Journal of Scientific Computing

, Volume 19, Issue 1–3, pp 95–122

TV Based Image Restoration with Local Constraints

  • M. Bertalmio
  • V. Caselles
  • B. Rougé
  • A. Solé
Article

Abstract

The problem of recovering an image that has been blurred and corrupted with additive noise is ill-posed. Among the methods that have been proposed to solve this problem, one of the most successful ones is that of constrained Total Variation (TV) image restoration, proposed by L. Rudin, S. Osher, and E. Fatemi. In its original formulation, to ensure the satisfaction of constraints, TV restoration requires the estimation of a global parameter λ (a Lagrange multiplier). We observe that if λ is global, the constraints of the method are also satisfied globally, but not locally. The effect is that the restoration is better achieved in some regions of the image than in others. To avoid this, we propose a variant of the TV restoration model including, instead of a single constraint λ, a set of constraints λi, each one corresponding to a region Oi of the image. We discuss the existence and uniqueness of solutions of the proposed model and display some numerical experiments.

image restoration Total Variation variational methods satellite images 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acar, R., and Vogel, C. R. (1994). Analysis of total variation penalty methods for Ill-posed problems. Inverse Problems 10, 1217-1229.Google Scholar
  2. 2.
    Alvarez, L., Gousseau, Y., and Morel, J. M. (1999). The size of objects in natural and artificial images. Advances in Imaging and Electron Physics 111.Google Scholar
  3. 3.
    Ambrosio, L., Fusco, N., and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.Google Scholar
  4. 4.
    Andreu, F., Ballester, C., Caselles, V., and Mazón, J. M. (2001). Minimizing total variation flow. Differential Integral Equations 14, 321-360.Google Scholar
  5. 5.
    Anzellotti, G. (1983). Pairings between measures and bounded functions and compensated compactness. Ann. di Matematica Pura ed Appl. IV 135, 293-318.Google Scholar
  6. 6.
    Bellettini, G., Caselles, V., and Novaga, M. The Total Variation Flow in ℝN, Preprint 2001.Google Scholar
  7. 7.
    Black, M., and Sapiro, G. (1999). Edges as Outliers: Anisotropic Smoothing using Local Image Statistics, Proceedings Scale-Space Conference, Corfu, Greece.Google Scholar
  8. 8.
    Chambolle, A., and Lions, P. L. (1995). Image Recovery via Total Variation Minimization and Related Problems, Preprint.Google Scholar
  9. 9.
    Chan, T. F., Golub, G. H., and Mulet, P. (1999). A nonlinear primal-dual method for total variation based image restoration. SIAM J. Sci. Comput. 20(6), 1964-1977.Google Scholar
  10. 10.
    Chan, T. F., Golub, G. H., and Mulet, P. (1997). Total Variation Image Restoration: Numerical Methods and Extensions, Proceedings International Conference on Image Processing, ICIP-97, October 26-29, Santa Barbara, California, Vol. III, pp. 384-387.Google Scholar
  11. 11.
    Ciarlet, P. G. (1988). Introduction to Numerical Linear Algebra and Optimization, Cambridge University Press.Google Scholar
  12. 12.
    Demoment, G. (1989). Image reconstruction and restoration: Overview of common estimation structures and problems, IEEE Trans. on Acoustics, Speech and Signal Proc. 37(12), 2024-2036.Google Scholar
  13. 13.
    Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D. (1995). Wavelet Shrinkage: Asymptopia? J. Roy. Statist. Soc. Ser. B 57, 301-369.Google Scholar
  14. 14.
    Durand, S., Malgouyres, F., and Rougé, B. (1999). Image Deblurring, Spectrum Interpolation and Application to Satellite Imaging, Mathematical Modelling and Numerical Analysis.Google Scholar
  15. 15.
    Evans, L. C., and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions, Studies in Advanced Math., CRC Press.Google Scholar
