.5 : correlated increments Application to synthetic monofractal surfaces Theoretical Predictions : τ (q) is linear : τ (q) = qh 2 multifractal spectrum is degenerated : D(h = H) = 2

14 Application to synthetic multifractal surfaces FISC (Fractionally Integrated Singular Cascades) model simple multiplicative model : p-model or multinomial model M p 1 M p 2 p 3 M p 4 M M p 1 p 1 M p 1 p 2 p M 1 p 1 p p3 4 M M fractional integration (Fourier domain) generalization : p is a random variable with < p >= 1/4 (conservative cascading process)

15 Application to synthetic multifractal surfaces Multifractal (Fractionally Integrated Singular Cascades) surfaces Theoretical predictions : τ (q) is non-linear τ (q) = 2 q(1 H ) log 2 (p q 1 + p q 2 ) singularity spectrum is a nondegenerated convex curve

16 classical multifractal formalism comes from turbulence The Navier-Stokes equations: t u + u. u = 1 p + f + ν u, ρ +.u = 0 + BC + CI turbulent regime : u. u >> ν u signal highly disorganized and structures at all scales, unpredictable as for details Statistical tools required

17 Multifractal description of intermittency in turbulence classical multifractal formalism comes from turbulence multifractal model for velocity : longitudinal structure functions based on (1D) velocity increments : S p (l) = < (e.δv(r, le)) p > l ζ p, p > 0 multifractal model for (1D surrogate) dissipation : RSH hypothesis S p (l) < ε p/3 l > l p/3 l τ ε(p/3)+p/3. Measure of the spectra τ ε (p) and f(α) with the box-counting method. multifractal formalism based on WT (WTMM method) for regularity analysis of functions, measures and distributions. generalization of the WTMM method 1D/2D 3D generalization to multidimensional vector fields. First application to velocity and vorticity fields from numerical turbulent flows.

18 3D data Tψ(r, a) = " # $! % 3D scalar WTMM method I φa x (r) I φ a y (r) I φ a z (r) = ( I φa ) (r)

19 3D scalar WTMM method : skeleton WTMM surfaces WTMM surfaces at 3 different scales WTMMM points Linking WTMMM : WT Skeleton (projection along z)

20 Recursive filters in 3D 1.0 e x2 /2σ y y x/σ ( 1 + x2 σ 2 )e x2 /2σ 2 x/σ filters with separated variables Approximate Gaussian filter e x2 /2σ 2 with h σ (x) x (a 0 cos(ω 0 σ ) + a x 1 sin(ω 0 σ )) exp b x 0 σ +(c 0 cos(ω 1 x σ ) + c 1 sin(ω 1 x σ )) exp b 1 4th order difference equation : y k = n 00 x k + n 11 x k 1 + n 22 x k 2 + n 33 x k 3 d 11 y k 1 d 22 y k 2 d 33 y k 3 d 44 y k 4 comparison FFT/recursive filters : computing time decrease in 3D case: 60 % for Gaussian filter 25 % for Mexican filter x σ

21 Test-application to synthetic 3D monofractal fields fractional Brownian fields : B H (r) H < 0.5 : anti-correlated increments H = 0.5 : non-correlated increments H > 0.5 : correlated increments Theoretical predictions : τ (q) is linear: τ (q) = qh 3 multifractal spectrum is degenerated: D(h = H) = 3

22 Test-application to synthetic 3D multifractal fields 3D multifractal fields (Fractionally Integrated Singular Cascades) Theoretical predictions : τ (q) = 2 q(1 H ) log 2 (p 1 q + p 2 q ). with p 1 + p 2 = 1 singularity spectrum is a non-degenerated convexe curve

23 3D dissipation field : isotropic turbulence DNS pseudo-spectral code, (512) 3 grid, R λ = 216 (M. Meneguzzi)

24 3D WTMM methodology vs Box-Counting algorithms Box-Counting algorithm, binomial fit with p 1 = 0.3 and p 2 = 0.7 p 1 + p 2 = 1 : diagnoses a conservative multiplicative structure 3D WTMM method reveals a non-conservative multiplicative structure : binomial fit with p 1 = 0.36 and p 2 = 0.80 p 1 + p 2 1 dissipation binomial model : τ (q) = 2 q log 2 (p 1 q +p 2 q ) remark: p 1 = p 1 p 1 +p 2 inconsistent box-counting!

25 3D WTMM methodology vs Box-Counting algorithms Box-Counting algorithm, binomial fit with p 1 = 0.3 and p 2 = 0.7 p 1 + p 2 = 1 : diagnoses a conservative multiplicative structure 3D WTMM method reveals a non-conservative multiplicative structure : binomial fit with p 1 = 0.36 and p 2 = 0.80 p 1 + p 2 1 dissipation binomial model : τ (q) = 2 q log 2 (p 1 q +p 2 q ) remark: p 1 = p 1 p 1 +p 2 inconsistent box-counting!

