Skip to main content
SpringerLink
Log in

Mode Decomposition Evolution Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Wei and Jia in Europhys. Lett. 59(6):814–819, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett. 6(7):165–167, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be easily used for secondary processing. Various simplifications of the proposed MoDEEs, including a linearized version, and an algebraic version, are discussed for computational convenience. The Fourier pseudospectral method, which is unconditionally stable for linearized high order MoDEEs, is utilized in our computation. Validation is carried out to mode separation of high frequency adjacent modes. Applications are considered to signal and image denoising, image edge detection, feature extraction, enhancement etc. It is hoped that this work enhances the understanding of high order PDEs and yields robust and useful tools for image and signal analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angenent, S., Pichon, E., Tannenbaum, A.: Mathematical methods in medical image processing. Bull. Am. Math. Soc. 43(3), 365–396 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Archibald, R., Gelb, A., Yoon, J.H.: Polynomial fitting for edge detection in irregularly sampled signals and images. SIAM J. Numer. Anal. 43(1), 259–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Archibald, R., Gelb, A., Saxena, R., Xiu, D.B.: Discontinuity detection in multivariate space for stochastic simulations. J. Comput. Phys. 228(7), 2676–2689 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barbu, T., Barbu, V., Biga, V., Coca, D.: A PDE variational approach to image denoising and restoration. Nonlinear Anal., Real World Appl. 10(3), 1351–1361 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bates, P.W., Chen, Z., Sun, Y.H., Wei, G.W., Zhao, S.: Geometric and potential driving formation and evolution of biomolecular surfaces. J. Math. Biol. 59(2), 193–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertalmio, M.: Strong-continuation, contrast-invariant inpainting with a third-order optimal PDE. IEEE Trans. Image Process. 15(7), 1934–1938 (2006)

    Article  Google Scholar 

  7. Bertozzi, A.L., Greer, J.B.: Low-curvature image simplifiers: global regularity of smooth solutions and Laplacian limiting schemes. Commun. Pure Appl. Math. 57(6), 764–790 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buxton, R.B.: Introduction to Functional Magnetic Resonance Imaging—Principles and Techniques. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  9. Canny, J.: A computational approach to edge-detection. IEEE Trans. Pattern Anal. Mach. Intell. 8(6), 679–698 (1986)

    Article  Google Scholar 

  10. Caselles, V., Morel, J.M., Sapiro, G., Tannenbaum, A.: Introduction to the special issue on partial differential equations and geometry-driven diffusion in image processing and analysis. IEEE Trans. Image Process. 7(3), 269–273 (1998)

    Article  Google Scholar 

  11. Catte, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge-detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chan, Y.: Wavelet Basics. Springer, Berlin (1995)

    Book  Google Scholar 

  14. Chan, T., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods. Society for Industrial Mathematics, Philadelphia (2005)

    MATH  Google Scholar 

  15. Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, K., Chen, X., Renaut, R., Alexander, G.E., Bandy, D., Guo, H., Reiman, E.M.: Characterization of the image-derived carotid artery input function using independent component analysis for the quantitation of 18f fluorodeoxyglucose positron emission tomography images. Phys. Med. Biol. 52(23), 7055–7071 (2007)

    Article  Google Scholar 

  17. Chen, Q.H., Huang, N., Riemenschneider, S., Xu, Y.S.: A B-spline approach for empirical mode decompositions. Adv. Comput. Math. 24(1–4), 171–195 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, Z., Baker, N.A., Wei, G.W.: Differential geometry based solvation models I: Eulerian formulation. J. Comput. Phys. 229, 8231–8258 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  20. Echeverria, J.C., Crowe, J.A., Woolfson, M.S., Hayes-Gill, B.R.: Application of empirical mode decomposition to heart rate variability analysis. Med. Biol. Eng. Comput. 39(4), 471–479 (2001)

    Article  Google Scholar 

  21. Farge, M.: Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. 24, 395–457 (1992)

    Article  MathSciNet  Google Scholar 

  22. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Trans. Image Process. 11(7), 689–703 (2002)

