The Spectral Graph Wavelet Transform: Fundamental Theory and Fast Computation

  • David K. HammondEmail author
  • Pierre Vandergheynst
  • Rémi Gribonval
Part of the Signals and Communication Technology book series (SCT)


The spectral graph wavelet transform (SGWT) defines wavelet transforms appropriate for data defined on the vertices of a weighted graph. Weighted graphs provide an extremely flexible way to model the data domain for a large number of important applications (such as data defined on vertices of social networks, transportation networks, brain connectivity networks, point clouds, or irregularly sampled grids). The SGWT is based on the spectral decomposition of the \(N\times N\) graph Laplacian matrix \(\mathscr {L}\), where N is the number of vertices of the weighted graph. Its construction is specified by designing a real-valued function g which acts as a bandpass filter on the spectrum of \(\mathscr {L}\), and is analogous to the Fourier transform of the “mother wavelet” for the continuous wavelet transform. The wavelet operators at scale s are then specified by \(T_g^s = g(s\mathscr {L})\), and provide a mapping from the input data \(f\in \mathbb {R}^N\) to the wavelet coefficients at scale s. The individual wavelets \(\psi _{s,n}\) centered at vertex n, for scale s, are recovered by localizing these operators by applying them to a delta impulse, i.e. \(\psi _{s,n} = T_g^s \delta _n\). The wavelet scales may be discretized to give a graph wavelet transform producing a finite number of coefficients. In this work we also describe a fast algorithm, based on Chebyshev polynomial approximation, which allows computation of the SGWT without needing to compute the full set of eigenvalues and eigenvectors of \(\mathscr {L}\).


