Journal of Mathematical Imaging and Vision

, Volume 20, Issue 1–2, pp 121–131 | Cite as

Image Sharpening by Flows Based on Triple Well Potentials

  • Guy Gilboa
  • Nir Sochen
  • Yehoshua Y. Zeevi


Image sharpening in the presence of noise is formulated as a non-convex variational problem. The energy functional incorporates a gradient-dependent potential, a convex fidelity criterion and a high order convex regularizing term. The first term attains local minima at zero and some high gradient magnitude, thus forming a triple well-shaped potential (in the one-dimensional case). The energy minimization flow results in sharpening of the dominant edges, while most noisy fluctuations are filtered out.

image filtering image enhancement image sharpening nonlinear diffusion hyper-diffusion variational image processing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Alvarez and L. Mazorra, "Signal and image restoration using shock filters and anisotropic diffusion," SIAM J. Numer. Anal., Vol. 31, No. 2, pp. 590–605, 1994.Google Scholar
  2. 2.
    J. Ball and R. James, "Proposed experimental tests of a theory of fine microstructure and the two-well problem," Phil. Trans. R. Soc. Lond. A, Vol. 338, pp. 389–450, 1992.Google Scholar
  3. 3.
    J.W. Cahn and J.E. Hilliard, "Free energy of a nonuniform system. I. Interfacial free energy," J. Chem. Phys.,Vol. 28, No. 2, pp. 258–267, 1958.Google Scholar
  4. 4.
    C. Carstensen and P. Plechac, "Adaptive mesh refinement in scalar non-convex variational problems," Berichtsreihe des Mathematischen Seminars Kiel,Vol. 97, No. 2, 1997.Google Scholar
  5. 5.
    F. Catte, P.L. Lions, J.M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear diffusion," SIAM J. Num. Anal.,Vol. 29, No. 1, pp. 182–193, 1992.Google Scholar
  6. 6.
    T.F. Chan and C. Wong, "Total variation blind deconvolution," IEEE Transactions on Image Processing,Vol. 7, pp. 370–375, 1998.Google Scholar
  7. 7.
    P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, "Two deterministic half-quadratic regularization algorithms for computed imaging," Proc. IEEE ICIP' 94,Vol. 2, pp. 168–172, 1994.Google Scholar
  8. 8.
    L. Dascal and N. Sochen, "The maximum principle in the Beltrami color flow," L.D. Griffin and M. Lillholm (Eds.), Scale-Space 2003, LNCS 2659, pp. 196–208, Springer, 2003.Google Scholar
  9. 9.
    S. Demoulini, "Young measure solutions for a nonlinear parabolic equation of forward-backward type," SIAM J. Math. Anal.,Vol. 27, pp. 376–403, 1996.Google Scholar
  10. 10.
    R. Deriche and O. Faugeras, "Les EDP en traitement des images et vision par ordinateur," Technical report, INRIA, November 1995.Google Scholar
  11. 11.
    J. Ericksen, "Some constrained elastic crystals," in Material In-stabilities in Continuum Mechanics and Related Problems, John Ball, Ed. Oxford University Press: Oxford, 1987, pp. 119–137.Google Scholar
  12. 12.
    G. Gilboa, N. Sochen, and Y.Y. Zeevi, "A forward-and-backward diffusion process for adaptive image enhancement and denoising," IEEE Transactions on Image Processing,Vol. 11, No. 7, pp. 689–703, 2002.Google Scholar
  13. 13.
    K. Höllig, "Existence of infinitely many solutions for a forward-backward heat equation," Trans. Amer. Math. Soc.,Vol. 278, pp. 299–316, 1983.Google Scholar
  14. 14.
    M.K. Gobbert and A. Prohl, "A survey of classical and new finite element methods for the computation of crystalline microstructure," IMA preprints, 1576, June 1998.Google Scholar
  15. 15.
    R. Kaftory, N. Sochen, and Y.Y. Zeevi, "Beltrami operator denoising and blind deconvolution of a color image," in Proc. ISSPA, Paris, July, 2003.Google Scholar
  16. 16.
    Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence. Springer-Verlag: New York, 1984.Google Scholar
  17. 17.
