Advanced Search
Article Contents
Article Contents

Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models

Abstract Related Papers Cited by
  • We present new continuous variants of the Geman--McClure model and the Hebert--Leahy model for image restoration, where the energy is given by the nonconvex function $x \mapsto x^2/(1+x^2)$ or $x \mapsto \log(1+x^2)$, respectively. In addition to studying these models' $\Gamma$-convergence, we consider their point-wise behaviour when the scale of convolution tends to zero. In both cases the limit is the Mumford-Shah functional.
    Mathematics Subject Classification: Primary: 94A08; Secondary: 49J45, 49N45, 68U10.


    \begin{equation} \\ \end{equation}
  • [1]

    R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.


    L. Ambrosio, Compactness for a special class of functions of bounded variation, Boll. Un. Mat. Ital. B, 3 (1989), 857-881.


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


    L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence, Comm. Pure Appl. Math., 43 (1990), 999-1036.doi: 10.1002/cpa.3160430805.


    G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variations, Second edition, Applied Mathematical Sciences, 147, Springer, New York, 2006.


    B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional, Numerische Math., 85 (2000), 609-646.doi: 10.1007/PL00005394.


    A. Braides and G. Dal Maso, Non-local approximation of the Mumford-Shah functional, Calc. Var. Partial Differential Equations, 5 (1997), 293-322.doi: 10.1007/s005260050068.


    A. Chambolle, Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations, SIAM J. Appl. Math., 55 (1995), 827-863.doi: 10.1137/S0036139993257132.


    A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional, M2AN Math. Model. Numer. Anal., 33 (1999), 261-288.doi: 10.1051/m2an:1999115.


    A. Chambolle and G. Dal Maso, Discrete approximation of the Mumford-Shah functional in dimension two, M2AN Math. Model. Numer. Anal., 33 (1999), 651-672.doi: 10.1051/m2an:1999156.


    M. Chipot, R. March, M. Rosati and G. Vergara Caffarelli, Analysis of a nonconvex problem related to signal selective smoothing, Math. Models Methods Appl. Sci., 7 (1997), 313-328.doi: 10.1142/S0218202597000189.


    G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49.


    G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993.doi: 10.1007/978-1-4612-0327-8.


    G. Dal Maso, J.-M. Morel and S. Solimini, A variational method in image segmentation: Existence and approximation results, Acta Math., 168 (1992), 89-151.doi: 10.1007/BF02392977.


    G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, {233}, Birkhäuser Verlag, Basel, 2005.


    E. De Giorgi and L. Ambrosio, New functionals in the calculus of variations, (Italian) Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 82 (1988), 199-210.


    L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.


    S. Geman and D. McClure, Bayesian image analysis. An application to single photon emission tomography, in Proceedings of the American Statistical Association, Statistical Computing Section, 1985, 12-18.


    M. Gobbino, Finite difference approximation of the Mumford-Shah functional, Comm. Pure Appl. Math., 51 (1998), 197-228.doi: 10.1002/(SICI)1097-0312(199802)51:2<197::AID-CPA3>3.0.CO;2-6.


    T. Hebert and R. Leahy, A generalised EM algorithm for $3$-D Bayesian reconstruction from Poisson data using Gibbs priors, IEEE Trans. Med. Imag., 8 (1989), 194-202.


    V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997.


    D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.doi: 10.1002/cpa.3160420503.


    M. Rosati, Asymptotic behavior of a Geman and McClure discrete model, Appl. Math. Optim., 41 (2000), 51-85.doi: 10.1007/s002459911004.


    L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), 259-268.doi: 10.1016/0167-2789(92)90242-F.


    J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals, Preprint, submitted, 2014.

  • 加载中

Article Metrics

HTML views() PDF downloads(49) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint