Advertisement

Numerical Algorithms

, Volume 64, Issue 1, pp 43–72 | Cite as

Blind image deconvolution using a banded matrix method

  • Antonios Danelakis
  • Marilena Mitrouli
  • Dimitrios Triantafyllou
Original Paper

Abstract

In this paper we study the blind image deconvolution problem in the presence of noise and measurement errors. We use a stable banded matrix based approach in order to robustly compute the greatest common divisor of two univariate polynomials and we introduce the notion of approximate greatest common divisor to encapsulate the above approach, for blind image restoration. Our method is analyzed concerning its stability and complexity resulting to useful conclusions. It is proved that our approach has better complexity than the other known greatest common divisor based blind image deconvolution techniques. Examples illustrating our procedures are given.

Keywords

Blind image deconvolution Blurring function Univariate polynomial Greatest Common Divisor Banded matrix Convolution 

Mathematics Subject Classifications (2010)

65F05 65G50 65Y20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beck, A., Eldar, Y.C.: Regularization in regression with bounded noise: a Chebyshev center approach. SIAM J. Matrix Anal. Appl. 29, 606–625 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ayers, G.R., Dainty, J.C.: Iterative blind deconvolution method and its applications. Opt. Lett. 13, 547–549 (1988)CrossRefGoogle Scholar
  3. 3.
    Barnett, S.: Greatest common divisor from generalized Sylvester resultant matrices. Linear Multilinear Algebra 8, 271–279 (1980)MATHCrossRefGoogle Scholar
  4. 4.
    Chatterjee, P.: Denoising using the K-SVD method. Course Web Pages, EE 264: Image Processing and Reconstruction, pp. 1–12 (2007)Google Scholar
  5. 5.
    Cooley, J.W., James W., Tukey W.J.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MATHCrossRefGoogle Scholar
  6. 6.
    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising with block-matching and 3D filtering. In: Proceedings of Electronic Imaging, Proc. SPIE 6064, No 6064A-30, vol. 14. San Jose, USA (2006)Google Scholar
  7. 7.
    Datta, N.B.: Numerical Linear Algebra and Applications, 2nd edn. SIAM, Philadelphia (2010)MATHCrossRefGoogle Scholar
  8. 8.
    El-Khamy, S.E., Hadhood, M.M., Dessouky, M.I., Salam, B.M., Abd El-Samie, F.E.: A Greatest common divisor approach to blind super-resolution reconstruction of images. J. Mod. Opt. 53, 1027–1039 (2006)MATHCrossRefGoogle Scholar
  9. 9.
    Fiori, S.: A fast fixed-point neural blind deconvolution algorithm. IEEE Trans. Neural Netw. 15, 455–459 (2004)CrossRefGoogle Scholar
  10. 10.
    Foi, A., Katkovnik, V., Egiazarian, K.: Pointwise shape-adaptive DCT for high-quality deblocking of compressed color images. IEEE Trans. Image Process. 16, 1057–7149 (2007)CrossRefGoogle Scholar
  11. 11.
    Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE 93, 216–231 (2005)CrossRefGoogle Scholar
  12. 12.
    Gentleman, W.M., Sande, G.: Fast fourier transforms-for fun and profit. Proc. AFIPS 29, 563–578 (1966)Google Scholar
  13. 13.
    Gonzalez, R., Woods, R.: Digital Image Processing. Addison-Wesley, New York (1992)Google Scholar
  14. 14.
    He, L., Marquina, A., Osher, S.J.: Blind deconvolution using TV regularization and bregman iteration. Int. J. Imag. Syst. Technol. 15, 74–83 (2005)CrossRefGoogle Scholar
  15. 15.
    Heindl, A.R.: Fourier transform, polynomial GCD, and image restoration. Master Thesis for Clemson University, Department of Mathematical Sciences (2005)Google Scholar
  16. 16.
    Heideman, M.T., Burrus, C.S.: On the number of multiplications necessary to compute a length-2n DFT. IEEE Trans. Acoust. Speech. Signal Process. 34, 91–95 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kervrann, C., Boulanger, J.: Local adaptivity to variable smoothness for exemplar-based image regularization and representation. Int. J. Comput. Vis. 79, 45–69 (2006)CrossRefGoogle Scholar
  18. 18.
    Kundur, D., Hatzinakos, D.: Blind image deconvolution. IEEE Signal Process. Mag. 13(3), 43–64 (1996)CrossRefGoogle Scholar
  19. 19.
    Kundur, D., Hatzinakos, D.: A novel blind deconvolution scheme for image restoration using recursive filtering. IEEE Trans. Signal Process. 46, 375–390 (1998)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mathis, H., Douglas, C.S.: Bussgang blind deconvolution for impulsive signals. IEEE Trans. Signal Process. 51, 1905–1915 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Karcanias, N., Mitrouli, M., Triantafyllou, D.: Matrix pencil methodologies for computing the greatest common divisor of polynomials: hybrid algorithms and their performance. Int. J. Control 79, 1447–1461 (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kim, Y., Kim, S., Kim, M.: The derivation of a new blind equalization algorithm. ETRI J. 18, 53–60 (1996)CrossRefGoogle Scholar
  23. 23.
    Lundy, T., Van Buskirk, J.: A new matrix approach to real FFTs and convolutions of length 2k. Computing 80, 23–45 (2007)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Triantafyllou, D., Mitrouli, M.: On rank and null space computation of the generalized Sylvester matrix. Numer. Algorithms 54, 297–324 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nagy, G.J., Palmer, K., Perrone, L.: Iterative methods for image deblurring: a Matlab object-oriented approach. Numer. Algorithms 36, 73–93 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pillai, S.U., Liang, B.: Blind image deconvolution using a robust GCD approach. IEEE Trans. Image Process. 8, 295–301 (1999)CrossRefGoogle Scholar
  27. 27.
    Abad, J.O., Morigi, S., Reichel, L., Sgallari, F.: Alternating Krylov subspace image restoration methods. J. Comput. Appl. Math. 236(8), 2049–2062 (2012)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Schatzman, C.J.: Accuracy of the discrete fourier transform and the fast fourier transform. SIAM J. Sci. Comput. 17, 1150–1166 (1996)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    van der Veen, A.J., Paulraj, A.: An analytical constant modulus algorithm. IEEE Trans. Signal Process. 44, 1–19 (1999)Google Scholar
  30. 30.
    Vermon, D.: Machine Vision. Prentice-Hall, Englewood Cliffs (1991)Google Scholar
  31. 31.
    Chan, F.T., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7, 370–375 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Antonios Danelakis
    • 1
  • Marilena Mitrouli
    • 2
  • Dimitrios Triantafyllou
    • 3
  1. 1.Department of Informatics & TelecommunicationsUniversity of AthensAthensGreece
  2. 2.Department of MathematicsUniversity of AthensAthensGreece
  3. 3.Section of Mathematics & Engineering Sciences, Department of Military SciencesHellenic Army AcademyVariGreece

Personalised recommendations