Curvelet-based non-linear adaptive subtraction with sparseness constraints. Felix J Herrmann, Peyman P Moghaddam (EOS-UBC)

Transcription

1 Curvelet-based non-linear adaptive subtraction with sparseness constraints Felix J Herrmann, Peyman P Moghaddam (EOS-UBC)

2 Context Spiky/minimal structure deconvolution (Claerbout, Ulrych, Oldenburg, Sacchi, Trad etc.) Sparse Radon (Ulrych, Sacchi, Trad, etc.) FFT/Focus transform-based Interpolation (Duyndam, Zwartjes, Verschuur, Berkhout) Redundant dictonaries/morphological component separation/pursuits (Mallat, Chen, Donoho, Starck, Elad) L2-migration (Nemeth, Chavent, de Hoop, Hu, Kuehl) 2-D/3-D Curvelets Non-linear synthesis (Durand, Starck, Candes, Demanet, Ying) Anisotropic Diffusion (Osher)

3 Goals Processing & imaging scheme increases resolution & SNR preserves edges = freq. content works with and extends existing noise removal/signal separation imaging schemes Develop the right language to deal with SNR 0...

4 General Framework Divide-and-conquer approach: 1. Sparseness with thresholding 2. Continuity with constrained optimization Main focus: use of Curvelets as optimal Frames (iterative) thresholding

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6 Wish list Seek a transform domain that is relative insensitive to local phase sparse & local (position/dip) optimal for curved events well-behaved under operators near diagonalizes Covariance Aim to bring out those high frequencies with ultra-low SNR<0!

7 Why curvelets Nonseparable Local in 2-D space Local in 2-D Fourier Anisotropic 2 j 2 2j Multiscale Almost orthogonal Tight frame B T B = I Optimal

8 Thanks to Demanet & Ying 3-D Curvelets

9 Thanks to Demanet & Ying 3-D Curvelets

10 Thanks to Demanet & Ying 3-D Curvelets

11 Why curvelets W j = {!, 2 j! 2 j+1, " " J # 2 j/2 } Fourier/SVD/KL f f m F m 1/2, m Wavelet f f W m m 1, m second dyadic partitioning Optimal data adaptive f f A m m 2, m Close to optimal Curvelet source: Candes 01, Stein 90 f f C m C m 2 (log m) 3, m

12 Why curvelets! $!! %!! "! #!!!"#$%&%'()*(+,-./01)*

13 Denoising Input data with noise Curvelet transform Threshold Bd Curvelet coeff. data S λ (Bd) λ λ Inv. curvelet transform denoised Curvelet coeff. Filtered input data

14 Denoising

15 Denoising

16 Wavelets

17 Curvelets

18 Coherent noise suppression Filtered input data with curvelets Bd Curvelet transform Curvelet coeff. data Input data with noise Threshold Γ Bˆn predicted noise Curvelet coeff. pred. noise Inv. curvelet transform Curvelet coeff. model S λγ (Bd)

19 Multiple suppression with curvelets Input with multiples

20 Multiple suppression with curvelets Output curvelet filtering with stronger threshold Preserved primaries

21 Denoising Denoising signal separation Multiple & Ground-roll removal 4-D difference cubes Migration denoising (H & M 04) Model noise Strategies: Weighted thresholding Iterated weighted thresholding

22 L2/L1-matched filter denoised {}}{ ˆn Matched filter: : min Φ = d }{{} noisy data p=1 enhances sparseness residue is the denoised data risk of over fitting matched filter {}}{ Φ t m }{{} pred. noise p Guitton& Verschuur 04 Loose primary reflection events...

23 Colored denoising Denoising: noisy data {}}{ d = m }{{} noise-free + col. noise {}}{ n ˆm = arg min m 1 2 C 1/2 n (d m) J(m) with covariance C n E{nn T } and both m, n related to PDE

24 Ground-roll removal with curvelets Radon 1000 Iterations=3

25 Weighted thresholding Covariance model & noise near diagonal: For ortho basis and app. noise prediction: equivalent to BC n or m B T Γ 2 ˆm = B T S λγ (Bd) ˆm = B T 1 ( d ) arg min m 2 Γ 1 m near diagonal m 1,λ d = Bd, m = Bm and Γ = [diag{diag{bˆn}}] 1/2

26 Multiple suppression with curvelets Output curvelet filtering with stronger threshold Preserved primaries

27 4-D difference

28 ! 3-D Curvelets! "! #!! #"! "! #!! #"!

29 Iterative thresholding Curvelets are Frames: redundant (factor 7.5-4) thresholding does not solve: ˆm = B T 1 ( d ) arg min m 2 Γ 1 m Alternative formulation by iterative thresholding! m 1,λ

30 Starck et al.; Daubechies et al., Donoho 2004 (also Zwartjes, Sacchi, Trad & Ulrich) Iterative thresholding Solves signal-separation problem: by minimizing LP-program with s = s 1 + s 2 + n and s = s 1 + s 2 ˆx = arg min x s Φx x 1 w1,1 + x 2 w2,1 x = [ x 1 x 2 ] T and Φ = [ Φ 1 Φ 2 ]

31 Example shallow water environment Leakage of L.S. subtraction Input data with multiples Result L.S. subtraction

32 Example shallow water environment Input data with multiples Result curvelet-based subtraction

33 Example shallow water environment Input data with multiples True primaries

34 Example shallow water environment Input data with multiples and noise Result curvelet-based subtraction

35 Global optimization After thresholding: remove artifacts & miss fires normal operator (inversion) Impose additional penalty functional prior information sparseness & continuity Defining the right norm is crucial...

36 Global optimization Formulate constrained optimization: ˆm : min m J(m) s.t. m ˆ m 0 µ e µ, µ with ( d ) ˆm 0 = B Θ λγ and with e µ threshold and noisedependent tolerance on curvelet coeff.

37 Imaging with Curvelets Least-squares migrated Image Constrained Optimization

38 Penalty functionals Anisotropic diffusion for imaging: J(m) = Λ 1/2 m p with 1 Λ[ c] = c 2 + 2β 2 {( ) x2 c ( x2 c ) x1 c + β 2 I} x1 c. brings out wavefront set.

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43 Conclusions Presented a framework that Stable signal separation via thresholding: Ground-roll & Multiple removal (Wednesday) Compute 4-D difference Cubes improves imaging & inversion: deals with incoherent noise & missing data extended to 3-D sparseness constrained imaging (H & P 04)

44 Acknowledgements Candes, Donoho, Demanet, Ying for making their Curvelet code available. Partially supported by a NSERC Grant.