Advertisement

Journal of Mathematical Imaging and Vision

, Volume 38, Issue 3, pp 182–196 | Cite as

An Augmented Lagrangian Method for TV g +L 1-norm Minimization

  • Jonas Koko
  • Stéphanie Jehan-Besson
Article

Abstract

In this paper, the minimization of a weighted total variation regularization term (denoted TV g ) with L 1 norm as the data fidelity term is addressed using the Uzawa block relaxation method. The unconstrained minimization problem is transformed into a saddle-point problem by introducing a suitable auxiliary unknown. Applying a Uzawa block relaxation method to the corresponding augmented Lagrangian functional, we obtain a new numerical algorithm in which the main unknown is computed using Chambolle projection algorithm. The auxiliary unknown is computed explicitly. Numerical experiments show the availability of our algorithm for salt and pepper noise removal or shape retrieval and also its robustness against the choice of the penalty parameter. This last property is useful to attain the convergence in a reduced number of iterations leading to efficient numerical schemes. The specific role of the function g in TV g is also investigated and we highlight the fact that an appropriate choice leads to a significant improvement of the denoising results. Using this property, we propose a whole algorithm for salt and pepper noise removal (denoted UBR-EDGE) that is able to handle high noise levels at a low computational cost. Shape retrieval and geometric filtering are also investigated by taking into account the geometric properties of the model.

