Skip to main content
SpringerLink
Log in

Deconvolution: a wavelet frame approach

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b) as wavelet frame based high resolution image reconstruction methods. We first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation, which completes the formulation given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). This formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties. These missing coefficients are recovered iteratively together with a built-in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating. This approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in Chan et al. (SIAM J. Sci. Comput. 24(4), 1408–1432, 2003; Appl. Comput. Hormon. Anal. 17, 91–115, 2004a; Int. J. Imaging Syst. Technol. 14, 91–104, 2004b). However, an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers. This paper is to establish the theoretical foundation of this wavelet frame approach. In particular, a proof of convergence, an analysis of the stability of algorithms and a study of the minimization property of solutions are given.

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. Beylkin G., Coifman R. and Rokhlin V. (1991). Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 44: 141–183

    Article  MATH  MathSciNet  Google Scholar 

  2. de Boor C., DeVore R. and Ron A. (1993). On the construction of multivariate (Pre)wavelet. Constr. Approx. 9: 123–166

    Article  MATH  MathSciNet  Google Scholar 

  3. Borup, L., Grivonbal, R., Nielsen, M.: Tight wavelet frames in Lebesgue and Sobolev spaces. J. Funct. Spaces Appl. 2(3) (2004)

  4. Borup L., Grivonbal R. and Nielsen M. (2004). Bi-framelet systems with few vanishing moments characterize Besov spaces. Appl. Comput. Harmon. Anal. 17: 3–28

    Article  MATH  MathSciNet  Google Scholar 

  5. Chan, R., Chan, T., Shen, L., Shen, Z.: A wavelet method for high-resolution image reconstruction with displacement errors. In: IEEE Signal Processing Society. Proceedings of the 2001 International Symposium of Intelligent Multimedia, Video and Speech Processing, Hong Kong, pp. 24–27. IEEE, USA (2001)

  6. Chan R., Chan T., Shen L. and Shen Z. (2003). Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24(4): 1408–1432

    Article  MATH  MathSciNet  Google Scholar 

  7. Chan R., Chan T., Shen L. and Shen Z. (2003). Wavelet deblurring algorithms for sparially varying blur from high-resolution image reconstruction. Linear Algebra Appl. 366: 139–155

    Article  MATH  MathSciNet  Google Scholar 

  8. Chan R., Riemenschneider S., Shen L. and Shen Z. (2004). Tight frame: the efficient way for high-resolution image reconstruction. Appl. Comput. Harmon. Anal. 17: 91–115

    Article  MATH  MathSciNet  Google Scholar 

  9. Chan R., Riemenschneider S., Shen L. and Shen Z. (2004). High-resolution image reconstruction with displacement errors: a frame approach. Int. J. Imaging Syst. Technol. 14: 91–104

    Article  MathSciNet  Google Scholar 

  10. Chan, R., Shen, Z., Xia, T.: Resolution enhancement for video clips: tight frame approach. In: Proceedings of IEEE International Conference on Advanced Video and Signal-Based Surveillance, Italy, pp. 406–410 (2005)

  11. Chan, R., Shen, Z., Xia, T.: A framelet algorithm for enchancing video stills. Appl. Comput. Harmon. Anal (2007, in press)

  12. Chan T., Shen J. and Zhou H. (2006). Total variation wavelet inpainting. J. Math. Imaging Vis 25(1): 107–125

    Article  MathSciNet  Google Scholar 

  13. Chen D. (2000). On the splitting trick and wavelet frame packets. SIAM J. Math. Anal. 31(4): 726–739

    Article  MATH  MathSciNet  Google Scholar 

  14. Chui C. and He W. (2000). Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8(3): 293–319

    Article  MATH  MathSciNet  Google Scholar 

  15. Chui C., He W. and Stöckler J. (2002). Compact supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal. 13: 224–262

    Article  MATH  MathSciNet  Google Scholar 

  16. Cohen A., Hoffmann M. and Reiss M. (2004). Adaptive wavelet Galerkin methods for linear inverse probelms. SIAM J. Numer. Anal. 42(4): 1479–1501

