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(Φ,Φ*) Image Decomposition Models and Minimization Algorithms

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Abstract

We propose in this paper minimization algorithms for image restoration using dual functionals and dual norms. In order to extract a clean image u from a degraded version f=Ku+n (where f is the observation, K is a blurring operator and n represents additive noise), we impose a standard regularization penalty Φ(u)=∫ φ(|Du|)dx<∞ on u, where φ is positive, increasing and has at most linear growth at infinity. However, on the residual f−Ku we impose a dual penalty Φ*(f−Ku)<∞, instead of the more standard \(\|f-\mathit{Ku}\|^{2}_{L^{2}}\) fidelity term. In particular, when φ is convex, homogeneous of degree one, and with linear growth (for instance the total variation of u), we recover the (BV,BV *) decomposition of the data f, as suggested by Y. Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Am. Math. Soc., Providence, 2001). Practical minimization methods are presented, together with theoretical, experimental results and comparisons to illustrate the validity of the proposed models. Moreover, we also show that by a slight modification of the associated Euler-Lagrange equations, we obtain well-behaved approximations and improved results.

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References

  1. Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217–1229 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, R.A.: Sobolev Spaces. Academic Press, San Diego (1975)

    MATH  Google Scholar 

  3. Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by non-linear diffusion 2. SIAM J. Numer. Anal. 29(3), 845–866 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, London (2000)

    MATH  Google Scholar 

  5. Andreu-Vaillo, F., Caselles, V., Mazon, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223. Birkhäuser, Basel (2004)

    MATH  Google Scholar 

  6. Aubert, G., Aujol, J.-F.: Modeling very oscillating signals. Application to image processing. Appl. Math. Optim. 51(2), 163–182 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Aubert, G., Vese, L.: A variational method in image recovery. SIAM J. Numer. Anal. 34(5), 1948–1979 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Aujol, J.-F., Chambolle, A.: Dual norms and image decomposition models. Int. J. Comput. Vis. 3(1), 85–104 (2005)

    Article  MathSciNet  Google Scholar 

  9. Aujol, J.-F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition application to SAR images. In: LNCS, vol. 2695, pp. 297–312 (2003)

  10. Aujol, J.-F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71–88 (2005)

    Article  Google Scholar 

  11. Bourgain, J., Brezis, H.: On the equation div Y=f and application to control of phases. J. Am. Math. Soc. 16, 393–426 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge-detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chung, G., Le, T.M., Lieu, L.H., Tanushev, N.M., Vese, L.A.: Computational methods for image restoration, image segmentation, and texture modeling. In: Bouman, C.A., Miller, E.L., Pollak, I. (eds.) Computational Imaging IV. Proc. of SPIE-IS&T Electronic Imaging, SPIE, vol. 6065, pp. 60650J-1–60650J-15 (2006)

  15. Daubechies, I., Teschke, G.: Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising. Appl. Comput. Harmon. Anal. 19(1), 1–16 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. De Pauw, T.: On SBV dual. Indiana Univ. Math. J. 47(1), 99–121 (1998)

    MATH  MathSciNet  Google Scholar 

  17. Demengel, F., Témam, R.: Convex functions of a measure and applications. Indiana Univ. Math. J. 33(5), 673–709 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)

    MATH  Google Scholar 

  19. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  20. Garnett, J.B., Le, T.M., Meyer, Y., Vese, L.A.: Image decompositions using bounded variation and homogeneous Besov spaces. Appl. Comput. Harmon. Anal. 23, 25–56 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Geman, D., Geman, S.: Bayesian image analysis. In: Bienenstock, E., Fogelman, F., Weisbuch, G. (eds.) Disordered Systems and Biological Organization, vol. F20. Springer, Berlin (1986)

    Google Scholar 

  22. Geman, D., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern. Anal. Mach. Intell. 14(3), 367–383 (1992)

    Article  Google Scholar 

  23. Geman, D., Yang, C.: Nonlinear image recovery with half-quadratic regularization. IEEE Trans. Image Process. 4(7), 932–946 (1995)

