An iterative algorithm for nonlinear wavelet thresholding: Applications to signal and image processing

Transcription

1 An iterative algorithm for nonlinear wavelet thresholding: Applications to signal and image processing Marie Farge, LMD-CNRS, ENS, Paris Kai Schneider, CMI, Université de Provence, Marseille Alexandre Azzalini, PhD student, Paris IMS, Singapore, 18 August 2004

2 Outline Denoising by nonlinear wavelet thresholding Motivation Classical algorithm Iterative algorithm Properties of the iterative algorithm Properties of the iteration function Convergence of the algorithm Application to academic example Numerical applications in physics Bose-Einstein condensate C.S. in two-dimensional turbulence C.S. in three-dimensional turbulence Another C.S. Conclusions and perspectives

3 Motivation We study turbulent flows in the limit of large Reynolds numbers, i.e. when the nonlinear term dominates the linear dissipative terms in Navier-Stokes equations. We observe the formation of coherent structures and we want to analyse their dynamics and characterize them statistically. Replace the Fourier representation by the wavelet representation, using : continous wavelet transform C. R. Acad. Sci. Paris, 307, J. Fluid Mech., 206, 1989 wavelet packets Fluid Dyn. Res., 10, Ann. Rev. Fluid Mech., J. Fluid Mech., 346, 1997 orthogonal wavelets IEEE Proceedings, 84(4), Theor. Comput. Fluid Dyn., 9, 1997 wavelet denoising ACHA, 3, Phys. Fluids 11(8), Phys. Fluids 15 (10), 2003 iterative wavelet denoising (IWD) Phys. Fluids 11(8), 1999, ACHA, submitted, 2003 //wavelets.ens.fr

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14 1D academic example (SNR = 20 db) PDF of wavelet coefficients PDF of denoised signal and noise

15 2D academic example (SNR = 10 db) PDF of wavelet coefficients PDF of denoised signal and noise

16 Extraction of Bose-Einstein condensate Optical density of Lithium atoms Iterative Wavelet Denoising (IWD) Median Absolute Deviation (MAD)

17 Extraction of Bose-Einstein condensate Iterative Wavelet Denoising (IWD) Median Absolute Deviation (MAD)

18 C. S. in two-dimensional turbulence 2D turbulent flow from laboratory experiment Total vorticity Coherent vorticity Incoherent vorticity

19 PDF of vorticity Total Gaussian Coherent Incoherent (wavelet) Incoherent (wavelet packet)

20 C. S. in two-dimensional turbulence 2D turbulent flow from numerical experiment QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image.

21 Extraction of coherent vortices using IWD 0.2 % of coefficients 99.8 % of kinetic energy 93.6 % of enstrophy 99.8 % of coefficients 0.2 % of kinetic energy 6.4 % of enstrophy Coherent vorticity 0.2 % N Incoherent vorticity 99.8% N Total vorticity N=512 2

22 PDF of vorticity P(ω) Coherent skewed exponential Incoherent unskewed Gaussian ω

23 Energy spectrum Coherent E(k) k -5 scaling, i.e. long-range correlation Incoherent k -1 scaling, i.e. enstrophy equipartition k

24 Results for 2D turbulence With only 0.2 % of the wavelet coefficients CVS retains : 99.8 % of the kinetic energy, 93.6 % of enstrophy, all coherent vortices of the flow field. We observe for both laboratory and numerical experiments : long-range correlation for the coherent flow, enstrophy equipartition and Gaussian vorticity for the incoherent flow. We conjecture that discarding the incoherent flow is sufficient to model turbulent dissipation.

25 C. S. in three-dimensional turbulence : Extraction of coherent vortices with IWD 3 % of coefficients 99% of kinetic energy 75% of enstrophy 97 % of coefficients 0.6 % of kinetic energy 25 % of enstrophy Coherent vorticity Incoherent vorticity Total vorticity

26 PDF of velocity P(V) CVS filter Gaussian incoherent flow, i.e. easy to model LES filter non-gaussian small scales, i.e. difficult to model

27 Energy spectrum E(k) Coherent k -5 scaling, i.e. long-range correlation k Incoherent k +1 scaling, i.e. energy equipartition

28 Results for 3D turbulence With only 3 % of the wavelet coefficients CVS retains 99 % of the kinetic energy, 75 % of enstrophy, all coherent vortices of the flow field. We observe long-range correlation for the coherent flow, energy equipartition and Gaussian velocity for the incoherent flow, in contrast to LES. We conjecture that discarding the incoherent flow is sufficient to model turbulent dissipation.

29 Coherent Vortex Simulation (CVS) Projection of vorticity onto an orthogonal wavelet basis. Coherence extraction using the IWD algorithm. Reconstruction of the coherent vorticity. Computation of the coherent velocity using Biot-Savart s law. Addition of a security zone in wavelet space. Integration of Navier-Stokes of in the reduced wavelet basis. Phys. Fluids,11(8),1999 Flow, Turbulence and Combustion, 66(4), 2001 Appl. Comp. Harmonic Anal. (12), 2002

30 2D vorticity field in wavelet space QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. The red interface corresponds to the threshold T of WD : coefficients > T (below the interface) are coherent, coefficients < T (above) are incoherent.

31 Evolution of the interface in wavelet space QuickTime et un décompresseur codec YUV420 sont requis pour visionner cette image. The isosurface corresponds to the threshold T of WD

32 Perspectives Test IWD with frames or curvelets to preserve isotropy, possibly geometry and topology of vortex tubes (Kaneda), Extract coherent structures in space-time ( movie ) applying IWD to 4D vector fields and check that there is: - no scale separation in space and time, - long-range correlation of the coherent structures, - decorrelation of the incoherent background flow. Perform adaptive computation in both space and time. Appl. Comp. Harmonic Anal., submitted, 2003 Papers and softwares available from : //wavelets.ens.fr

33 Another C. S. : Claudia Schiffer