Keywords

Total variation L1 norm Augmented Lagrangian Fenchel duality Uzawa methods Salt and pepper noise removal Shape retrieval Geometric filtering 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alliney, S.: Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992) zbMATHCrossRefGoogle Scholar
  2. 2.
    Alliney, S.: Recursive median filters of increasing order: a variational approach. IEEE Trans. Signal Process. 44(6), 1346–1354 (1996) CrossRefGoogle Scholar
  3. 3.
    Alliney, S.: A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Trans. Signal Process. 45(4), 913–917 (1997) CrossRefGoogle Scholar
  4. 4.
    Aujol, J.F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 63(1), 85–104 (2005) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Aujol, J.F., Gilboa, G., Chan, T.F., Osher, S.: Structure-texture image decomposition - modeling, algorithms, and parameter selection. Int. J. Comput. Vis. 67(1), 111–136 (2006) CrossRefGoogle Scholar
  6. 6.
    Bar, L., Sochen, N.A., Kiryati, N.: Image deblurring in the presence of salt-and-pepper noise. In: Kimmel, R., Sochen, N.A., Weickert, J., Scale-Space. Lecture Notes in Computer Science, vol. 3459, pp. 107–118. Springer, Berlin (2005) Google Scholar
  7. 7.
    Bertsekas, D.: Constrained Optimization and Lagrange Multipliers Methods. Academic Press, New York (1982) Google Scholar
  8. 8.
    Bonnans, J.F., Gilbert, J., Lemaréchal, C., Sagastizabal, C.: Numerical Optimization: Theoretical and Numerical Aspects. Springer, Berlin (2003) zbMATHGoogle Scholar
  9. 9.
    Bresson, X., Chan, T.F.: Active contours based on Chambolle’s mean curvature motion. In: ICIP (1), pp. 33–36. IEEE (2007) Google Scholar
  10. 10.
    Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J.P., Osher, S.: Fast global minimization of the active contour/snake model. J. Math. Imaging Vis. 28, 151–167 (2007) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Cai, J., Chan, R., Nikolova, M.: Two-phase methods for deblurring images corrupted by impulse plus Gaussian noise. Inverse Probl. Imaging 2, 187–204 (2008) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Cai, J., Chan, R., Nikolova, M.: Fast two-phase image deblurring under impulse noise. J. Math. Imaging Vis. 36(1), 46–53 (2009) CrossRefGoogle Scholar
  13. 13.
    Caselles, V., Catte, F., Coll, T., Dibos, F.: A geometric model for active contours. Numer. Math. 66, 1–31 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Computer Vis. 22, 61–79 (1997) zbMATHCrossRefGoogle Scholar
  15. 15.
    Chambolle, A.: An algorithm for total variation minimization and applications J. Math. Imaging Vis. 20(1–2), 89–97 (2004) MathSciNetGoogle Scholar
  16. 16.
    Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 136–152 (2005) Google Scholar
  17. 17.
    Chambolle, A.: Total variation minimization and class of binary MRF models. In: Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 3757, pp. 136–152. Springer, Berlin (2005) CrossRefGoogle Scholar
  18. 18.
    Chan, R., Ho, C., Nikolova, M.: Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization. IEEE Trans. Image Process. 14(15), 1479–1485 (2005) CrossRefGoogle Scholar
  19. 19.
    Chan, R., Hu, C., Nikolova, M.: An iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett. pp. 921–924 (2004) Google Scholar
  20. 20.
    Chan, T., Golub, G., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized L1 function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2004) CrossRefMathSciNetGoogle Scholar
  22. 22.
    Chan, T.F., Vese, L.A.: Active contour without edges. IEEE Trans. Image Process. 10, 266–277 (2001) zbMATHCrossRefGoogle Scholar
  23. 23.
    Chen, T., Wu, H.: Space variant median filters for the restoration of impulse noise corrupted images. IEEE Trans. Circuits Syst. II 48(8), 784–789 (2001) zbMATHCrossRefGoogle Scholar
  24. 24.
    Darbon, J.: Total variation minimization with L1 data fidelity as a contrast invariant filter. In: International Symposium on Image and Signal Processing and Analysis, pp. 221–226. Zagreb, Croatia (2005) Google Scholar
  25. 25.
    Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part I: Fast and exact optimization. J. Math. Imaging Vis. 26(3), 261–271 (2006) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part II: Levelable functions, convex priors and non convex cases. J. Math. Imaging Vis. 26(3), 277–291 (2006) CrossRefMathSciNetGoogle Scholar
  27. 27.
    De Haan, G., Lodder, R.: De-interlacing of video data using motion vectors and edge information. In: International Conference on Consumer Electronics, pp. 70–71 (2002) Google Scholar
  28. 28.
    Duval, V., Aujol, J., Gousseau, Y.: the TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ekland, I., Temam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999) Google Scholar
  30. 30.
    Eng, H., Ma, K.: Noise adaptive soft-switching median filter. IEEE Trans. Image Process. 10(2), 242–251 (2001) zbMATHCrossRefGoogle Scholar
  31. 31.
    Fleming, W., Rishel, R.: An integral formula for total gradient variation. Arch. Math. 11, 218–222 (1960) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems. North-Holland, Amsterdam (1983) Google Scholar
  33. 33.
    Fu, H., Ng, M.K., Nikolova, M., Barlow, J.L.: Efficient minimization methods of mixed l2-l1 and l1-l1 norms for image restoration. SIAM J. Sci. Comput. 27(6), 1881–1902 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Glowinski, R., Tallec, P.L.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989) zbMATHGoogle Scholar
  35. 35.
    Grasmair, M.: Locally adaptive total variation regularization. In: Scale Space and Variational methodsin computer Vision, pp. 331–342 (2009) Google Scholar
  36. 36.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1, 321–332 (1988) CrossRefGoogle Scholar
  37. 37.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: International Conference on Computer Vision, pp. 810–815. Boston, USA (1995) Google Scholar
  38. 38.
    Koko, J.: Uzawa block relaxation domain decomposition method for the two-body contact problem with Tresca friction. Comput. Methods. Appl. Mech. Engrg. 198, 420–431 (2008) CrossRefMathSciNetGoogle Scholar
  39. 39.
    Luenberger, D.: Linear and Nonlinear Programming. Addison-Wesley, Reading (1989) Google Scholar
  40. 40.
    Mumford, D., Shah, J.: Boundary detection by minimizing functionals. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 22–26 (1985) Google Scholar
  41. 41.
    Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms, application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1–2), 99–120 (2004) CrossRefMathSciNetGoogle Scholar
  43. 43.
    Nikolova, M., Esedoglu, S., Chan, T.F.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. App. Math. 66(5), 1632–1648 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Osher, S.J., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, Berlin (2003) zbMATHGoogle Scholar
  45. 45.
    Rudin, L., Osher, S.: Total variation based image restoration with free local constraints. In: ICIP, vol. 1, pp. 31–35. Austin, Texas (1994) Google Scholar
  46. 46.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992) zbMATHCrossRefGoogle Scholar
  47. 47.
    Soille, P.: Morphological Image Analysis. Springer, Berlin/Heidelberg (1999) zbMATHGoogle Scholar
  48. 48.
    Wang, Z., Zhang, D.: Progressive switching median filter for the removal of impulse noise from highly corrupted images. IEEE Trans. Circuits Syst. II 46(1), 78–80 (1999) CrossRefGoogle Scholar
  49. 49.
    Weiss, P., Aubert, G., Blanc-Feraud, L.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31(3), 2047–2080 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Yin, W., Goldfarb, D., Osher, S.: The total variation regularized L1 model for multiscale decomposition. Multiscale Model. Simul. 6(1), 190–211 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Zhang, S., Karim, M.: A new impulse detector for switching median filter. IEEE Signal Process. Lett. 9(11), 360–363 (2002) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.LIMOSUniversité Blaise Pascal, CNRS UMR 6158AubiereFrance

Personalised recommendations