    Article  Google Scholar 

  23. Gilboa, G., Sochen, N., Zeevi, Y.Y.: Image sharpening by flows based on triple well potentials. J. Math. Imaging Vis. 20(1–2), 121–131 (2004)

    Article  MathSciNet  Google Scholar 

  24. Greer, J.B., Bertozzi, A.L.: H-1 solutions of a class of fourth order nonlinear equations for image processing. Discrete Contin. Dyn. Syst. 10(1–2), 349–366 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Greer, J.B., Bertozzi, A.L.: Traveling wave solutions of fourth order PDEs for image processing. SIAM J. Math. Anal. 36(1), 38–68 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grimm, V., Henn, S., Witsch, K.: A higher-order PDE-based image registration approach. Numer. Linear Algebra Appl. 13(5), 399–417 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gu, Y., Wei, G.W.: Conjugate filter approach for shock capturing. Commun. Numer. Methods Eng. 19(2), 99–110 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Guan, S., Lai, C., Wei, G.: A wavelet method for the characterization of spatiotemporal patterns. Physica D 163(1–2), 49–79 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Guo, H., Renaut, R., Chen, K.: An input function estimation method for FDG-PET human brain studies. Nucl. Med. Biol. 34(5), 483–492 (2007)

    Article  Google Scholar 

  30. Guo, H., Renaut, R.A., Chen, K., Reiman, E.: FDG-PET parametric imaging by total variation minimization. Comput. Med. Imaging Graph. 33(4), 295–303 (2009)

    Article  Google Scholar 

  31. Haacke, E.M., Brown, R.W., Thompson, M.R., Venkatesan, R.: Magnetic Resonance Imaging: Physical Principles and Sequence Design. Wiley, New York (1999)

    Google Scholar 

  32. Huang, N.E., Long, S.R., Shen, Z.: The mechanism for frequency downshift in nonlinear wave evolution. Adv. Appl. Mech. 32, 59 (1996)

    Article  Google Scholar 

  33. Huang, N.E., Shen, Z., Long, S.R.: A new view of nonlinear water waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31, 417–457 (1999)

    Article  MathSciNet  Google Scholar 

  34. Huang, N.E., Shen, Z., Long, S.R., Wu, M.L.C., Shih, H.H., Zheng, Q.N., Yen, N.C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc., Math. Phys. Eng. Sci. 454(1971), 903–995 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jain, A.K.: Partial-differential equations and finite-difference methods in image-processing. 1. image representation. J. Optim. Theory Appl. 23(1), 65–91 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. Jin, J.H., Shi, J.J.: Feature-preserving data compression of stamping tonnage information using wavelets. Technometrics 41(4), 327–339 (1999)

    Article  Google Scholar 

  37. Jin, Z.M., Yang, X.P.: Strong solutions for the generalized Perona-Malik equation for image restoration. Nonlinear Anal., Theory Methods Appl. 73(4), 1077–1084 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Karras, D.A., Mertzios, G.B.: New PDE-based methods for image enhancement using SOM and Bayesian inference in various discretization schemes. Meas. Sci. Technol. 20(10), 8 (2009)

    Article  Google Scholar 

  39. Kopsinis, Y., McLaughlin, S.: Development of EMD-based denoising methods inspired by wavelet thresholding. IEEE Trans. Signal Process. 57(4), 1351–1362 (2009)

    Article  MathSciNet  Google Scholar 

  40. Li, S.: Markov Random Field Modeling in Image Analysis. Springer, New York (2009)

    MATH  Google Scholar 

  41. Liang, H.L., Lin, Q.H., Chen, J.D.Z.: Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease. IEEE Trans. Biomed. Eng. 52(10), 1692–1701 (2005)

    Article  Google Scholar 

  42. Lin, L., Wang, Y., Zhou, H.: Iterative filtering as an alternative algorithm for empirical mode decomposition. Adv. Adapt. Data Anal. 1(4), 543–560 (2009)

