Skip to main content
SpringerLink
Log in

Optimizing Wavelet Bases for Sparser Representations

  • Conference paper
  • First Online:
Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10746))

Abstract

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.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chapa, J., Rao, R.: Algorithms for designing wavelets to match a specified signal. Trans. Sig. Proc. 48(12), 3395–3406 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45(5), 485–560 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Daubechies, I.: Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1992). https://doi.org/10.1137/1.9781611970104

    Book  MATH  Google Scholar 

  8. Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41(7), 909–996 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Haar, A.: Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69(3), 331–371 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989)

    Article  MATH  Google Scholar 

  11. Mallat, S.: A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edn. Academic Press, Cambridge (2008)

    MATH  Google Scholar 

  12. Mallat, S., Peyr, G.: A review of Bandlet methods for geometrical image representation. Num. Algorithms 44(3), 205–234 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Meyer, Y.: Principe d’incertitude, bases hilbertiennes et algbres d’oprateurs. Sminaire Bourbaki, vol. 28, pp. 209–223 (1985)

    Google Scholar 

  14. Ophir, B., Lustig, M., Elad, M.: Multi-scale dictionary learning using wavelets. IEEE J. Sel. Topics Sig. Process. 5(5), 1014–1024 (2011)

    Article  Google Scholar 

  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 2009

    Google Scholar 

  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). https://doi.org/10.1007/978-3-540-72823-8_55

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sulam, J., Ophir, B., Zibulevsky, M., Elad, M.: Trainlets: dictionary learning in high dimensions. IEEE Trans. Sig. Process. 64(12), 3180–3193 (2016)

    Article  MathSciNet  Google Scholar 

  19. Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmonic Anal. 3(2), 186–200 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  21. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol. 58, 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Graphics and Vision, TU Graz, Graz, Austria

    Thomas Grandits & Thomas Pock

Authors
  1. Thomas Grandits
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Pock
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Grandits .

Editor information

Editors and Affiliations

  1. Ca’ Foscari University of Venice, Venice, Italy

    Marcello Pelillo

  2. University of York, York, United Kingdom

    Edwin Hancock

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Grandits, T., Pock, T. (2018). Optimizing Wavelet Bases for Sparser Representations. In: Pelillo, M., Hancock, E. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2017. Lecture Notes in Computer Science(), vol 10746. Springer, Cham. https://doi.org/10.1007/978-3-319-78199-0_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-78199-0_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78198-3

  • Online ISBN: 978-3-319-78199-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation