International Journal of Computer Vision

, Volume 70, Issue 3, pp 279–298 | Cite as

Image Deblurring in the Presence of Impulsive Noise

  • Leah Bar
  • Nahum Kiryati
  • Nir Sochen


Consider the problem of image deblurring in the presence of impulsive noise. Standard image deconvolution methods rely on the Gaussian noise model and do not perform well with impulsive noise. The main challenge is to deblur the image, recover its discontinuities and at the same time remove the impulse noise. Median-based approaches are inadequate, because at high noise levels they induce nonlinear distortion that hampers the deblurring process. Distinguishing outliers from edge elements is difficult in current gradient-based edge-preserving restoration methods. The suggested approach integrates and extends the robust statistics, line process (half quadratic) and anisotropic diffusion points of view. We present a unified variational approach to image deblurring and impulse noise removal. The objective functional consists of a fidelity term and a regularizer. Data fidelity is quantified using the robust modified L 1 norm, and elements from the Mumford-Shah functional are used for regularization. We show that the Mumford-Shah regularizer can be viewed as an extended line process. It reflects spatial organization properties of the image edges, that do not appear in the common line process or anisotropic diffusion. This allows to distinguish outliers from edges and leads to superior experimental results.


image deblurring restoration impulse noise salt and pepper noise variational methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Acar, R. and Vogel, C.R., 1994. “Analysis of Total Variation Penalty Methods”, Inverse Problems, Vol. 10, pp. 1217–1229.MATHMathSciNetCrossRefGoogle Scholar
  2. Alicandro, R., Braides, A. and Shah, J., 1999. “Free-Discontinuity Problems via Functionals Involving the L1-Norm of the Gradient and their Approximation”, Interfaces and Free Boundaries, Vol. 1, pp. 17–37.MATHMathSciNetGoogle Scholar
  3. Ambrosio, L. and Tortorelli, V.M., 1990. “Approximation of Functionals Depending on Jumps by Elliptic Functionals via γ-Convergence”, Communications on Pure and Applied Mathematics, Vol. XLIII, pp. 999–1036.MathSciNetGoogle Scholar
  4. Arce, G.R., Paredes, J.L. and Mullan, J., 2000. “Nonlinear Filtering for Image Analysis and Enhancement”, in Bovik, A.L. (Ed.), Handbook of Image & Video Processing, Academic Press.Google Scholar
  5. Aubert, G. and Kornprobst, P., 2002. Mathematical Problems in Image Processing, Springer, New York.MATHGoogle Scholar
  6. Aubert, G., Blanc-Féraud, L. and March, R., 2004. “γ-Convergence of Discrete Functionals with Nonconvex Perturbation for Image Classification”, SIAM Journal of Numerical Analysis, Vol. 42, pp. 1128–1145.MATHCrossRefGoogle Scholar
  7. Banham, M. and Katsaggelos, A., 1997. “Digital Image Restoration”, IEEE Signal Processing Mag., Vol. 14, pp. 24-41.CrossRefGoogle Scholar
  8. Bar, L., Sochen, N. and Kiryati, N., 2004. “Variational Pairing of Image Segmentation and Blind Restoration”, Proc. ECCV′2004, Prague, Czech Republic, Part II: LNCS #3022, pp. 166–177, Springer.Google Scholar
  9. Bar, L., Sochen, N. and Kiryati, N., 2005. “Image Deblurring in the Presence of Salt and Pepper Noise”, Proc. Scale-Space 2005, Hofgeismar, Germany: LNCS #3459, pp. 107–118, Springer.Google Scholar
  10. Bect, J., Blanc-Féraud, L., Aubert, G. and Chambolle, A., 2004. “A L1-Unified Variational Framework for Image Restoration”, Proc. ECCV′2004, Prague, Czech Republic, Part IV: LNCS #3024, pp. 1–13, Springer.Google Scholar
  11. Black, M.J. and Rangarajan, A., 1996. “On the Unification of Line Processes, Outlier Rejection, and Robust Statistics with Applications in Early Vision”, International Journal of Computer Vision, Vol. 19, pp. 57–92.CrossRefGoogle Scholar
  12. Black, M.J., Sapiro, G., Marimont, D., and Heeger, D., 1998. “Robust Anisotropic Diffusion”, IEEE Trans. Image Processing, Vol. 7, pp. 421–432.CrossRefGoogle Scholar
  13. Braides, A., 1998. Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics, Vol. 1694, Springer-Verlag.Google Scholar
  14. Brook, A., Kimmel, R. and Sochen, N., 2003. “Variational Segmentation for Color Images”, International Journal of Computer Vision, Vol. 18, pp. 247–268.MATHMathSciNetGoogle Scholar
  15. Brox, T., Bruhn, A., Papenberg, N. and Weickert, J., 2004. “High Accuracy Optical Flow Estimation Based on a Theory for Warping”, Proc. ECCV′2004, Prague, Czech Republic, Part IV: LNCS #3024, pp. 25–36, Springer.Google Scholar
  16. Catté, F., Lions, P.L., Morel, J.M., and Coll, T., 1992. “Image Selective Smoothing and Edge Detection by Nonlinear Diffusion”, SIAM Journal of Numerical Analysis, Vol. 29, pp. 182–193.MATHCrossRefGoogle Scholar
  17. Chan, R.H., Ho, C. and Nikolova, M., 2005. “Salt-and-Pepper Noise Removal by Median-type Noise Detectors and Detail-preserving Regularization”, IEEE Transactions on Image Processing, 14:1479–1485.CrossRefGoogle Scholar
  18. Chan, T.F. and Wong, C., 1998. “Total Variation Blind Deconvolution”, IEEE Trans. Image Processing, Vol. 7, pp. 370–375CrossRefGoogle Scholar
  19. Charbonnier, P., Blanc-Féraud, L., Aubert, G., and Barlaud, M., 1997. “Deterministic Edge-Preserving Regularization in Computed Imaging”, IEEE Trans. Image Processing, Vol. 6, pp. 298–311.CrossRefGoogle Scholar
  20. Chen, T. and Wu, H.R., 2001. “Space Variant Median Filters for the Restoration of Impulse Noise Corrupted Images”, IEEE Trans. Circuits and Systems II, Vol. 48, pp. 784–789.MATHMathSciNetCrossRefGoogle Scholar
  21. Chipot, M., March, R., Rosati, M., and Vergara Caffarelli, G., 1997. “Analysis of a Nonconvex Problem Related to Signal Selective Smoothing”, Mathematical Models and Methods in Applied Science, Vol. 7, pp. 313–328.MATHMathSciNetCrossRefGoogle Scholar
  22. Dal Maso, G., 1993. An Introduction to γ-Convergence, Progress in Nonlinear Differential Equations and their Applications, Birkhauser.Google Scholar
  23. Durand, S. and Froment, J., 2003. “Reconstruction of Wavelet Coefficients Using Total Variation Minimization”, SIAM Journal of Scientific Computing, Vol. 24, pp. 1754–1767.MATHMathSciNetCrossRefGoogle Scholar
  24. Durand, S. and Nikolova, M., 2003. “Restoration of Wavelet Coefficients by Minimizing a Specially Designed Objective Function”, Proc. IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision, pp. 145–152.Google Scholar
  25. Geman, D. and Reynolds, G., 1992. “Constrained Restoration and the Recovery of Discontinuities”, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 14, pp. 367–383.CrossRefGoogle Scholar
  26. Geman, S. and Geman, D., 1984. “Stochastic Relaxation, Gibbs Distributions and Bayesian Restoration of Images”, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 6, pp. 721–741.MATHCrossRefGoogle Scholar
  27. Geman, S. and McClure, D.E., 1987. “Statistical Methods for Tomographic Image Reconstruction”, Bulletin of the International Statistical Institute, LII–4, pp. 5–21.MathSciNetGoogle Scholar
  28. Huber, P.J., 1981. Robust Statistics, John Wiley and Sons, New York.MATHGoogle Scholar
  29. Hwang, H. and Haddad, R.A., 1995. “Adaptive Median Filters: New Algorithms and Results”, IEEE Trans. Image Processing, Vol. 4, pp. 499–502.CrossRefGoogle Scholar