  16. 16.
    Geman, D., and Reynolds, G. (1992). Constrained image restoration and recovery of discontinuities. IEEE Trans. Pattern Anal. Machine Intell. 14, 367-383.Google Scholar
  17. 17.
    Groetsch, C. W. (1984). The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind, Pitman, Boston.Google Scholar
  18. 18.
    Ito, K., and Kunisch, K. (1999). An Active Set Strategy Based On The Augmented Lagrangian Formulation For Image Restoration, Mathematical Modelling and Numerical Analysis, Vol. 33, No. 1, pp. 1-21Google Scholar
  19. 19.
    Koepfler, G., Lopez, C., and Morel, J.-M. (1994). A multiscale algorithm for image segmentation by variational method. SIAM J. Numer. Anal. 31, 282-299.Google Scholar
  20. 20.
    Lions, P. L., Osher S., and Rudin, L. (1992). Denoising and Deblurring using Constrained Nonlinear Partial Differential Equations, Tech. Repport, Cognitech, Santa Monica, CA, submitted to SINUM.Google Scholar
  21. 21.
    Moisan, L. (2001). Extrapolation de spectre et variation totale ponderée, Preprint.Google Scholar
  22. 22.
    Morel, J. M., and Solimini, S. (1994). Variational Methods in Image Processing, Birkhäuser.Google Scholar
  23. 23.
    Mumford, D., and Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math., 17, 577-685.Google Scholar
  24. 24.
    Nikolova, M. (2000). Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math 61, 633-658.Google Scholar
  25. 25.
    Peressini, A. L., Sullivan, F. E., Uhl, J. J., Jr. (1988). The Mathematics of Nonlinear Programming, Springer Verlag.Google Scholar
  26. 26.
    Rohatgi, V. K. (1976). An Introduction to Probability Theory and Mathematical Statistics, Wiley.Google Scholar
  27. 27.
    Rougé, B. (1998). Théorie de l'échantillonage et satellites d'observation de la terre, Analyse de Fourier et traitement d'images, Journées X-UPS.Google Scholar
  28. 28.
    Rosen, J. G. (1961). The gradient projection method for nonlinear programming. Part II. Nonlinear constraints. J. Soc. Indust. Appl. Math. 9, 514-532.Google Scholar
  29. 29.
    Rudin, L. I. (1987). Images, Numerical Analysis of Singularities and Shock Filters, Ph.D. dissertation, Caltech, Pasadena, California n5250:TR.Google Scholar
  30. 30.
    Rudin, L., and Osher, S. (1994). Total Variation based Image Restoration with Free Local Constraints, Proc. of the IEEE ICIP-94, Vol. 1, Austin, TX, pp. 31-35.Google Scholar
  31. 31.
    Rudin, L. I., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Phys. D 60, 259-269.Google Scholar
  32. 32.
    Strong, D., Blomgren, P., and Chan, T. Spatially Adaptative Local Feature Driven Total Variation Minimizing Image Restoration, CAM Report.Google Scholar
  33. 33.
    Strong, D., and Chan, T. F. (1996). Spatially and Adaptative Total Variation Based Regularization and Anisotropic Diffusion in Image Processing, CAM Report, UCLA.Google Scholar
  34. 34.
    Tikhonov, A. N., and Arsenin, V. Y. (1977). Solutions of Ill-Posed Problems, John Wiley, New York.Google Scholar
  35. 35.
    Twomey, S. (1965). The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements. J. Franklin Inst. 297, 95-109.Google Scholar
  36. 36.
    Vogel, C. R., and Oman, M. E. (1996). Iterative methods for total variation denoising. SIAM J. Sci. Computing 17(1), 227-238.Google Scholar
  37. 37.
    Vogel, C. R., and Oman, M. E. (1995). Fast Total Variation Based Image Reconstruction, Proceedings of the 1995 ASME Design Engineering Conferences, Vol. 3, pp. 1009-1015.Google Scholar
  38. 38.
    Ziemer, W. P. (1989). Weakly Differentiable Functions, GTM 120, Springer-Verlag.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • M. Bertalmio
    • 1
  • V. Caselles
    • 1
  • B. Rougé
    • 2
  • A. Solé
    • 1
  1. 1.Department of TecnologiaUniversitat Pompeu FabraBarcelonaSpain
  2. 2.CNESToulouse Cedex 4France

Personalised recommendations