Empirical Mode Decomposition (EMD) on Surfaces

1. Geometric Modeling and Processing 2012 Empirical Mode Decomposition (EMD) on Surfaces Hui Wang1,2 Zhixun Su1 Junjie Cao1 Ye Wang3 Hao Zhang2 1Dalian University of Technology 2Simon Fraser University 3Harbin Institute of Technology

2. Motivation Generalize signal processing methods to surfaces Original surface Low-pass filtering Enhancement filtering

3. Previous Works • Parameterization-based methods Geometry images (Gu et al. 2002) Filtering by spherical harmonics (Zhou et al. 2004) … • Surface-based methods Fourier transform (Taubin 1995) Subdivision wavelet (Valette and Prost 2004, Wang and Tang 2009) Detail editing via Laplacian coordinates (Wang et al. 2011) Mexican Hat Wavelet (Hou and Qin 2012) …

4. Our work Generalize multi-scaleEmpirical Mode Decomposition (EMD) to surfaces = + + Original scalar function IMF 1 IMF 2 Smoothed residue Details at different scales IMF: Intrinsic Mode Function

5. Contents • 1. 1D EMD • 2. Our generalized EMD on surfaces • 3. Feature-preserving smoothing by EMD • 4. Conclusion and future works

6. Contents • 1. 1D EMD • 2. Our generalized EMD on surfaces • 3. Feature-preserving smoothing by EMD • 4. Conclusion and future works

7. 1D EMD Empirical Mode Decomposition (EMD) and Hilbert-Huang Transform (HHT) Cited by 5675!

8. Comparison The basis is data-driven and adaptive. Work well for non-linear and non-stationary signals. Motivate potential applications in geometry processing.

9. 1D EMD Example Data: x = IMF 1: d1 + IMF 2: d2 + IMF 3: d3 + IMF 4: d4 + IMF 5: d5 + IMF 6: d6 + Residue: r6

10. What is the 1D IMF? Images taken from [Huang et al. 1998] Typical example of 1D IMF Similar to the harmonic function Use “Sifting Process” to extract each IMF

11. Sifting Process Original data: x

12. Sifting Process Local maximum

13. Sifting Process Local maximum and minimum

14. Sifting Process Envelopes Interpolated by the Cubic Spline

15. Sifting Process Mean of envelopes of x: m0

16. Sifting Process h1 = x - m0

17. Sifting Process h1

18. Sifting Process Mean of envelopes of h1: m1

19. Sifting Process h2 = h1 - m1

20. Sifting Process h2

21. Sifting Process h1 = x - m0 h2 = h1 – m1 … hk = hk-1 – mk-1 How to stop the sifting process?

22. Stopping Criterion of Sifting Process The number of zero-crossings and extrema of hk are the same or differ at most by one. AND The stander deviation of hk and hk-1 is smaller than a pre-set value. IMF 1: d1 = hk

23. First Scale EMD IMF 1: d1 = h5 Data: x = IMF 1: d1 + Residue 1: r1

24. Second Scale EMD Data: x = IMF 1: d1 + IMF 2: d2 + Residue 2: r2

25. 1D EMD x r1 r2 rk-1 rk … … d1 d2 dk How to stop the EMD?

26. Stop Criterion of EMD The residue or IMF becomes so small. OR The residue becomes a monotonic function or constant.

27. Finial EMD Data: x = IMF 1: d1 + IMF 2: d2 + IMF 3: d3 + IMF 4: d4 + IMF 5: d5 + IMF 6: d6 + Residue: r6

28. Contents • 1. 1D EMD • 2. Our generalized EMD on surfaces • 3. Feature-preserving smoothing by EMD • 4. Conclusion and future works

29. Generalized EMD on Surfaces = + + Original scalar function IMF 1 IMF 2 Smoothed residue The principle is similar to that of 1D Local extrema detection and interpolation method

30. Local Extrema Detection Local maximum: functional value isn’t smaller than that of 1-ring neighbors Local minimum: functional value isn’t larger than that of 1-ring neighbors

31. Interpolated Method on Surfaces We minimize the linearized thin-plate energy: The Euler-Lagrange equation is:

32. Interpolated Method on Surfaces A bi-harmonic field with Dirichlet boundary conditions:

33. Result of EMD on surfaces Original scalar function IMF 1 IMF 2 IMF 3 IMF 4 IMF 5 Residue

34. Application: Filtering Original scalar function: IMFs represent details at different scales Filtering result: Enhancing Smoothing

35. Filtering Scalar Function Original function Enhancing result Smoothing result

36. Filtering the Surface Filtering the three coordinates functions respectively. Original surface High enhancement Band enhancement Band smoothing Smoothing (1, 0, 0) (0, 0, 0)

37. Surface Denoising Original surface Corrupted by noise The first residue

38. Contents • 1. 1D EMD • 2. Our generalized EMD on surfaces • 3. Feature-preserving smoothing by EMD • 4. Conclusion and future works

39. Feature-preserving Smoothing? Our generalized EMD cannot preserve sharp features. First, the interpolation method is not feature aware. Second, the three coordinates functions are processed separately. We still propose a feature-preserving smoothing method based on EMD.

40. Edge-preserving Multiscale Image Decomposition Images taken from [Subr et al. 2009] K. Subr, C. Soler, F. Durand, Edge-preserving multiscale image decomposition based on local extrema, ACM Transactions on Graphics 28 (5) (2009) 1–9.

41. 1D Edge-preserving Smoothing Enlarge the extrema-location kernel Images taken from [Subr et al. 2009] Feature-preserving interpolation No sifting

42. Our Generalization • Extrema identification: Local extrema of Gaussian curvature of k-ring neighbors at the k-th scale • Feature-preserving interpolation:

43. Feature-preserving Smoothing Result Original noisy surface The 8th level smoothed result

44. Feature-preserving Smoothing Result Surface with real world noise The first level smoothed result

45. Compare with the Bilateral Filtering Bilateral Filtering Our EMD-based method Need a more robust feature-aware interpolation

46. Contents • 1. 1D EMD • 2. Our generalized EMD on surfaces • 3. Feature-preserving smoothing by EMD • 4. Conclusion and future works

47. Conclusions • We first introduce the EMD from Euclidean space to the setting of surfaces. • We also make a first try for feature-preserving surface smoothing based EMD.

48. Limitations and Future Works • Our work is the first step toward generalizing the EMD to geometric processing, some problems needed be improved or investigated: The behaviour of IMFs on surface Generalize the Hilbert-Huang Transform (HHT) to surfaces More possible applications of the generalized EMD More robust anisotropic extrema detection and feature-aware interpolation …

49. Thanks! 谢谢！