    Article  Google Scholar 

  24. Geman, S., McClure, D.E.: Statistical methods for tomographic image reconstruction. In: Proc. 46th Sess. Int. Stat. Inst. Bulletin ISI, vol. 52 (1987)

  25. Haddad, A.: Méthodes variationnelles en traitement d’image. Ph.D. Thesis Ecole Normale Supérieure Cachan, June 2005

  26. Kindermann, S., Osher, S., Xu, J.: Denoising by BV-duality. J. Sci. Comput. 28(2–3), 411–444 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Le, T.M., Vese, L.A.: Image decomposition using total variation and div(BMO). Multiscale Model. Simul. 4(2), 390–423 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Levine, S.E.: An adaptive variational model for image decomposition. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition. 5th International Workshop, EMMCVPR 2005, St. Augustine, FL, 9–11 November 2005. LNCS, vol. 3757, pp. 382–397. Springer, Berlin (2005)

    Chapter  Google Scholar 

  29. Lieu, L., Vese, L.: Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces. Appl. Math. Optim. 58, 167–193 (2008)

    Article  MathSciNet  Google Scholar 

  30. Marquina, A., Osher, S.: A new time dependent model based on level set motion for nonlinear deblurring and noise removal. In: LNCS, vol. 1682, pp. 429–434 (1999)

  31. Marquina, A., Osher, S.: Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal. SIAM J. Sci. Comput. 22(2), 387–405 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series, vol. 22. Am. Math. Soc., Providence (2001)

    MATH  Google Scholar 

  33. Meyer, Y.: Contenu informatif des images numériques, Public Lecture, CMLA ENS-Cachan (2004)

  34. Mumford, D., Gidas, B.: Stochastic models for generic images. Q. Appl. Math. 59(1), 85–111 (2001)

    MATH  MathSciNet  Google Scholar 

  35. Mumford, D., Shah, J.: Optimal approximations by piecewise-smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  36. Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the H −1 norm. Multiscale Model. Simul. 1(3), 349–370 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern. Anal. Mach. Intell. 12(7), 629–639 (1990)

    Article  Google Scholar 

  38. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MATH  Google Scholar 

  39. Rudin, L.I., Osher, S.: Total variation based image restoration with free local constraints. In: Proceedings of IEEE ICIP-94, vol. 1, pp. 31–35 (1994)

  40. Strang, G.: L 1 and L ∞ and approximation of vector fields in the plane. In: Fujita, H., Lax, P., and Strang, G. (eds.) Nonlinear Partial Differential Equations in Applied Science. Lecture Notes in Num. Appl. Anal., vol. 5, pp. 273–288 (1982)

  41. Vese, L.: A study in the BV space of a denoising-deblurring variational problem. Appl. Math. Optim. 44(2), 131–161 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Vese, L., Osher, S.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(13), 553–572 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  43. Vese, L.A., Osher, S.J.: Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vis. 20, 7–18 (2004)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, USA

    Triet M. Le

  2. Department of Mathematics, University of California, Davis, Davis, CA, USA

    Linh H. Lieu

  3. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA

    Luminita A. Vese

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  1. Triet M. Le
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  2. Linh H. Lieu
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  3. Luminita A. Vese
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Correspondence to Luminita A. Vese.

Additional information

This work was supported in part by an Alfred P. Sloan fellowship, by the UCLA Institute for Pure and Applied Mathematics, and by the National Science Foundation grants ITR/ACI-0113439, DMS-0312222, DMS-0714945, and DMS-0809270.

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Le, T.M., Lieu, L.H. & Vese, L.A. (Φ,Φ*) Image Decomposition Models and Minimization Algorithms. J Math Imaging Vis 33, 135–148 (2009). https://doi.org/10.1007/s10851-008-0130-1

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