26 Comparative 3D WTMM analysis of dissipation and enstrophy Dissipation, non-conservative multiplicative structure : binomial fit with p 1 = 0.36 and p 2 = 0.80 p 1 + p 2 = 1.16 Enstrophy, non-conservative multiplicative structure : binomial fit with p 1 = 0.38 and p 2 = 0.94 p 1 + p 2 = 1.32 Intermittency coefficients: Dissipation : C Enstrophy : C

27 Self-similar multifractal vector-valued measure (2D case) Falconer and O Neil (1995) log µ(b(r,l)) scalar measure {r : lim = α}, α = h + 2 l 0 log l Z(q, l) = i µ q i (l) l τ µ(q) vector-valued measure { r : lim l 0 Z(q, l) = i Φ l µ q i l τ µ(q) log B(r,l) Φ lµ(s) dl d (s) log l = α }, α = h + 2

28 Self-similar multifractal vector-valued measure (2D case) τ µ (q) = log(pq 1 +pq 2 +pq 3 +pq 4 ) log 2 D µ (h) = f µ (α 2) = inf q (qh τ µ (q)).

29 Tensorial wavelet transform (2D case) 1. Tensorial wavelet transform of field V = (V 1, V 2 ) : T ψ [V](b, a) = (T ψi [V j ](b, a)) = T ψ 1 [V 1 ] T ψ1 [V 2 ] T ψ2 [V 1 ] T ψ2 [V 2 ] ( ) T ψi [V j ](b, a) = a 2 d 2 xr ψ i a 1 (r b) V j (r), j = 1, 2 2. Direction of greatest variation of vector field : T ψ [V] = sup C 0 T ψ [V].C C 3. Singular value decomposition of WT tensor: T ψ [V] = 4. Tensorial wavelet transform : ( ) ( ) T G.( σ max 0 0 σ min ). D T ψ,max [V](b, a) = σ max G σmax

30 Tensorial 2D WTMM methodology Data Tensorial wavelet transform T ψ,max [V](b, a) = σ max G σmax Modulus Maxima σ max chains of tensorial wavelet transform at scale a : { (b, a)/ σ max 2 } σ max = 0 et < 0 G max G 2 max

31 Tensorial 2D WTMM methodology: Skeleton WTMM Chains at 3 different scales WTMM Chains WTMMM linking: WT Skeleton WTMMM

32 Monofractal 2D vector fields fractional Brownian fields : B H (r) Spectral method simulation Theoretical predictions : linear τ (q): τ (q) = qh 2 degenerated singularity spectrum: D(h = H) = 2

33 2D self-similar multifractal vector-valued measures Self-similar multifractal vector-valued measures (Falconer and O Neil s model) Theoretical predictions: vectorial box-counting vectorial 2D WTMM method τ (q) = log 2 (p 1 q +p 2 q +p 3 q +p 4 q ) p 1 = p 4 = 0.5, p 2 = 2 and p 3 = 1 vectorial box-counting is less accurate

34 Tensorial 3D WTMM method: turbulent velocity field (R λ = 140)

35 Tensorial 3D WTMM method: singularity spectrum of velocity 1D increments method: parabolic fit : τ (q) = C 0 C 1 q C 2 q 2 2 intermittency coefficient C 2 = ± longitudinal : C 2 (δv L ) transverse : C 2 (δv T ) 0.040

36 Tensorial 3D WTMM method: turbulent velocity field (R λ = 140)

37 Tensorial 3D WTMM method: singularity spectrum of vorticity vorticity D v (h + 1) spectrum translated velocity same 3D intermittency coefficient!

38 Tensorial 3D WTMM method: singularity spectrum of vorticity vorticity D v (h + 1) spectrum translated velocity same 3D intermittency coefficient!

39 Conclusion assessment: WTMM multifractal analysis: moving towards vector fields outlooks : better understanding of the information embedded in the WT tensor. identification of coherent structures in turbulence using WT tensor s smallest singular value: vorticity filaments or sheets. others applications : astrophysics (interstellar medium, interstellar turbulence), MHD, geophysics,... Thanks : E. Lévêque, Laboratoire de Physique, ENS Lyon (Turbulent flows DNS). References : P. Kestener and A. Arneodo, Phys. Rev. Lett., 91:194501, P. Kestener and A. Arneodo, Phys. Rev. Lett., 93:044501, 2004.

40 Scalar 3D WTMM method: Galaxy distribution simulation data Preliminary results : Isosurface plots of data and 3D WT modulus (Gaussian filtering) : z = 5 z = 2 z = 0

41 3D Box-Counting method: singularity spectrum (z = 0 and z = 5) NO CLEAR SCALING!!! perhaps scaling at large scales??

