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Image Sharpening by Flows Based on Triple Well Potentials

Abstract

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.

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References

  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.

  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.

  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.

  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.

  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.

  6. 6.

    T.F. Chan and C. Wong, "Total variation blind deconvolution," IEEE Transactions on Image Processing,Vol. 7, pp. 370–375, 1998.

  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.

  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.

  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.

  10. 10.

    R. Deriche and O. Faugeras, "Les EDP en traitement des images et vision par ordinateur," Technical report, INRIA, November 1995.

  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.

  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.

  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.

  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.

  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.

  16. 16.

    Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence. Springer-Verlag: New York, 1984.

  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.

  18. 18.

    M. Kerckhove (Ed.), "Scale-Space and morphology in computer-vision," Scale-Space 2001,Vol. LNCS 2106, Springer-Verlag 2001.

  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.

  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.

  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.

  22. 22.

    P. Pedregal, "On the numerical analysis of non-convex varia-tional problems," Numer. Math.,Vol. 74, No. 3, p. 325, 1996.

  23. 23.

    P. Pedregal, "Optimization, relaxation and Young measures," Bull. Amer. Math. Soc.,Vol. 36, pp. 27–58, 1999.

  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.

  25. 25.

    B.M. ter Haar Romeny (Ed.), Geometry Driven Diffusion in Computer Vision. Kluwer Academic Publishers, 1994.

  26. 26.

    M. Rost and J. Krug, "A practical model for the Kuramoto-Sivashinsky equation," Physica D,Vol. 88, pp. 1–13, 1995.

  27. 27.

    T. Roubicek, Relaxation in Optimization Theory and Variational Calculus,Walter de Gruyter, Berlin, New York, 1997.

  28. 28.

    L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D,Vol. 60, pp. 259–268, 1992.

  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.

  30. 30.

    G.I. Sivashinsky, "Instabilities, pattern formation, and turbulence in flames," Ann. Rev. Fluid Mech.,Vol. 15, pp. 179–199, 1983.

  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.

  32. 32.

    G.W. Wei, "Generalized Perona-Malik equation for image restoration," IEEE Signal Processing Lett.,Vol. 6, pp.165–167, 1999.

  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.

  34. 34.

    J. Weickert, "Coherence-enhancing diffusion filtering," International Journal of Computer Vision,Vol. 31, pp. 111–127, 1999.

  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.

  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.

  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.

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Gilboa, G., Sochen, N. & Zeevi, Y.Y. Image Sharpening by Flows Based on Triple Well Potentials. Journal of Mathematical Imaging and Vision 20, 121–131 (2004). https://doi.org/10.1023/B:JMIV.0000011320.81911.38

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