Advances in Computational Mathematics

, Volume 31, Issue 1–3, pp 87–113 | Cite as

Convergence analysis of tight framelet approach for missing data recovery

  • Jian-Feng Cai
  • Raymond H. Chan
  • Lixin Shen
  • Zuowei Shen
Article

Abstract

How to recover missing data from an incomplete samples is a fundamental problem in mathematics and it has wide range of applications in image analysis and processing. Although many existing methods, e.g. various data smoothing methods and PDE approaches, are available in the literature, there is always a need to find new methods leading to the best solution according to various cost functionals. In this paper, we propose an iterative algorithm based on tight framelets for image recovery from incomplete observed data. The algorithm is motivated from our framelet algorithm used in high-resolution image reconstruction and it exploits the redundance in tight framelet systems. We prove the convergence of the algorithm and also give its convergence factor. Furthermore, we derive the minimization properties of the algorithm and explore the roles of the redundancy of tight framelet systems. As an illustration of the effectiveness of the algorithm, we give an application of it in impulse noise removal.

Keywords

Tight frame Missing data Inpainting Impulse noise 

Mathematics Subject Classifications (2000)

42C40 65T60 68U10 94A08 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abreu, E., Lightstone, M., Mitra, S., Arakawa, K.: A new efficient approach for the removal of impulse noise from highly corrupted images. IEEE Trans. Image Process. 5(3), 1012–1025 (1996)CrossRefGoogle Scholar
  2. 2.
    Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH, pp. 417–424. New Orleans, LA (2000)Google Scholar
  3. 3.
    Cai, J.-F., Chan, R., Di Fiore, C.: Minimization of a detail-preserving regularization functional for impulse noise removal. J. Math. Imaging Vision 29(1), 79–91 (2007)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Cai, J.-F., Chan, R., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24(2), 131–149 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chai, A., Shen, Z.: Deconvolution: a wavelet frame approach. Numer. Math. 106, 529–587 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan, R., Chan, T., Shen, L., Shen, Z.: Wavelet algorithms for high-resolution image reconstruction. SIAM J. Sci. Comput. 24(4), 1408–1432 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chan, R., Ho, C.-W., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans. Image Process. 14, 1479–1485 (2005)CrossRefGoogle Scholar
  8. 8.
    Chan, R., Hu, C., Nikolova, M.: An iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett. 11, 921–924 (2004)CrossRefGoogle Scholar
  9. 9.
    Chan, R., Riemenschneider, S.D., Shen, L., Shen, Z.: Tight frame: the efficient way for high-resolution image reconstruction. Appl. Comput. Harmon. Anal. 17(1), 91–115 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chan, R., Shen, L., Shen, Z.: A framelet-based approach for image in painting. Technical Report 2005-4. The Chinese University of Hong Kong, Feb. (2005)Google Scholar
  11. 11.
    Chan, T., Shen, J.: Nontexture inpainting by curvature driven diffusion (CDD). J. Visul Comm. Image Rep. 12, 436–449 (2001)CrossRefGoogle Scholar
  12. 12.
    Chen, T., Wu, H.: Space variant median filters for the restoration of impulse noise corrupted images. IEEE Trans. Circuits and Systems II 48, 784–789 (2001)MATHCrossRefGoogle Scholar
  13. 13.
    Combettes, P., Wajs, V.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Daubechies, I.: Ten lectures on wavelets. CBMS Conference Series in Applied Mathematics, vol. 61, SIAM, Philadelphia (1992)MATHGoogle Scholar
  15. 15.
    Donoho, D., Johnstone, I.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Duffin, R., Schaeffer, A.: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Flaig, A., Arce, G., Barner, K.: Affine order statistics filters: “medianization” of linear FIR filters. IEEE Trans. Signal Process. 46, 2101–2112 (1998)CrossRefGoogle Scholar
  18. 18.
    Gonzalez, R., Woods, R.: Digital Image Processing. Addison-Wesley, Boston, MA (1993)Google Scholar
  19. 19.
    Guleryuz, O.G.: Nonlinear approximation based image recovery using adaptive sparse reconstruction and iterated denoising: part I—theory. IEEE Trans. Image Process. 15(3), 539–554 (2006)CrossRefGoogle Scholar
  20. 20.
    Guleryuz, O.G.: Nonlinear approximation based image recovery using adaptive sparse reconstruction and iterated denoising: part II—adaptive algorithms. IEEE Trans. Image Process. 15(3), 555–571 (2006)CrossRefGoogle Scholar
  21. 21.
    Hwang, H., Haddad, R.: Adaptive median filters: new algorithms and results. IEEE Trans. Image Process. 4, 499–502 (1995)CrossRefGoogle Scholar
  22. 22.
    Ko, S., Lee, Y.: Adaptive center weighted median filter. IEEE Trans. Circuits Syst. 38, 984–993 (1998)CrossRefGoogle Scholar
  23. 23.
    Ng, M., Chan, R., Tang, W.: A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21, 851–866 (2000)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vision 20, 99–120 (2004)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Pok, G., Liu, J.-C., Nair, A.S.: Selective removal of impulse noise based on homogeneity level information. IEEE Trans. Image Process. 12, 85–92 (2003)CrossRefGoogle Scholar
  26. 26.
    Ron, A., Shen, Z.: Affine system in \(L_2(R^d)\): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sun, T., Neuvo, Y.: Detail-preserving based filters in image processing. Pattern Recogn. Lett. 15, 341–347 (1994)CrossRefGoogle Scholar
  28. 28.
    Windyga, P.S.: Fast impulsive noise removal. IEEE Trans. Image Process. 10, 173–179 (2001)CrossRefGoogle Scholar
  29. 29.
    Yin, L., Yang, R., Gabbouj, M., Neuvo, Y.: Weighted median filters: a tutorial. IEEE Trans. Circuit Theory 41, 157–192 (1996)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Jian-Feng Cai
    • 1
  • Raymond H. Chan
    • 2
  • Lixin Shen
    • 3
  • Zuowei Shen
    • 1
  1. 1.Temasek Laboratories and Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Department of MathematicsThe Chinese University of Hong KongHong KongPeople’s Republic of China
  3. 3.Department of MathematicsSyracuse UniversitySyracuseUSA

Personalised recommendations