Skip to main content
SpringerLink
Log in

Mathematical Analysis of a Inf-Convolution Model for Image Processing

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We deal with a second-order image decomposition model to perform denoising and texture extraction that was previously presented. We look for the decomposition of an image as the summation of three different order terms. For highly textured images, the model gives a two-scale texture decomposition: The first-order term can be viewed as a macro-texture (larger scale) which oscillations are not too large, and the zero-order term is the micro-texture (very oscillating) that contains the noise. Here, we perform mathematical analysis of the model and give qualitative properties of solutions using the dual problem and inf-convolution formulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Applications to PDEs and optimization (2006)

  4. Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

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

    Article  MathSciNet  Google Scholar 

  7. 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 

  8. 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 

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

    MathSciNet  Google Scholar 

  10. 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 

  11. Bergounioux, M., Piffet, L.: A second-order model for image denoising. Set-Valued Var. Anal. 18(3–4), 277–306 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bergounioux, M., Piffet, L.: A full second order variational model for multiscale texture analysis. Comput. Optim. Appl. 54, 215–237 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Bergounioux, M.: On poincaré-wirtinger inequalities in bv - spaces. Control Cybern 4(40), 921–929 (2011)

    MathSciNet  Google Scholar 

  15. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bredies, K., Kunisch, K., Valkonen, T.: Properties of \(l^1\)-\({\rm tgv}^2\): the one-dimensional case. J. Math. Anal. Appl. 398(1), 438–454 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bredies, K., Holler, M.: Regularization of linear inverse problems with total generalized variation. J. Inverse Ill Probl. 22(6), 871–913 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Ambrosio, L., Faina, L., March, R.: Variational approximation of a second order free discontinuity problem in computer vision. SIAM J. Math. Anal. 32(6), 1171–1197 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Carriero, M., Leaci, A., Tomarelli, F.: Uniform density estimates for the blake & zisserman functional. Discrete Contin. Dyn. Syst. 31(4), 1129–1150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Carriero, M., Leaci, A., Tomarelli, F.: Free gradient discontinuity and image inpainting. J. Math. Sci. 181(6), 805–819 (2012)

  21. Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, volume 22 of University Lecture Series. AMS, (2001)

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variations, volume 147 of Applied Mathematical Sciences. Springer, Berlin (2006)

    Google Scholar 

  24. Ziemer, W.P.: Weakly Differentiable Functions - Sobolev Space and Functions of Bounded Variation. Springer, Berlin (1980)

    Google Scholar 

  25. Carriero, M., Leaci, A., Tomarelli, F.: Special bounded hessian and elastic-plastic plate. Rend. Ac. Naz. Sci. XV I(13), 223–258 (1992)

  26. Moreau, J.-J.: Inf-convolution des fonctions numériques sur un espace vectoriel. C. R. Acad. Sci. Paris 256, 50475049 (1963)

    Google Scholar 

  27. Hiriart-Urruty, J.-B., Phelps, R.R.: Subdifferential calculus using \(\varepsilon \) -subdifferentials. J. Funct. Anal 118, 150–166 (1993)

    MathSciNet  Google Scholar 

  28. Ekeland, I., Temam, R.: Convex analysis and variational problems. In: Classics in Applied Mathematics, vol. 1. Society for Industrial and Applied Mathematics (1999)

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

    Article  MathSciNet  Google Scholar 

  30. Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the tv denoising problem and some extensions. SIAM Multiscale Model. Simul. 6(3), 879–894 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Bergounioux, M.: Inf-convolution model : numerical experiment. Technical report, hal.archives-ouvertes.fr/hal-01002958v2, (2014)

  32. Ring, W.: Structural properties of solutions of total variation regularization problems. ESAIM Math Model. Numer. Anal. 34, 799–840 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UFR Sciences, Math., Labo. MAPMO, UMR 7349, Université d’Orléans, Route de Chartres, BP 6759, 45067, Orléans cedex 2, France

    M. Bergounioux

Authors
  1. M. Bergounioux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Bergounioux.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bergounioux, M. Mathematical Analysis of a Inf-Convolution Model for Image Processing. J Optim Theory Appl 168, 1–21 (2016). https://doi.org/10.1007/s10957-015-0734-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0734-8

Keywords

Mathematics Subject Classification

Search

Navigation