Empirical Wavelet Transform

Transcription

1 Jérôme Gilles Department of Mathematics, UCLA Adaptive Data Analysis and Sparsity Workshop January 31th, 013

2 Outline Introduction - EMD 1D Empirical Wavelets Definition Experiments D Extensions Tensor product case Ridgelet case Experiments

3 Time-frequency signal analysis Time-Frequency representations are useful to analyze signals.

4 Time-frequency signal analysis Time-Frequency representations are useful to analyze signals. Short-time Fourier transform: Ff W (m, n) = f (s)g(s nt 0 )e ımω0s ds.

5 Time-frequency signal analysis Time-Frequency representations are useful to analyze signals. Short-time Fourier transform: Ff W (m, n) = f (s)g(s nt 0 )e ımω0s ds. Wavelet transform: WT f (m, n) = a m/ 0 f (t)ψ(a m 0 t nb 0 )dt.

6 Time-frequency signal analysis Time-Frequency representations are useful to analyze signals. Short-time Fourier transform: Ff W (m, n) = f (s)g(s nt 0 )e ımω0s ds. Wavelet transform: WT f (m, n) = a m/ 0 f (t)ψ(a m 0 t nb 0 )dt. Wigner-Ville transform (quadratic nonlinear + interference terms).

7 Time-frequency signal analysis Time-Frequency representations are useful to analyze signals. Short-time Fourier transform: Ff W (m, n) = f (s)g(s nt 0 )e ımω0s ds. Wavelet transform: WT f (m, n) = a m/ 0 f (t)ψ(a m 0 t nb 0 )dt. Wigner-Ville transform (quadratic nonlinear + interference terms). Hilbert-Huang transform (EMD + Hilbert transform)

8 Empirical Mode Decomposition (EMD): Principle Goal: decompose a signal f (t) into a finite sum of Intrinsic Mode Functions (IMF) f k (t): N f (t) = f k (t) where an IMF is an AM-FM signal: k=0 f k (t) = F k (t) cos (ϕ k (t)) where F k (t), ϕ k (t) > 0 t. Main assumption: F k and ϕ k vary much slower than ϕ k. Huang et al. 1 propose a pure algorithmic method to extract the different IMF. 1 The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. Royal Society London A., vol.454, pp ,1998

9 Empirical Mode Decomposition (EMD): Algorithm Initialization: f 0 = f while all IMF are no extracted do r0 k = f k while rn k is not an IMF (Sifting process) do Upper envelope ū(t) (maxima + spline) of rn k (t) Lower envelope l(t) (minima + spline) of rn k (t) Mean envelope m(t) = (ū(t) + l(t))/ IMF candidate rn+1 k (t) = r n k (t) m(t) end while f k+1 = f k rn+1 k end while

10 Empirical Mode Decomposition (EMD): Algorithm Initialization: f 0 = f while all IMF are no extracted do r0 k = f k while rn k is not an IMF (Sifting process) do Upper envelope ū(t) (maxima + spline) of rn k (t) Lower envelope l(t) (minima + spline) of rn k (t) Mean envelope m(t) = (ū(t) + l(t))/ IMF candidate rn+1 k (t) = r n k (t) m(t) end while f k+1 = f k rn+1 k end while

11 Empirical Mode Decomposition (EMD): Algorithm Initialization: f 0 = f while all IMF are no extracted do r0 k = f k while rn k is not an IMF (Sifting process) do Upper envelope ū(t) (maxima + spline) of rn k (t) Lower envelope l(t) (minima + spline) of rn k (t) Mean envelope m(t) = (ū(t) + l(t))/ IMF candidate rn+1 k (t) = r n k (t) m(t) end while f k+1 = f k rn+1 k end while

12 Empirical Mode Decomposition (EMD): Algorithm Initialization: f 0 = f while all IMF are no extracted do r0 k = f k while rn k is not an IMF (Sifting process) do Upper envelope ū(t) (maxima + spline) of rn k (t) Lower envelope l(t) (minima + spline) of rn k (t) Mean envelope m(t) = (ū(t) + l(t))/ IMF candidate rn+1 k (t) = r n k (t) m(t) end while f k+1 = f k rn+1 k end while

13 Example of EMD: input signals f Sig1 (t) = 6t + cos(8πt) cos(40πt) f Sig (t) = 6t + cos(10πt + 10πt ) + χ(t > 0.5) cos(80πt 15π) +χ(t 0.5) cos(60πt) cos(3πt + cos(64πt)) Sig3 1.+cos(πt) sin(πt) f (t) =

14 Example of EMD: f Sig

15 Example of EMD: f Sig

16 Example of EMD: f Sig

17 Example of EMD: f Sig4 - ECG

18 Hilbert-Huang Transform Hilbert transform H f (t) = 1 π p.v. + Property: if f k (t) = F k (t) cos (ϕ k (t)) then f (τ) t τ dτ f k (t) = f k(t) + ıh fk (t) = F k (t)e ıϕ k (t) we can extract F k (t) and the instantaneous frequency dϕ k dt (t).