    Article  MathSciNet  Google Scholar 

  43. Liu, B., Riemenschneider, S., Xu, Y.: Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum. Mech. Syst. Signal Process. 20(3), 718–734 (2006)

    Article  Google Scholar 

  44. Lysaker, M., Lundervold, A., Tai, X.C.: Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)

    Article  Google Scholar 

  45. Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  46. Mao, D., Rockmore, D., Wang, Y., Wu, Q.: EMD analysis for visual stylometry. Preprint

  47. Mao, D., Wang, Y., Wu, Q.: A new approach for analyzing physiological time series. Preprint

  48. Marr, D., Hildreth, E.: Theory of edge-detection. Proc. R. Soc. Lond. B, Biol. Sci. 207(1167), 187–217 (1980)

    Article  Google Scholar 

  49. Meyer, F.G., Coifman, R.R.: Brushlets: a tool for directional image analysis and image compression. Appl. Comput. Harmon. Anal. 4(2), 147–187 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  50. Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 826–833 (1992)

    Article  Google Scholar 

  51. Oppenheim, A.V., Schafer, R.W.: Digital Signal Process. Prentice-Hall, Englewood Cliffs (1989)

    Google Scholar 

  52. Perona, P., Malik, J.: Scale-space and edge-detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  53. Pesenson, M., Roby, W., McCollum, B.: Multiscale astronomical image processing based on nonlinear partial differential equations. Astrophys. J. 683(1), 566–576 (2008)

    Article  Google Scholar 

  54. Radke, R.J., Andra, S., Al-Kofahi, O., Roysam, B.: Image change detection algorithms: a systematic survey. IEEE Trans. Image Process. 14(3), 294–307 (2005)

    Article  MathSciNet  Google Scholar 

  55. Rezaei, D., Taheri, F.: Experimental validation of a novel structural damage detection method based on empirical mode decomposition. Smart Mater. Struct. 18(4) (2009)

  56. Rilling, G., Flandrin, P., Goncalves, P., Lilly, J.M.: Bivariate empirical mode decomposition. IEEE Signal Process. Lett. 14, 936–939 (2007)

    Article  Google Scholar 

  57. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MATH  Google Scholar 

  58. Saxena, R., Gelb, A., Mittelmann, H.: A high order method for determining the edges in the gradient of a function. Commun. Comput. Phys. 5(2–4), 694–711 (2009)

    MathSciNet  Google Scholar 

  59. Shih, Y., Rei, C., Wang, H.: A novel PDE based image restoration: convection-diffusion equation for image denoising. J. Comput. Appl. Math. 231(2), 771–779 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. Siddiqi, K., Kimia, B.B., Shu, C.W.: Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution. Graph. Models Image Process. 59(5), 278–301 (1997)

    Article  Google Scholar 

  61. Spedding, G.R., Browand, F.K., Huang, N.E., Long, S.R.: A 2D complex wavelet analysis of an unsteady wind-generated surface wavelet analysis of an unsteady wind-generated surface wave field. Dyn. Atmos. Ocean. 20, 55–77 (1993)

    Article  Google Scholar 

  62. Sun, Y.H., Wu, P.R., Wei, G., Wang, G.: Evolution-operator-based single-step method for image processing. Int. J. Biomed. Imaging 83847, 1 (2006)

    Article  Google Scholar 

  63. Sun, Y.H., Zhou, Y.C., Li, S.G., Wei, G.W.: A windowed Fourier pseudospectral method for hyperbolic conservation laws. J. Comput. Phys. 214(2), 466–490 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  64. Tanaka, T., Mandic, D.P.: Complex empirical mode decomposition. IEEE Signal Process. Lett. 14(2), 101–104 (2007)

    Article  Google Scholar 

  65. Tang, Y.-W., Tai, C.-C., Su, C.-C., Chen, C.-Y., Chen, J.-F.: A correlated empirical mode decomposition method for partial discharge signal denoising. Meas. Sci. Technol. 21, 085106 (2010)

    Article  Google Scholar 

  66. Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface processing via normal maps. ACM Trans. Graph. 22(4), 1012–1033 (2003)

