Extended Mumford-Shah Regularization in Bayesian Estimation for Blind Image Deconvolution and Segmentation

  • Hongwei Zheng
  • Olaf Hellwich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4040)


We present an extended Mumford-Shah regularization for blind image deconvolution and segmentation in the context of Bayesian estimation for blurred, noisy images or video sequences. The Mumford-Shah functional is extended to have cost terms for the estimation of blur kernels via a newly introduced prior solution space. This functional is minimized using Γ-convergence approximation in an embedded alternating minimization within Neumann conditions. Accurate blur identification is the basis of edge-preserving image restoration in the extended Mumford-Shah regularization. One output of the finite set of curves and object boundaries are grouped and partitioned via a graph theoretical approach for the segmentation of blurred objects. The chosen regularization parameters using the L-curve method is presented. Numerical experiments show that the proposed algorithm is efficiency and robust in that it can handle images that are formed in different environments with different types and amounts of blur and noise.


Point Spread Function Bayesian Estimation Image Restoration Degraded Image Blur 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.


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  1. 1.
    Hadamard, J.: Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)Google Scholar
  2. 2.
    Tikhonov, A., Arsenin, V.: Solution of Ill-Posed Problems. Wiley, Winston (1977)Google Scholar
  3. 3.
    Geman, S., Geman, D.: Stochastic relaxation, gibbs distribution and the bayesian restoratio of images. IEEE Trans. Pattern Anal. Machine Intell., 721–741 (1984)Google Scholar
  4. 4.
    Katsaggelos, A., Biemond, J., Schafer, R., Mersereau, R.: A regularized iterative image restoration algorithm. IEEE Tr. on Signal Processing 39, 914–929 (1991)CrossRefGoogle Scholar
  5. 5.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total varition based noise removal algorithm. Physica D 60, 259–268 (1992)MATHCrossRefGoogle Scholar
  6. 6.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics 42, 577–684 (1989)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Aubert, l., Kornprobst, P.: Mathematical problems in image processing: partical differential equations and the Calculus of Variations. Springer, Heidelberg (2002)Google Scholar
  8. 8.
    Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. on Image Processing, 10, 266–277 (2001)MATHCrossRefGoogle Scholar
  9. 9.
    Bar, L., Sochen, N., Kiryati, N.: Variational pairing of image segmentation and blind restoration. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3022, pp. 166–177. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)Google Scholar
  11. 11.
    Molina, R., Ripley, B.: Using spatial models as priors in astronomical image analysis. J. App. Stat. 16, 193–206 (1989)CrossRefGoogle Scholar
  12. 12.
    Green, P.: Bayesian reconstruction from emission tomography data using a modified em algorithm. IEEE Tr. Med. Imaging 9, 84–92 (1990)CrossRefGoogle Scholar
  13. 13.
    Bouman, C., Sauer, K.: A generalized gaussian image model for edge-preserving map estimation. IEEE Transactions of Image Processing 2, 296–310 (1993)CrossRefGoogle Scholar
  14. 14.
    Molina, R., Katsaggelos, A., Mateos, J.: Bayesian and regularization methods for hyperparameters estimate in image restoration. IEEE Tr. on S.P. 8 (1999)Google Scholar
  15. 15.
    Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via γ-convergence. Communications on Pure and Appled Mathematics 43, 999–1036 (1990)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bar, L., Sochen, N., Kiryati, N.: Image deblurring in the presence of salt-and-pepper noise. In: Pajdla, T., Matas, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 107–118. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Banham, M., Katsaggelos, A.: Digital image restoration. IEEE S.P. 14, 24–41 (1997)CrossRefGoogle Scholar
  18. 18.
    Kundur, D., Hatzinakos, D.: Blind image deconvolution. IEEE Signal Process. Mag., 43–64 (May 1996)Google Scholar
  19. 19.
    You, Y., Kaveh, M.: A regularization approach to joint blur identification and image restoration. IEEE Tr. on Image Processing 5, 416–428 (1996)CrossRefGoogle Scholar
  20. 20.
    Luenberger, D.G.: Linear and nonlinear programming. Addison-Wesley Publishing Company, Reading (1984)MATHGoogle Scholar
  21. 21.
    Chan, T., Wong, C.: Total variation blind deconvolution. IEEE Trans. on Image Processing 7, 370–375 (1998)CrossRefGoogle Scholar
  22. 22.
    Pothen, A., Simon, H.D.: Partitioning sparse matrices with eigenvectors of graphs. SIAM J. Matrix Analytical Applications 11, 430–452 (1990)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 888–905 (August 2000)Google Scholar
  24. 24.
    Perona, P., Freeman, W.T.: A factorization approach to grouping. In: Burkhardt, H., Neumann, B. (eds.) ECCV 1998. LNCS, vol. 1407, pp. 655–670. Springer, Heidelberg (1998)Google Scholar
  25. 25.
    Vogel, C.R., Oman, M.E.: Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. on Image Processing 7, 813–824 (1998)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hansen, P., O’Leary, D.: The use of the l-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14, 1487–1503 (1993)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hongwei Zheng
    • 1
  • Olaf Hellwich
    • 1
  1. 1.Computer Vision & Remote SensingBerlin University of TechnologyBerlin

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