    Article  MATH  MathSciNet  Google Scholar 

  17. Daubechies I.: Ten lectures on wavelets. CBMS Conference Series in Applied Mathematics 61, SIAM, Philadelphia (1992)

  18. Daubechies I., Defrise M. and De Mol C. (2004). An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57: 1413–1457

    Article  MATH  MathSciNet  Google Scholar 

  19. Daubechies I., Han B., Ron A. and Shen Z. (2003). Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14: 1–46

    Article  MATH  MathSciNet  Google Scholar 

  20. De Mol, C., Defrise, M.: A note on wavelet-based inversion algorithms. In: Nashed, M., Scherzer, O. (eds.) Inverse problems, image analysis, and medical imaging, New Orleans, LA, 2001, pp. 85–96. Contemp. Math. 313, Amer. Math. Soc., Providence, RI (2002)

  21. Dong B. and Shen Z. (2007). Pseudo-spline, wavelets and framelets. Appl. Comput. Harmon. Anal. 22(1): 78–104

    Article  MATH  MathSciNet  Google Scholar 

  22. Donoho D. (1995). Nonlinear solution of linear inverse problems by Wavelet-Vaguelette decomposition. Appl. Comput. Harmon. Anal. 2: 101–126

    Article  MATH  MathSciNet  Google Scholar 

  23. Donoho D. and Raimondo M. (2004). Translation invariant deconvolution in a periodic setting. Int. J. Wavelets Multiresolut. Inf. Process. 2(4): 415–431

    Article  MATH  MathSciNet  Google Scholar 

  24. Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht, Boston (1996)

  25. Foster M. (1961). An application of the Wiener–Kolmogorov smoothing theory to matrix inversion. J. SIAM 9(3): 387–392

    MATH  MathSciNet  Google Scholar 

  26. Jia R. and Shen Z. (1994). Multiresolution and wavelets. Proc. Edinburgh Math. Soc. 37: 271–300

    Article  MATH  MathSciNet  Google Scholar 

  27. Jia, R., Micchelli, C.: Using the refinement equations for the construction of pre-wavelets. II. Powers of two. In: Laurent, P., Mehaute, A., Schumaker, L. (ed.) First International Conference on Curves and Surfaces, Chamonix-Mont-Blanc, 1990. Curves and Surfaces, pp. 209–246. Academic, Boston (1991)

  28. Kalifa J., Mallat S. and Rougé B. (2003). Deconvolution by thresholding in mirror wavelet bases. IEEE Trans. Image Process. 12(4): 446–457

    Article  MathSciNet  Google Scholar 

  29. Kalifa J. and Mallat S. (2003). Thresholding estimators for linear inverse problems and deconvolutions. Ann. Stat. 31(1): 58–109

    Article  MATH  MathSciNet  Google Scholar 

  30. Long R. and Chen W. (1997). Wavelet basis packets and wavelet frame packets. J. Fourier Anal. Appl. 3(3): 239–256

    MATH  MathSciNet  Google Scholar 

  31. Ron A. and Shen Z. (1997). Affine Systems in \(L_2({\mathbb{R}}^d):\) the analysis of the analysis operatorJ. Funct. Anal. 148: 408–447

    Article  MATH  MathSciNet  Google Scholar 

  32. Ron A. and Shen Z. (1997). Affine systems in \(L_2({\mathbb{R}}^d)\) II: dual systemsJ. Fourier Anal. Appl. 3: 617–637

    MathSciNet  Google Scholar 

  33. Tikhonov A.N. (1963). On the solution of incorrectly put problems and the regularization method. Soviet Math. Doklady 4: 1035–1038

    Google Scholar 

  34. Wiener N. (1949). Extrapolation, Interpolation and Smoothing of Stationary Time Series. Wiley, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, 94305, USA

    Anwei Chai

  2. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Singapore

    Anwei Chai & Zuowei Shen

Authors
  1. Anwei Chai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zuowei Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anwei Chai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chai, A., Shen, Z. Deconvolution: a wavelet frame approach. Numer. Math. 106, 529–587 (2007). https://doi.org/10.1007/s00211-007-0075-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-007-0075-0

Mathematics Subject Classification (2000)

Search

Navigation