Optimizing Wavelet Bases for Sparser Representations

  • Thomas GranditsEmail author
  • Thomas Pock
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10746)


Optimization in the wavelet domain has been a very prominent research topic both for denoising, as well as compression, reflected in its use in the JPEG-2000 standard. Its performance depends to a great extent on the wavelet \(\psi \) itself, represented in the form of a filter in the case of the discrete wavelet transform. While other works solely optimize the coefficients in the wavelet domain, we will use a combined approach, optimizing the wavelet \(\psi \) and the coefficients simultaneously in order to adapt both to a given image, resulting in a better reconstruction of an image from less coefficients. We will use several orthonormal wavelet bases as a starting point, but we will also demonstrate that we can create wavelets from white Gaussian noise with our approach, which are in some cases even better in terms of performance. Experiments will be conducted on several images, demonstrating how the optimization algorithm adapts to textured, as well as more homogeneous images.


  1. 1.
    Adelson, E.H., Anderson, C.H., Bergen, J.R., Burt, P.J., Ogden, J.M.: Pyramid methods in image processing. RCA Eng. 29(6), 33–41 (1984)Google Scholar
  2. 2.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chambolle, A., Vore, R.A.D., Lee, N.Y., Lucier, B.J.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chang, S.G., Yu, B., Vetterli, M.: Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9(9), 1532–1546 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chapa, J., Rao, R.: Algorithms for designing wavelets to match a specified signal. Trans. Sig. Proc. 48(12), 3395–3406 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45(5), 485–560 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1992). CrossRefzbMATHGoogle Scholar
  8. 8.
    Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41(7), 909–996 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haar, A.: Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69(3), 331–371 (1910)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989)CrossRefzbMATHGoogle Scholar
  11. 11.
    Mallat, S.: A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edn. Academic Press, Cambridge (2008)zbMATHGoogle Scholar
  12. 12.
    Mallat, S., Peyr, G.: A review of Bandlet methods for geometrical image representation. Num. Algorithms 44(3), 205–234 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Meyer, Y.: Principe d’incertitude, bases hilbertiennes et algbres d’oprateurs. Sminaire Bourbaki, vol. 28, pp. 209–223 (1985)Google Scholar
  14. 14.
    Ophir, B., Lustig, M., Elad, M.: Multi-scale dictionary learning using wavelets. IEEE J. Sel. Topics Sig. Process. 5(5), 1014–1024 (2011)CrossRefGoogle Scholar
  15. 15.
    Ouarti, N., Peyr, G.: Best basis denoising with non-stationary wavelet packets. In: 2009 16th IEEE International Conference on Image Processing (ICIP), pp. 3825–3828, November 2009Google Scholar
  16. 16.
    Peyré, G.: Texture synthesis and modification with a patch-valued wavelet transform. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 640–651. Springer, Heidelberg (2007). CrossRefGoogle Scholar
  17. 17.
    Pock, T., Sabach, S.: Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems. SIAM J. Imaging Sci. 9(4), 1756–1787 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sulam, J., Ophir, B., Zibulevsky, M., Elad, M.: Trainlets: dictionary learning in high dimensions. IEEE Trans. Sig. Process. 64(12), 3180–3193 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmonic Anal. 3(2), 186–200 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tewfik, A.H., Sinha, D., Jorgensen, P.: On the optimal choice of a wavelet for signal representation. IEEE Trans. Inf. Theory 38(2), 747–765 (1992)CrossRefzbMATHGoogle Scholar
  21. 21.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol. 58, 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Unser, M., Chenouard, N., Van De Ville, D.: Steerable pyramids and tight wavelet frames in \(L_2(R^d)\). IEEE Trans. Image Process. 20(10), 2705–2721 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, L., Wu, X., Buades, A., Li, X.: Color demosaicking by local directional interpolation and nonlocal adaptive thresholding. J. Electron. Imaging 20(2), 023016–023016-16 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Computer Graphics and VisionTU GrazGrazAustria

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