Numerische Mathematik

, Volume 106, Issue 4, pp 529–587 | Cite as

Deconvolution: a wavelet frame approach

Article

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.

Mathematics Subject Classification (2000)

42C40 65T60 68U99 

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References

  1. 1.
    Beylkin G., Coifman R. and Rokhlin V. (1991). Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 44: 141–183 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    de Boor C., DeVore R. and Ron A. (1993). On the construction of multivariate (Pre)wavelet. Constr. Approx. 9: 123–166 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Borup, L., Grivonbal, R., Nielsen, M.: Tight wavelet frames in Lebesgue and Sobolev spaces. J. Funct. Spaces Appl. 2(3) (2004)Google Scholar
  4. 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 MATHCrossRefMathSciNetGoogle Scholar
  5. 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)Google Scholar
  6. 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 MATHCrossRefMathSciNetGoogle Scholar
  7. 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 MATHCrossRefMathSciNetGoogle Scholar
  8. 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 MATHCrossRefMathSciNetGoogle Scholar
  9. 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 CrossRefMathSciNetGoogle Scholar
  10. 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)Google Scholar
  11. 11.
    Chan, R., Shen, Z., Xia, T.: A framelet algorithm for enchancing video stills. Appl. Comput. Harmon. Anal (2007, in press)Google Scholar
  12. 12.
    Chan T., Shen J. and Zhou H. (2006). Total variation wavelet inpainting. J. Math. Imaging Vis 25(1): 107–125 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen D. (2000). On the splitting trick and wavelet frame packets. SIAM J. Math. Anal. 31(4): 726–739 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Chui C. and He W. (2000). Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8(3): 293–319 MATHCrossRefMathSciNetGoogle Scholar
  15. 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 MATHCrossRefMathSciNetGoogle Scholar
  16. 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 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Daubechies I.: Ten lectures on wavelets. CBMS Conference Series in Applied Mathematics 61, SIAM, Philadelphia (1992)Google Scholar
  18. 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 MATHCrossRefMathSciNetGoogle Scholar
  19. 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 MATHCrossRefMathSciNetGoogle Scholar
  20. 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)Google Scholar
  21. 21.
    Dong B. and Shen Z. (2007). Pseudo-spline, wavelets and framelets. Appl. Comput. Harmon. Anal. 22(1): 78–104 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Donoho D. (1995). Nonlinear solution of linear inverse problems by Wavelet-Vaguelette decomposition. Appl. Comput. Harmon. Anal. 2: 101–126 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Donoho D. and Raimondo M. (2004). Translation invariant deconvolution in a periodic setting. Int. J. Wavelets Multiresolut. Inf. Process. 2(4): 415–431 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht, Boston (1996)Google Scholar
  25. 25.
    Foster M. (1961). An application of the Wiener–Kolmogorov smoothing theory to matrix inversion. J. SIAM 9(3): 387–392 MATHMathSciNetGoogle Scholar
  26. 26.
    Jia R. and Shen Z. (1994). Multiresolution and wavelets. Proc. Edinburgh Math. Soc. 37: 271–300 MATHMathSciNetCrossRefGoogle Scholar
  27. 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)Google Scholar
  28. 28.
    Kalifa J., Mallat S. and Rougé B. (2003). Deconvolution by thresholding in mirror wavelet bases. IEEE Trans. Image Process. 12(4): 446–457 CrossRefMathSciNetGoogle Scholar
  29. 29.
    Kalifa J. and Mallat S. (2003). Thresholding estimators for linear inverse problems and deconvolutions. Ann. Stat. 31(1): 58–109 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Long R. and Chen W. (1997). Wavelet basis packets and wavelet frame packets. J. Fourier Anal. Appl. 3(3): 239–256 MATHMathSciNetGoogle Scholar
  31. 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 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Ron A. and Shen Z. (1997). Affine systems in \(L_2({\mathbb{R}}^d)\) II: dual systemsJ. Fourier Anal. Appl. 3: 617–637 MathSciNetGoogle Scholar
  33. 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. 34.
    Wiener N. (1949). Extrapolation, Interpolation and Smoothing of Stationary Time Series. Wiley, New York MATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore

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