    A. Kurganov, D. Levy, and P. Rosenau, "On Burgers-type Equations with Non-monotonic Dissipative Fluxes," Communications on Pure and Applied Mathematics,Vol. 51, pp. 443–473, 1998.Google Scholar
  18. 18.
    M. Kerckhove (Ed.), "Scale-Space and morphology in computer-vision," Scale-Space 2001,Vol. LNCS 2106, Springer-Verlag 2001.Google Scholar
  19. 19.
    M. Luskin, "Approximation of a laminated microstructure for a rotationally invariant, double well energy density," Numer. Math.,Vol. 75, pp. 205–221, 1997.Google Scholar
  20. 20.
    J. Munoz and P. Pedregal, "Explicit solutions of nonconvex variational problems in dimension one," Applied Math. and Optimization, Vol. 41, No. 1, pp. 129–140, 2000.Google Scholar
  21. 21.
    M. Nikolova, "Minimizers of cost-functions involving non-smooth data-fidelity terms. Application to the processing of out-liers," to appear in SIAM Journ. on Numerical Analysis.Google Scholar
  22. 22.
    P. Pedregal, "On the numerical analysis of non-convex varia-tional problems," Numer. Math.,Vol. 74, No. 3, p. 325, 1996.Google Scholar
  23. 23.
    P. Pedregal, "Optimization, relaxation and Young measures," Bull. Amer. Math. Soc.,Vol. 36, pp. 27–58, 1999.Google Scholar
  24. 24.
    P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE Trans. Pat. Anal. Machine Intel., Vol. 12, No. 7, pp. 629–639, 1990.Google Scholar
  25. 25.
    B.M. ter Haar Romeny (Ed.), Geometry Driven Diffusion in Computer Vision. Kluwer Academic Publishers, 1994.Google Scholar
  26. 26.
    M. Rost and J. Krug, "A practical model for the Kuramoto-Sivashinsky equation," Physica D,Vol. 88, pp. 1–13, 1995.Google Scholar
  27. 27.
    T. Roubicek, Relaxation in Optimization Theory and Variational Calculus,Walter de Gruyter, Berlin, New York, 1997.Google Scholar
  28. 28.
    L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D,Vol. 60, pp. 259–268, 1992.Google Scholar
  29. 29.
    C. Samson, L. Blanc-Féraud, G. Aubert, and J. Zerubia, "A variational model for image classification and restoration," IEEE Trans. Pat. Anal. Machine Intel.,Vol. 22, No. 5, pp. 460–472, 2000.Google Scholar
  30. 30.
    G.I. Sivashinsky, "Instabilities, pattern formation, and turbulence in flames," Ann. Rev. Fluid Mech.,Vol. 15, pp. 179–199, 1983.Google Scholar
  31. 31.
    N. Sochen, R. Kimmel, and R. Malladi, "A general framework for low level vision," IEEE Trans. on Image Processing,Vol. 7, pp. 310–318, 1998.Google Scholar
  32. 32.
    G.W. Wei, "Generalized Perona-Malik equation for image restoration," IEEE Signal Processing Lett.,Vol. 6, pp.165–167, 1999.Google Scholar
  33. 33.
    J. Weickert, "A review of nonlinear diffusion filtering," Scale-Space Theory in Computer Vision, B. ter Haar Romeny, L. Florack, J. Koenderink, and M. Viergever (Eds.), LNCS 1252, Springer, Berlin, 1997, pp. 3–28.Google Scholar
  34. 34.
    J. Weickert, "Coherence-enhancing diffusion filtering," International Journal of Computer Vision,Vol. 31, pp. 111–127, 1999.Google Scholar
  35. 35.
    J. Weickert and B. Benhamouda, "A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox," Advances in Computer Vision, F.Solina (Ed.), Springer, Wien, 1997, pp. 1–10.Google Scholar
  36. 36.
    T.P. Witelski, "The structure of internal layers for unstable non-linear diffuison equations," Studies in Applied Mathematics, Vol. 96, pp. 277–300, 1996.Google Scholar
  37. 37.
    Y. Yuo, W. Xu, A. Tannenbaum, and M. Kaveh, "Behavioral analysis of anisotropic diffusion in image processing," IEEE Trans. Image Process,Vol. 5, No. 11, 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Guy Gilboa
    • 1
  • Nir Sochen
    • 2
  • Yehoshua Y. Zeevi
    • 1
    • 3
  1. 1.Department of Electrical EngineeringTechnion—Israel Institute of TechnologyTechnion City, HaifaIsrael
  2. 2.Department of Applied MathematicsUniversity of Tel-Aviv Ramat-AvivTel-AvivIsrael
  3. 3.Department of Biomedical EngineeringColumbia UniversityNew YorkUSA

Personalised recommendations