PSFVIP - 4, S5 Avanced Image Processing
A wavelet based method for multifractal analysis of 3D random fields : application to turbulence simulation data
Pierre Kestener Alain Arneodo
Laboratoire de Physique, ENS Lyon, France.
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
Contents of the talk
☞ 3D Wavelet Transform Modulus Maxima (WTMM) Methodology ☞ WT Skeleton ☞ Multifractal Formalism ☞ Test-Applications to Monofractal and Multifractal 3D Fields ☞ Application to Turbulence Simulation Data (3D dissipation)
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
3D WTMM Methodology 3D Data
!
%
$ #
"
↓
I∗
Tψ (r, a) = I∗ I∗
∂φa (r) ∂x ∂φa (r) ∂y ∂φa (r) ∂z
June 3-5, 2003. Chamonix, France
= ∇ I ∗ φa (r)
PSFVIP - 4, S5 Avanced Image Processing
3D WTMM Methodology : Skeleton WTMM Surfaces
WTMMM points
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
3D WTMM Methodology : Skeleton WTMM Surfaces
WTMMM points
June 3-5, 2003. Chamonix, France
WTMM Surfaces at 3 different increasing scales
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PSFVIP - 4, S5 Avanced Image Processing
3D WTMM Methodology : Skeleton WTMM Surfaces
WTMM Surfaces at 3 different increasing scales
Linking WTMMM points : WT Skeleton (projection along z) WTMMM points
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
¨ Characterizing local roughness : Holder exponent
h(r0 )
f (r0 + λu) − f (r0 ) ∼ λ
June 3-5, 2003. Chamonix, France
f (r0 + u) − f (r0 )
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PSFVIP - 4, S5 Avanced Image Processing
¨ Characterizing local roughness : Holder exponent
h(r0 )
f (r0 + λu) − f (r0 ) ∼ λ ☞ 3D Monofractal field
M ∼ ah
June 3-5, 2003. Chamonix, France
f (r0 + u) − f (r0 )
☞ 3D Multifractal field
M ∼ ah , ah , ah
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PSFVIP - 4, S5 Avanced Image Processing
From Partition Functions to Multifractal Spectra D(h)
D
Singularity Spectrum:
3
D(h) = dH r ∈ R , h(r) = h
D(h)
D
D D h
June 3-5, 2003. Chamonix, France
h
h h
h
h
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PSFVIP - 4, S5 Avanced Image Processing
From Partition Functions to Multifractal Spectra D(h)
D
Singularity Spectrum:
3
D(h)
D(h) = dH r ∈ R , h(r) = h
D
D D h
h
h h
h
h
Analogy with statistical physics : computation of the Partition Functions
Z(q, a) =
X
L(a)
Legendre transform:
D(h) = minq qh − τ (q)
H(q, a) =
∼ aτ (q)
X
ln |Mψ (r, a)| Wψ (r, a) ∼ ah(q)
X
ln |Wψ (r, a)| Wψ (r, a) ∼ aD(q)
L(a)
D(q, a) =
L(a)
June 3-5, 2003. Chamonix, France
Mψ (r, a)
q
PSFVIP - 4, S5 Avanced Image Processing
Test-Application to Synthetic 3D Monofractal Fields Fractional Brownian Fields : BH (r) ➳ H < 0.5: anti-correlated increments ➳ H = 0.5: un-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
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
Test-Application to Synthetic 3D Multifractals Fields 3D Multifractals Fields (Fractionally Integrated Singular Cascades)
Theoretical predictions:
☞ τ (q) = −2 − q(1 − H ∗ ) − log2 (p1 q + p2 q ). with p1 + p2 = 1 ☞ singularity spectrum is a non-degenerated convex curve
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
3D Dissipation Field : isotropic turbulence DNS from M. Meneguzzi pseudospectral code, (512)3 grid , Rλ
June 3-5, 2003. Chamonix, France
= 150
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PSFVIP - 4, S5 Avanced Image Processing
3D WTMM methodology vs Box-Counting Algorithms ☞ “3D WTMM” methodology reveals a non-conservative multiplicative structure : p-model parameters estimates p1 = 0.36 and p2 = 0.78 ⇒ p1 + p2 6= 1
τ (q) = −2−q−log2 (p1 q +p2 q )
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
3D WTMM methodology vs Box-Counting Algorithms ☞ “3D WTMM” methodology reveals a non-conservative multiplicative structure : p-model parameters estimates p1 = 0.36 and p2 = 0.78 ⇒ p1 + p2 6= 1 ☞ “Box-Counting” algorithms intrinsically limited, it can only reach p = p 1 /(p1 + p2 ) ⇒ misleading conservative diagnosis !!!
τ (q) = −2−q−log2 (p1 q +p2 q )
June 3-5, 2003. Chamonix, France
10
PSFVIP - 4, S5 Avanced Image Processing
3D WTMM methodology vs Box-Counting Algorithms ☞ “3D WTMM” methodology reveals a non-conservative multiplicative structure : p-model parameters estimates p1 = 0.36 and p2 = 0.78 ⇒ p1 + p2 6= 1 ☞ “Box-Counting” algorithms intrinsically limited, it can only reach p = p 1 /(p1 + p2 ) ⇒ misleading conservative diagnosis !!!
τ (q) = −2−q−log2 (p1 q +p2 q )
June 3-5, 2003. Chamonix, France
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PSFVIP - 4, S5 Avanced Image Processing
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Conclusion
References :
☞ Kestener P. and Arneodo A. (2003). A wavelet-based method for multifractal analysis of 3D random fields : application to turbulence simulation data. Proceedings of PSFVIP-4, June 3-5, 2003, Chamonix, France.
☞ Kestener P. and Arneodo A. (2003). A three-dimensional wavelet based multifractal method : about the need of revisiting the multifractal description of turbulence dissipation data. Submitted to Phys. Rev. Lett. Thanks : M. Meneguzzi
June 3-5, 2003. Chamonix, France