  30. Malgouyres, F., 2002. “Minimizing the Total Variation Under a General Convex Constraint”, IEEE Trans. Image Processing, Vol. 11, pp. 1450–1456.MathSciNetCrossRefGoogle Scholar
  31. Mumford, D. and Shah, J., 1989. “Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems”, Communications on Pure and Applied Mathematics, Vol. 42, pp. 577–684.MATHMathSciNetGoogle Scholar
  32. Nikolova, M., 2002. “Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms: Application to the Processing of Outliers”, SIAM Journal on Numerical Analysis, Vol. 40, pp. 965–994.MATHMathSciNetCrossRefGoogle Scholar
  33. Nikolova, M., 2004. “A Variational Approach to Remove Outliers and Impulse Noise”, Journal of Mathematical Imaging and Vision, Vol. 20, pp. 99–120.MathSciNetCrossRefGoogle Scholar
  34. Nordstrom, N., 1990. “Biased Anisotropic Diffusion - A Unified Regularization and Diffusion Approach to Edge Detection”, Proc. 1st European Conference on Computer Vision, pp. 18–27, Antibes, France.Google Scholar
  35. Perona, P. and Malik, J., 1990. “Scale Space and Edge Detection Using Anisotropic Diffusion”, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 12, pp. 629–639.CrossRefGoogle Scholar
  36. Pok, G., Liu, J.–C. and Nair, A.S., 2003. “Selective Removal of Impulse Noise based on Homogeneity Level Information”, IEEE Trans. Image Processing, Vol. 12, pp. 85–92.CrossRefGoogle Scholar
  37. Rondi, L. and Santosa, F., 2001. “Enhanced Electrical Impedance Tomography via the Mumford-shah Functional”, ESAIM: Control, Optimization and Calculus of Variations, Vol. 6, pp. 517–538.MATHMathSciNetCrossRefGoogle Scholar
  38. Rosati, M., 2000. “Asymptotic Behavior of a Geman and McClure Discrete Model”, Applied Math. Optim., Vol. 41, pp. 51–85.MATHMathSciNetCrossRefGoogle Scholar
  39. Rudin, L. and Osher, S., 1994. “Total Variation Based Image Restoration with Free Local Constraints”, Proc. IEEE ICIP, Vol. 1, pp. 31–35, Austin TX, USA.Google Scholar
  40. Rudin, L., Osher, S. and Fatemi, E., 1992. “Non Linear Total Variation Based Noise Removal Algorithms”, Physica D, Vol. 60, pp. 259–268.MATHCrossRefGoogle Scholar
  41. Shah, J., 1996. “A Common Framework for Curve Evolution, Segmentation and Anisotropic Diffusion”, Proc. IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, pp. 136–142.Google Scholar
  42. Sochen, N., Kimmel, R. and Malladi, R., 1998. “A General Framework for Low level Vision” IEEE Trans. Image Processing, Vol. 7, pp. 310–318.MATHMathSciNetCrossRefGoogle Scholar
  43. Teboul, S., Blanc-Féraud, L., Aubert, G., and Barlaud, M., 1998. “Variational Approach for Edge-Preserving Regularization Using Coupled PDE’s”, IEEE Trans. Image Processing, Vol. 7, pp. 387–397.CrossRefGoogle Scholar
  44. Tikhonov, A. and Arsenin, V., 1997. Solutions of Ill-posed Problems, New York.Google Scholar
  45. Vogel, C. and Oman, M., 1998. “Fast, Robust Total Variation-based Reconstruction of Noisy, Blurred Images”, IEEE Trans. Image Processing, Vol. 7, pp. 813–824.MATHMathSciNetCrossRefGoogle Scholar
  46. Weickert, J., 1994. “Anisotropic Diffusion Filters for Image Processing Based Quality Control”, Proc. Seventh European Conference on Mathematics in Industry, Teubner, Stuttgart, pp. 355–362.Google Scholar
  47. Weickert, J., 1999. “Coherence-Enhancing Diffusion Filtering”, International Journal of Computer Vision, Vol. 31, pp. 111–127.CrossRefGoogle Scholar
  48. Weisstein, E.W. et al, “Minimal Residual Method”, from MathWorld–A Wolfram Web Resource.

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

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

Personalised recommendations