Skip to main content
SpringerLink
Log in

Cell average image transform algorithms with exact error control

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper we present non-separable two-dimensional multi-resolution algorithms based on Harten’s cell average framework for multi-resolution which guarantee a priori prescribed quality in the reconstructed image, hence being suitable for applications where quality control is most important, yet performing economically in terms of storage and speed of computation. Moreover, after applying the algorithm the exact error between the original and the decoded images measured in the L 2 discrete norm is known without being necessary to decode the image.

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. Amat, S., Aràndiga, F., Cohen, A., Donat, R.: Tensor product multiresolution analysis with error control for compact image representation. Signal Proc. 4, 587–608 (2002)

    Article  Google Scholar 

  2. Amat, S., Aràndiga, F., Cohen, A., Donat, R., Garcia, G., von Oehsen, M.: Data compression with ENO schemes: a case study. Appl. Comput. Harmon. Anal. 11(2), 273–288 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amat, S., Busquier, S., Trillo, J.C.: Harten’s multiresolution on the quincunx pyramid. J. Comput. Appl. Math. 189, 555–567 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Amat, S., Dadourian, K., Liandrat, J., Ruiz, J., Trillo, J.C.: A family of stable nonlinear nonseparable multiresolution schemes in 2D. J. Comput. Appl. Math. 234(4), 1277–1290 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Amat, S., Ruiz, J., Trillo, J.C.: Adaptive interpolation of images using a new nonlinear cell-average scheme. Math. Comput. Simul 82(9), 1586–1596 (2012)

    Article  MathSciNet  Google Scholar 

  6. Aràndiga, F., Baccou, J., Doblas, M., Liandrat, J.: Image compression based on a multi-directional map-dependent algorithm. Appl. Comput. Harmon. Anal. 23, 181–197 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Aràndiga, F., Baeza, A., Belda, A.M., Mulet, P.: Analysis of WENO schemes for full and global accuracy. SIAM J. Numer. Anal. 49(2), 893–915 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Aràndiga, F., Belda, A.M.: Weighted ENO interpolation and applications. Commun. Nonlinear Sci. Numer. Simul. 9(2), 187–195 (2003)

    Article  Google Scholar 

  9. Aràndiga, F., Belda, A.M., Mulet, P.: Point-value WENO multiresolution applications to stable image compression. J. Sci. Comput. 43(2), 158–182 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Arandiga, F., Cohen, A., Donat, R., Dyn, N., Matei, B.: Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques. Appl. Comput. Harmon. Anal. 24(2), 225–250 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Aràndiga, F., Donat, R.: Nonlinear multiscale decompositions: the approach of A. Harten. Numer. Algorithms 23(2-3), 175–216 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Arándiga, F., Donat, R.: Stability through syncronization in nonlinear multiscale transformations. SIAM J. Sci. Comput. 29(1), 265–289 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Aràndiga, F., Donat, R., Mulet, P.: Adaptive interpolation of images. Sig. Proc. 83, 459–464 (2003)

    Article  MATH  Google Scholar 

  14. Aràndiga, F., Mulet, P., Renau, V.: Non-separable two-dimensional weighted ENO interpolation. Appl. Numer. Math. 62(8), 975–987 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Aràndiga, F., Mulet, P., Renau, V.: Lossless and near-lossless image compression based on multiresolution analysis. J. Comput. Appl. Math. 242(0), 70–81 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bihari, B.L., Harten, A.: Application of generalized wavelets: an adaptive multiresolution scheme. J. Comput. Appl. Math. 61(3), 275–321 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bloom, C.: Solving the problems of context modeling. http://www.cbloom.com/papers/ (1998)

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Cohen, A., Dyn, N., Matei, B.: Quasilinear subdivision schemes with applications to eno interpolation. Appl. Comput. Harmon. Anal. 15, 89–116 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Harizanov, P., Oswald, S.: Stability of nonlinear subdivision and multiscale transforms. Constr. Approx. 31(3), 359–393 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Harten, A.: Discrete multiresolution analysis and generalized wavelets. J. Appl. Num. Math. 12, 153–193 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Harten, A.: Multiresolution representation of data: A general framework. SIAM J. Numer. Anal. 33, 1205–1256 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys. 71(2), 231–303 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu, X-D, Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Matei, B., Meignen, S., Zakharova, A.: Smoothness characterization and stability of nonlinear and non-separable multiscale representations. J. Approx. Theory 163(11), 1707–1728 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Matei, B., Meignen, S., Zakharova, A.: Smoothness of non-linear and non-separable subdivision schemes. Asymptot. Analisis 74(3-4), 229–247 (2011)

    MATH  MathSciNet  Google Scholar 

  27. Plonka, G., Iske, A., Tenorth, S.: Optimal representation of piecewise holder smooth bivariate functions by the easy path wavelet transform. J. Approx. Theory 176, 42–67 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Matemàtica Aplicada, Universitat de València, Valencia, Spain

    Francesc Aràndiga, Pep Mulet & Vicent Renau

Authors
  1. Francesc Aràndiga
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pep Mulet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vicent Renau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pep Mulet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aràndiga, F., Mulet, P. & Renau, V. Cell average image transform algorithms with exact error control. Numer Algor 69, 75–93 (2015). https://doi.org/10.1007/s11075-014-9882-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-014-9882-4

Keywords

Search

Navigation