  1. 1.
    A. Grossmann, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41(7), 909–996 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    S.G. Mallat, A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989)zbMATHCrossRefGoogle Scholar
  4. 4.
    Y. Meyer, Orthonormal wavelets, Wavelets. Inverse Problems and Theoretical Imaging (Springer, Berlin, 1989)Google Scholar
  5. 5.
    G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44(2), 141–183 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    D.L. Donoho, I.M. Johnstone, Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    S. Chang, B. Yu, M. Vetterli, Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9, 1532–1546 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    L. Sendur, I. Selesnick, Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Process. 50, 2744–2756 (2002)CrossRefGoogle Scholar
  9. 9.
    J. Portilla, V. Strela, M.J. Wainwright, E.P. Simoncelli, Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12, 1338–1351 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    I. Daubechies, G. Teschke, Variational image restoration by means of wavelets: simultaneous decomposition, deblurring, and denoising. Appl. Comput. Harmon. Anal. 19(1), 1–16 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    F. Luisier, C. Vonesch, T. Blu, M. Unser, Fast interscale wavelet denoising of poisson-corrupted images. Signal Process. 90(2), 415–427 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    J. Shapiro, Embedded image coding using zerotrees of wavelet coefficients. IEEE Trans. Signal Process. 41, 3445–3462 (1993)zbMATHCrossRefGoogle Scholar
  13. 13.
    A. Said, W. Pearlman, A new, fast, and efficient image codec based on set partitioning in hierarchical trees. IEEE Trans. Circuits Syst. Video Technol. 6, 243–250 (1996)CrossRefGoogle Scholar
  14. 14.
    M. Hilton, Wavelet and wavelet packet compression of electrocardiograms. IEEE Trans. Biomed. Eng. 44, 394–402 (1997)CrossRefGoogle Scholar
  15. 15.
    R. Buccigrossi, E. Simoncelli, Image compression via joint statistical characterization in the wavelet domain. IEEE Trans. Image Process. 8, 1688–1701 (1999)CrossRefGoogle Scholar
  16. 16.
    D. Taubman, M. Marcellin, JPEG2000: Image Compression Fundamentals, Standards and Practice (Kluwer Academic Publishers, Dordrecht, 2002)CrossRefGoogle Scholar
  17. 17.
    J.-L. Starck, A. Bijaoui, Filtering and deconvolution by the wavelet transform. Signal Process. 35(3), 195–211 (1994)zbMATHCrossRefGoogle Scholar
  18. 18.
    D.L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2(2), 101–126 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    E. Miller, A.S. Willsky, A multiscale approach to sensor fusion and the solution of linear inverse problems. Appl. Comput. Harmon. Anal. 2(2), 127–147 (1995)zbMATHCrossRefGoogle Scholar
  20. 20.
    R. Nowak, E. Kolaczyk, A statistical multiscale framework for Poisson inverse problems. IEEE Trans. Inf. Theory 46, 1811–1825 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    J. Bioucas-Dias, Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors. IEEE Trans. Image Process. 15, 937–951 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    J.R. Wishart, Wavelet deconvolution in a periodic setting with long-range dependent errors. J. Stat. Plan. Inference 143(5), 867–881 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    F.K. Chung, Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92 (AMS Bookstore, Providence, 1997)Google Scholar
  24. 24.
    M. Crovella, E. Kolaczyk, Graph wavelets for spatial traffic analysis, in INFOCOM 2003. Twenty-Second Annual Joint Conference of the IEEE Computer and Communications Societies, Jan 2003, vol. 3 (IEEE, 2003), pp. 1848–1857Google Scholar
  25. 25.
    A. Smalter, J. Huan, G. Lushington, Graph wavelet alignment kernels for drug virtual screening. J. Bioinform. Comput. Biol. 7, 473–497 (2009)CrossRefGoogle Scholar
  26. 26.
    M. Jansen, G.P. Nason, B.W. Silverman, Multiscale methods for data on graphs and irregular multidimensional situations. J. R. Stat. Soc. Ser. (Stat. Methodol.) 71(1), 97–125 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    F. Murtagh, The Haar wavelet transform of a dendrogram. J. Classif. 24, 3–32 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    A.B. Lee, B. Nadler, L. Wasserman, Treelets - an adaptive multi-scale basis for sparse unordered data. Ann. Appl. Stat. 2, 435–471 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    R.R. Coifman, M. Maggioni, Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53–94 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    M. Maggioni, H. Mhaskar, Diffusion polynomial frames on metric measure spaces. Appl. Comput. Harmon. Anal. 24(3), 329–353 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    D. Geller, A. Mayeli, Continuous wavelets on compact manifolds. Mathematische Zeitschrift 262, 895–927 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    D.K. Hammond, P. Vandergheynst, R. Gribonval, Wavelets on graphs via spectral graph theory. Appl. Comput. Harmon. Anal. 30(2), 129–150 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    D. Thanou, D.I. Shuman, P. Frossard, Learning parametric dictionaries for signals on graphs. IEEE Trans. Signal Process. 62, 3849–3862 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    W.H. Kim, D. Pachauri, C. Hatt, M.K. Chung, S. Johnson, V. Singh, Wavelet based multi-scale shape features on arbitrary surfaces for cortical thickness discrimination, in Advances in Neural Information Processing Systems 25 ed. by F. Pereira, C.J.C. Burges, L. Bottou, K.Q. Weinberger (Curran Associates Inc., 2012), pp. 1241–1249Google Scholar
  35. 35.
    W.H. Kim, M.K. Chung, V. Singh, Multi-resolution shape analysis via non-Euclidean wavelets: applications to mesh segmentation and surface alignment problems, in 2013 IEEE Conference on Computer Vision and Pattern Recognition, June 2013 (2013), pp. 2139–2146Google Scholar
  36. 36.
    N. Tremblay, P. Borgnat, Multiscale community mining in networks using spectral graph wavelets, in 21st European Signal Processing Conference (EUSIPCO 2013), Sept 2013 (2013), pp. 1–5Google Scholar
  37. 37.
    M. Zhong, H. Qin, Sparse approximation of 3d shapes via spectral graph wavelets. Visual Comput. 30, 751–761 (2014)CrossRefGoogle Scholar
  38. 38.
    M. Reed, B. Simon, Methods of Modern Mathematical Physics Volume 1: Functional Analysis (Academic Press, London, 1980)Google Scholar
  39. 39.
    I. Daubechies, Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics (1992)Google Scholar
  40. 40.
    C.E. Heil, D.F. Walnut, Continuous and discrete wavelet transforms. SIAM Rev. 31(4), 628–666 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    D. Watkins, The Matrix Eigenvalue Problem - GR and Krylov Subspace Methods. Society for Industrial and Applied Mathematics (2007)Google Scholar
  42. 42.
    G.L.G. Sleijpen, H.A.V. der Vorst, A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17(2), 401–425 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    K.O. Geddes, Near-minimax polynomial approximation in an elliptical region. SIAM J. Numer. Anal. 15(6), 1225–1233 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    W. Fraser, A survey of methods of computing minimax and near-minimax polynomial approximations for functions of a single independent variable. J. Assoc. Comput. Mach. 12, 295–314 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    G.M. Phillips, Interpolation and Approximation by Polynomials. CMS Books in Mathematics (Springer, Berlin, 2003)Google Scholar
  46. 46.
    I. Daubechies, M. Defrise, C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    J.R. Shewchuk, An introduction to the conjugate gradient method without the agonizing pain. Technical report, Pittsburgh, PA, USA, 1994Google Scholar
  48. 48.
    N. Leonardi, D.V.D. Ville, Tight wavelet frames on multislice graphs. IEEE Trans. Signal Process. 61, 3357–3367 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    J.B. Tenenbaum, V.d. Silva, J.C. Langford, A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)Google Scholar
  50. 50.
    P. Wessel, W.H.F. Smith, A global, self-consistent, hierarchical, high-resolution shoreline database. J Geophys. Res. 101(B4), 8741–8743 (1996)CrossRefGoogle Scholar
  51. 51.
    N. Perraudin, J. Paratte, D. Shuman, L. Martin, V. Kalofolias, P. Vandergheynst, D.K. Hammond, GSPBOX: a toolbox for signal processing on graphs, ArXiv e-prints (2014)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David K. Hammond
    • 1
    Email author
  • Pierre Vandergheynst
    • 2
  • Rémi Gribonval
    • 3
  1. 1.Oregon Institute of Technology - Portland MetroWilsonvilleUSA
  2. 2.Ecole Polytechnique Federale de LausanneLausanneSwitzerland
  3. 3.Univ Rennes, Inria, CNRS, IRISARennesFrance

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