19 Hilbert-Huang Transform Hilbert transform H f (t) = 1 π p.v. + Property: if f k (t) = F k (t) cos (ϕ k (t)) then f (τ) t τ dτ f k (t) = f k(t) + ıh fk (t) = F k (t)e ıϕ k (t) we can extract F k (t) and the instantaneous frequency dϕ k dt (t). HHT For each IMF k, we extract F k and dϕ k dt (t) and accumulate the information in the time-frequency plane.

20 HHT of f sig frequency (rad/s) time

21 HHT of f sig4 - ECG frequency (rad/s) time

22 EMD: Issues and Properties Useful to analyze real signals. Implementation dependent. Problem: it s a nonlinear algorithm which has no mathematical theory difficult to predict and understand its output and behavior in the general case. Experimental property: seems to behave as an adaptive filter bank (Flandrin et al. ) 004 Empirical mode decomposition as a filter bank, IEEE Signal Processing Letters, vol.11, No., pp ,

23 Key ideas about wavelets Wavelets filtering WT f (m, n) = a m/ 0 = a m/ 0 ( ) s where ψ m (s) = ψ. a m 0 = (f ψ m )(na m 0 b 0) f (t)ψ(a m 0 t nb 0 )dt ( t na m ) f (t)ψ 0 b 0 dt a m 0

24 Key ideas about wavelets Wavelets filtering WT f (m, n) = a m/ 0 = a m/ 0 ( ) s where ψ m (s) = ψ. a m 0 = (f ψ m )(na m 0 b 0) f (t)ψ(a m 0 t nb 0 )dt ( t na m ) f (t)ψ 0 b 0 dt a m 0 Wavelets can be built both in the temporal or Fourier domains.

25 Empirical wavelet transform (EWT): Concept Idea: Combining the strength of wavelet s formalism with the adaptability of EMD.

26 Empirical wavelet transform (EWT): Concept Idea: Combining the strength of wavelet s formalism with the adaptability of EMD. Wavelets are equivalent to filter banks (dyadic) decomposition of the Fourier line... π/8 π/4 π/ π ω

27 Empirical wavelet transform (EWT): Concept Idea: Combining the strength of wavelet s formalism with the adaptability of EMD. Wavelets are equivalent to filter banks (dyadic) decomposition of the Fourier line... π/8 π/4 π/ π ω Does not necessarily correspond to modes positions.

28 Empirical wavelet transform (EWT): Concept Idea: Combining the strength of wavelet s formalism with the adaptability of EMD. Wavelets are equivalent to filter banks (dyadic) decomposition of the Fourier line... π/8 π/4 π/ π ω Does not necessarily correspond to modes positions. EWT adaptive decomposition of the Fourier line ω 1 ω ω 3 ω n ω n+1 π

29 EWT: finding the modes Fourier spectrum segmentation: Find the local maxima. Take support boundaries as the middle between successive maxima ˆfsig1

30 EWT: finding the modes Fourier spectrum segmentation: Find the local maxima. Take support boundaries as the middle between successive maxima ˆfsig1

31 EWT: finding the modes Fourier spectrum segmentation: Find the local maxima. Take support boundaries as the middle between successive maxima ˆfsig 8

32 EWT: filter bank construction (1/3) Notations ω n : support boundaries τ n : half the length of the transition phase 1 τ 1 τ τ 3 τ n τ n+1 τ N ω 1 ω ω 3 ω n ω n+1 π

33 EWT: filter bank construction (1/3) Notations ω n : support boundaries τ n : half the length of the transition phase 1 τ 1 τ τ 3 τ n τ n+1 τ N ω 1 ω ω 3 ω n ω n+1 π In practice we choose τ n = γω n

34 EWT: filter bank construction (/3) Scaling function spectrum 1 [ ( )] if ω (1 γ)ω n π ˆφ n(ω) = cos β 1 γω n ( ω (1 γ)ω n) if (1 γ)ω n ω (1 + γ)ω n 0 otherwise Wavelet spectrum 1 [ ( if (1 + γ)ω )] n ω (1 γ)ω n+1 e ı ω cos π β 1 ( ω (1 γ)ω γω n+1 ) n+1 ˆψ n(ω) = [ ( if (1 )] γ)ω n+1 ω (1 + γ)ω n+1 e ı ω sin π β 1 γω n ( ω (1 γ)ω n) if (1 γ)ω n ω (1 + γ)ω n 0 otherwise

35 EWT: filter bank construction (3/3) Scaling function spectrum for ω n = 1 and γ = Π Π Wavelet spectrum for ω n = 1, ω n+1 =.5 and γ = Π Π

36 EWT: property and example (1/) Proposition If γ < min n ( ωn+1 ω n ω n+1 +ω n ), then the set {φ 1 (t), {ψ n (t)} N n=1 } is an orthonormal basis of L (R).