42 Scalar 3D WTMM method: singularity spectrum (z = 0 and z = 5) parabolic fit : τ (q) = C 0 C 1 q C 2 q 2 2 intermittency coefficients C 2 = 1.32 ± 0.05 C 2 = 0.22 ± 0.02

43 Scalar 3D WTMM method: singularity spectra of galaxy distribution Scalar 3D WTMM method:

44 Box-Counting method: singularity spectra of galaxy distribution Box-Counting method: Box-counting intermittency coefficients are smaller than those computed using 3D WTMM method!!! (To be continued).

45 Cancellation exponent basics Signed singular measure : A, B A/ µ s (A)µ s (B) < 0 (ex : magnetic field component,...) ln Cancellation exponent : κ = lim ε 0 where {I i,ε } is a tiling of measure s support. using the wavelet transform: κ = D F τ (q = 1) choice of the analyzing wavelet. µ(i i i,ε), ln(1/ε) link to the notion of conservativity of a multiplicative cascade: κ = D F τ (q = 1) = ln<m> ln b rate of the measure from scale a to scale a/b conservative cascade κ = 0 non-conservative cascade κ 0 : transfert 1.0 D(h) h

46 Recursive filter technics in 3D 1.0 e x2 /2σ y y x/σ ( 1 + x2 σ 2 )e x2 /2σ 2 x/σ Gaussian filtere x2 /2σ 2 approximated by h σ (x) x (a 0 cos(ω 0 σ ) + a x 1 sin(ω 0 σ )) exp b x 0 σ +(c 0 cos(ω 1 x σ ) + c 1 sin(ω 1 x σ )) exp b 1 Coefficients a i, b i, c i and ω i are estimated by minimizing the relative quadratic error: ɛ 2 = 10σ i=1 (g σ (i) h σ (i)) 2 10σ i=1 g σ (i) 2 4th order recursive equation: y k = n 00 x k + n 11 x k 1 + n 22 x k 2 + n 33 x k 3 d 11 y k 1 d 22 y k 2 d 33 y k 3 d 44 y k 4 computing time decrease in 3D: 60 % for Gaussian filter and 25 % for Mexican filter x σ

47 Box-counting algorithms : multifractal measure µ: probability measure whose support E R d log µ(b(l,x)) singularity exponent: α(x) = lim l 0 log l method: tile support of the measure with boxes of size l i = L/2 i partition functions: S q (l i ) = µ(b) 0 multifractal spectrum: τ (q) = lim l 0 log S q (l) log l ; [ µ(b) ] q =< µ q > f(α) = D(h = α d) = min(αq dq τ (q)) q

48 Singularity spectra of velocity and vorticity To each velocity v field singularity h(r 0 ) corresponds a vorticity ω = v singularity h(r 0 ) 1: f(x 0 + l) = f(x 0 ) + ( x )f(x 0 )l ( x ) n f(x 0 )l (n) + l h(x 0) C(l) l l l x f(x 0 + l) = x f(x 0 ) + x ( x f) (x 0 )l n 1 x ( x f) (x 0 )l (n 1) l + h l h 1 u l C(l)

49 Mammography and breast anatomy Goals : using WTMM method to diagnosis help of breast cancer

50 What is breast cancer? malignant tumor of mammal gland incidence : new case each year in France prevention is very difficult (as opposed to lung cancer) hereditarity : 5 to 10 % only (BRCA1/2 genes) forecast depends on the tumoral volume at diagnosis SCREENING using mammography

51 Radiological anomalies Opacities Calcifications Architectural distorcions

52 Digitalized mammographies : texture analysis dense breasts : more difficult to diagnose only 2 classes of monofractal properties Digital Database for Screening Mammography: Dense breast Fatty breast

53 Application of 2D WTMM methodology in mammography Tissue classification : dense vs fatty Dense breast : monofractal, H = 0.65 persitent correlations Fatty breast : monofractal, H = 0.30 anti-persitent correlations

54 Application of 2D WTMM methodology in mammography Tissue classification : dense vs fatty Dense breast : monofractal, H = 0.65 persitent correlations Fatty breast : monofractal, H = 0.30 anti-persitent correlations

55 Application to digitalized mammographies Colored Maps : segmentation of dense h > 0.52 areas and fatty h < 0.38 areas

56 Application to digitalized mammographies Colored Maps : segmentation of dense h > 0.52 areas and fatty h < 0.38 areas

57 Microcalcifications detection Segmentation of WT skeleton lines : microcalcifications vs background texture Background lines Microcalcifications almost-punctual objects behave like Dirac shapes (h = 1)

58 Cluster of microcalcifications Study of microcalcification spatial distribution Partition functions : D F = 1.3 observation : fractal ramification of cluster of microcalcifications (1 < D F < 2) seems to be correlated to the pathology s malignancy

59 Conclusions and prospects (1) the 2D WTMM method provides a framework for an automated measure of the breast radio-density and for studying the fractal geometry of clusters of microcalcifications. further study is necessary to validate quantitatively how far measuring the fractal dimension D F could improve computer-aided diagnosis systems benign/malignant