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A Second-Order Model for Image Denoising

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Abstract

We present a variational model for image denoising and/or texture identification. Noise and textures may be modelled as oscillating components of images. The model involves a L 2-data fitting term and a Tychonov-like regularization term. We choose the BV 2 norm instead of the classical BV norm. Here BV 2 is the bounded hessian function space that we define and describe. The main improvement is that we do not observe staircasing effects any longer, during denoising process. Moreover, texture extraction can be performed with the same method. We give existence results and present a discretized problem. An algorithm close to the one set by Chambolle (J Math Imaging Vis 20:89–97, 2004) is used: we prove convergence and present numerical tests.

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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  MathSciNet  MATH  Google Scholar 

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

  3. Attouch, H., Briceño-Arias, L.M., Combettes, P.L.: A parallel splitting method for coupled monotone inclusions. SIAM J. Control Optim. 48, 3246 (2010)

    Article  MATH  Google Scholar 

  4. Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in sobolev and BV spaces: applications to PDEs and optimization. MPS-SIAM series on optimization. Philadelphia, ISBN 0-89871-600-4 (2006)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Aubert, G., Kornprobst, P.: Mathematical problems in image processing, partial differential equations and the calculus of variations. Applied Mathematical Sciences, vol. 147. Springer Verlag (2006)

  8. Aujol, J.F.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34, 307–327 (2009)

    Article  MathSciNet  Google Scholar 

  9. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)

    Article  MathSciNet  Google Scholar 

  10. Demengel, F.: Fonctions à hessien borné. Annales de l’institut Fourier, Tome 34(2), 155–190 (1984)

    MathSciNet  MATH  Google Scholar 

  11. Echegut, R., Piffet, L.: A variational model for image texture identification (preprint). http://hal.archives-ouvertes.fr/hal-00439431/fr/

  12. Ekeland, I., Temam, R.: Convex Analysis and Variational problems. SIAM Classic in Applied Mathematics, vol. 28 (1999)

  13. Fadili, J., Peyré, G.: Total variation projection with first order schemes (preprint). http://hal.archives-ouvertes.fr/hal-00380491/fr/

  14. Hinterberger, W., Scherzer, O.: Variational methods on the space of functions of bounded Hessian for convexification and denoising. Computing 76(1–2), 109–133 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22. AMS (2002)

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Osher, S., Vese, L.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1–3), 553–572 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Osher, S.J., Vese, L.A.: Image denoising and decomposition with total variation minimization and oscillatory functions. Special issue on mathematics and image analysis. J. Math. Imaging Vis. 20(1–2), 7–18 (2004)

    MathSciNet  Google Scholar 

  20. Piffet, L.: Modèles variationnels pour l’extraction de textures 2D. Ph.D. Thesis, Orléans (2010)

  21. Weiss, P., Blanc-Féraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31(n°3), 2047–2080 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. J. Vis. Commun. Image 18, 240–252 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. UFR Sciences, Math., Labo. MAPMO, UMR 6628, Université d’Orléans, Route de Chartres, BP 6759, 45067, Orléans cedex 2, France

    Maïtine Bergounioux & Loic Piffet

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  1. Maïtine Bergounioux
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  2. Loic Piffet
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Correspondence to Maïtine Bergounioux.

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Bergounioux, M., Piffet, L. A Second-Order Model for Image Denoising. Set-Valued Anal 18, 277–306 (2010). https://doi.org/10.1007/s11228-010-0156-6

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