Variational Pairing of Image Segmentation and Blind Restoration

  • Leah Bar
  • Nir Sochen
  • Nahum Kiryati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3022)


Segmentation and blind restoration are both classical problems, that are known to be difficult and have attracted major research efforts. This paper shows that the two problems are tightly coupled and can be successfully solved together. Mutual support of the segmentation and blind restoration processes within a joint variational framework is theoretically motivated, and validated by successful experimental results. The proposed variational method integrates Mumford-Shah segmentation with parametric blur-kernel recovery and image deconvolution. The functional is formulated using the Γ-convergence approximation and is iteratively optimized via the alternate minimization method. While the major novelty of this work is in the unified solution of the segmentation and blind restoration problems, the important special case of known blur is also considered and promising results are obtained.


Image Segmentation Image Restoration Variational Pairing Blur Kernel Isotropic Gaussian Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ambrosio, L., Tortorelli, V.M.: Approximation of Functionals Depending on Jumps by Elliptic Functionals via Γ-Convergence. Communications on Pure and Applied athematics XLIII, 999–1036 (1990)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Springer, New York (2002)zbMATHGoogle Scholar
  3. 3.
    Banham, M., Katsaggelos, A.: Digital Image Restoration. IEEE Signal Processing Mag. 14, 24–41 (1997)CrossRefGoogle Scholar
  4. 4.
    Carasso, A.S.: Direct Blind Deconvolution. SIAM J. Applied Math. 61, 1980–2007 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chambolle, A.: Image Segmentation by Variational Methods: Mumford and Shah functional, and the Discrete Approximation. SIAM Journal of Applied Mathematics 55, 827–863 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan, T., Wong, C.: Total Variation Blind Deconvolution. IEEE Trans. Image Processing 7, 370–375 (1998)CrossRefGoogle Scholar
  7. 7.
    Kim, J., Tsai, A., Cetin, M., Willsky, A.S.: A Curve Evolution-based Variational Approach to Simultaneous Image Restoration and Segmentation. Proc. IEEE ICIP 1, 109–112 (2002)Google Scholar
  8. 8.
    Kundur, D., Hatzinakos, D.: Blind Image Deconvolution. Signal Processing Mag. 13, 43–64 (1996)CrossRefGoogle Scholar
  9. 9.
    Kundur, D., Hatzinakos, D.: Blind Image Deconvolution Revisited. Signal Processing Mag. 13, 61–63 (1996)CrossRefGoogle Scholar
  10. 10.
    Mumford, D., Shah, J.: Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems. Communications on Pure and Applied Mathematics 42, 577–684 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Nakagaki, R., Katsaggelos, A.: A VQ-Based Blind Image Restoration Algorithm. IEEE Trans. Image Processing 12, 1044–1053 (2003)CrossRefGoogle Scholar
  12. 12.
    Richardson, T., Mitter, S.: Approximation, Computation and Distortion in the Variational Formulation. In: ter Harr Romeny, B.M. (ed.) Geometery-Driven Diffusion in Computer Vision, pp. 169–190. Kluwer, Boston (1994)Google Scholar
  13. 13.
    Rudin, L., Osher, S., Fatemi, E.: Non Linear Total Variation Based Noise Removal Algorithms. Physica D 60, 259–268 (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    Rudin, L., Osher, S.: Total Variation Based Image Restoration with Free Local Constraints. In: Proc. IEEE ICIP, Austin TX, USA, vol. 1, pp. 31–35 (1994)Google Scholar
  15. 15.
    Samson, C., Blanc-Féraud, L., Aubert, G., Zerubia, J.: Multiphase Evolution and Variational Image Classification, Technical Report No. 3662, INRIA Sophia Antipolis (April 1999)Google Scholar
  16. 16.
    Sonka, M., Hlavac, V., Boyle, R.: Image Processing, Analysis and Machine Vision. PWS Publishing (1999)Google Scholar
  17. 17.
    Sroubek, F., Flusser, J.: Multichannel Blind Iterative Image Restoration. IEEE. Trans. Image Processing 12, 1094–1106 (2003)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Tikhonov, A., Arsenin, V.: Solutions of Ill-posed Problems, New York (1977)Google Scholar
  19. 19.
    Vese, L.A., Chan, T.F.: A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model. International Journal of Computer Vision 50, 271–293 (2002)zbMATHCrossRefGoogle Scholar
  20. 20.
    Vogel, C., Oman, M.: Fast, Robust Total Variation-based Reconstruction of Noisy, Blurred Images. IEEE Trans. Image Processing 7, 813–824 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    You, Y., Kaveh, M.: A Regularization Approach to Joint Blur Identification and Image Restoration. IEEE Trans. Image Processing 5, 416–428 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Leah Bar
    • 1
  • Nir Sochen
    • 2
  • Nahum Kiryati
    • 1
  1. 1.School of Electrical Engineering 
  2. 2.Dept. of Applied MathematicsTel Aviv UniversityTel AvivIsrael

Personalised recommendations