37 EWT: property and example (1/) Proposition ( ωn+1 ω If γ < min n n ω n+1 +ω n ), then the set {φ 1 (t), {ψ n (t)} N n=1 } is an orthonormal basis of L (R). Filter Bank for ω n {0, 1.5,,.8, π} with γ = 0.05 <

38 EWT: property and example (/) Detail coefficients: Wf E (n, t) = f, ψ n = f (τ)ψ n (τ t)dτ = (ˆf (ω) ˆψ n (ω)), Approximation coefficients (convention Wf E (0, t): Wf E (0, t) = f, φ 1 = f (τ)φ 1 (τ t)dτ = (ˆf (ω) ˆφ1 (ω)), The reconstruction: N f (t) = Wf E (0, t) φ 1 (t) + Wf E (n, t) ψ n (t) ( = Ŵf E (0, ω) ˆφ 1 (ω) + n=1 N Ŵf E (n, ω) ˆψ n (ω)). n=1

39 EWT: algorithm Input: f, N (number of scales) 1 Fourier transform of f ˆf. Compute the local maxima of ˆf on [0, π] and find the set {ω n }. 3 Choose γ < min n ( ωn+1 ω n ω n+1 +ω n ). 4 Build the filter bank. 5 Filter the signal to get each component.

40 Experiment: f Sig

41 Experiment of EMD: f Sig

42 Experiment of EMD: f Sig

43 Experiment of EMD: ECG

44 Time-Frequency representation of f sig frequency (rad/s) time

45 Time-Frequency representation of f sig frequency (rad/s) time

46 D - Extension joint work with Giang Tran and Stan Osher

47 D Extension - Tensor product approach Like the classic wavelet transform process rows then columns but...

48 D Extension - Tensor product approach Like the classic wavelet transform process rows then columns but... The number of detected scales can be different for each row

49 D Extension - Tensor product approach Like the classic wavelet transform process rows then columns but... The number of detected scales can be different for each row The position of the Fourier boundaries can vary a lot from one row to the next ( information discontinuity )

50 D Extension - Tensor product approach Like the classic wavelet transform process rows then columns but... The number of detected scales can be different for each row The position of the Fourier boundaries can vary a lot from one row to the next ( information discontinuity ) Idea: Mean Filter Banks

51 D Extension - Tensor product algorithm

52 D Extension - Tensor product algorithm 1DFFT

53 D Extension - Tensor product algorithm x DFFT Average

54 D Extension - Tensor product algorithm x DFFT Average MF B R

55 D Extension - Tensor product algorithm x DFFT Average MF B R 1DFFT x 10 4 Average MF BC

56 D Extension - Tensor product algorithm x DFFT Average MF B R 1DFFT MF B R... x 10 4 Average MF BC MF B C MF B C MF B C

57 D Extension - Example

58 D Extension - Ridgelet approach Classic Ridgelets ω t (j, t) f F P W 1,t θ F 1,ω θ θ W R f (j, t) t ω θ W 1,t F 1,ω θ θ F P f W R f

59 D Extension - Ridgelet approach Empirical Ridgelets ω ω f ω F P average θ θ t mfb ω W E W ER f θ θ WE θ F P f W ER f

60 D Extension - Ridgelet: a first example

61 D Extension - Ridgelet: a first example

62 D Extension - Ridgelet: a first example

63 D Extension - Ridgelet: a first example

64 D Extension - Ridgelet: a noisy example

65 D Extension - Ridgelet: a noisy example

66 D Extension - Ridgelet: a noisy example

67 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work

68 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum.

69 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands.

70 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands. Generalization to any kind of Fourier based wavelets (e.g. Splines).

71 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands. Generalization to any kind of Fourier based wavelets (e.g. Splines). D (nd) extension: finish ridgelet idea, curvelet, true spectrum segmentation.

72 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands. Generalization to any kind of Fourier based wavelets (e.g. Splines). D (nd) extension: finish ridgelet idea, curvelet, true spectrum segmentation. Explore the applications (denoising, deconvolution,... ).

73 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands. Generalization to any kind of Fourier based wavelets (e.g. Splines). D (nd) extension: finish ridgelet idea, curvelet, true spectrum segmentation. Explore the applications (denoising, deconvolution,... ). Solve PDEs (cf. Stan s talk).

74 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands. Generalization to any kind of Fourier based wavelets (e.g. Splines). D (nd) extension: finish ridgelet idea, curvelet, true spectrum segmentation. Explore the applications (denoising, deconvolution,... ). Solve PDEs (cf. Stan s talk). Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin,... ); adaptive harmonic analysis.

75 Conclusion - Future work Conclusion Get the ability of EMD under the Wavelet theory. Fast algorithm. Adaptive wavelets is not a completely new idea: Malvar-Wilson wavelets, Brushlets. Future work Investigate different strategies to segment the spectrum. Possibility of using a best-basis pursuit approach to find the optimal number of subbands. Generalization to any kind of Fourier based wavelets (e.g. Splines). D (nd) extension: finish ridgelet idea, curvelet, true spectrum segmentation. Explore the applications (denoising, deconvolution,... ). Solve PDEs (cf. Stan s talk). Impact on existing functional spaces (Sobolev, Besov, Triebel-Lizorkin,... ); adaptive harmonic analysis....

76 THANK YOU! PS: Jack, I m from UCLA and on the job market ;-)