    Article  Google Scholar 

  67. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Oxford University Press, London (1948)

    Google Scholar 

  68. Wang, Y., Zhao, Y.B., Wei, G.W.: A note on the numerical solution of high-order differential equations. J. Comput. Appl. Math. 159(2), 387–398 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  69. Wang, Z., Vemuri, B.C., Chen, Y., Mareci, T.H.: A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Trans. Med. Imaging 23, 930 (2004)

    Article  Google Scholar 

  70. Wang, Y., Wei, G., Yang, S.: Iterative filtering decomposition based on local spectral evolution kernel. J. Sci. Comput. (2011, accepted). doi:10.1007/s10915-011-9496-0

  71. Wang, Y., Wei, G., Yang, S.: Partial differential equation transform—variational formulation and Fourier analysis. Int. J. Numer. Methods Biomed. Eng. (2011, accepted). doi:10.1002/cnm.1452

  72. Wei, G.W.: Generalized Perona-Malik equation for image restoration. IEEE Signal Process. Lett. 6(7), 165–167 (1999)

    Article  Google Scholar 

  73. Wei, G.W.: Wavelets generated by using discrete singular convolution kernels. J. Phys. A, Math. Gen. 33, 8577–8596 (2000)

    Article  MATH  Google Scholar 

  74. Wei, G.W.: Oscillation reduction by anisotropic diffusions. Comput. Phys. Commun. 144, 417–342 (2002)

    Article  Google Scholar 

  75. Wei, G.W., Jia, Y.Q.: Synchronization-based image edge detection. Europhys. Lett. 59(6), 814–819 (2002)

    Article  Google Scholar 

  76. Westin, C.-F., Maier, S.E., Mamata, H., Nabavi, A., Jolesz, F.A., Kikinis, R.: Processing and visualization of diffusion tensor MRI. Med. Image Anal. 6, 93 (2002)

    Article  Google Scholar 

  77. Witelski, T.P., Bowen, M.: ADI schemes for higher-order nonlinear diffusion equations. Appl. Numer. Math. 45(2–3), 331–351 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  78. Witkin, A.: Scale-space filtering: a new approach to multi-scale description. In: Proceedings of IEEE International Conference on Acoustic Speech Signal Processing, vol. 9, pp. 150–153. Institute of Electrical and Electronics Engineers, New York (1984)

    Google Scholar 

  79. Wu, J.Y., Ruan, Q.Q., An, G.Y.: Exemplar-based image completion model employing PDE corrections. Informatica 21(2), 259–276 (2010)

    MATH  Google Scholar 

  80. Xu, M., Zhou, S.L.: Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation. J. Math. Anal. Appl. 325(1), 636–654 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  81. Yang, S., Zhou, Y.C., Wei, G.W.: Comparison of the discrete singular convolution algorithm and the Fourier pseudospectral method for solving partial differential equations. Comput. Phys. Commun. 143(2), 113–135 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  82. Yang, S., Coe, J., Kaduk, B., Martínez, T.: An “optimal” spawning algorithm for adaptive basis set expansion in nonadiabatic dynamics. J. Chem. Phys. 130, 134113 (2009)

    Article  Google Scholar 

  83. You, Y., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2002)

    Article  MathSciNet  Google Scholar 

  84. Zhao, S., Wei, G.W.: Comparison of the discrete singular convolution and three other numerical schemes for solving fisher’s equation. SIAM J. Sci. Comput. 25(1), 127–147 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  85. Zhao, S., Wei, G.W.: Matched interface and boundary (MIB) for the implementation of boundary conditions in high-order central finite differences. Int. J. Numer. Methods Eng. 77(12), 1690–1730 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Yang Wang, Guo-Wei Wei & Siyang Yang

  2. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Guo-Wei Wei

Authors
  1. Yang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guo-Wei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siyang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-Wei Wei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Y., Wei, GW. & Yang, S. Mode Decomposition Evolution Equations. J Sci Comput 50, 495–518 (2012). https://doi.org/10.1007/s10915-011-9509-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-011-9509-z

Keywords

Search

Navigation