# American Institute of Mathematical Sciences

August  2015, 9(3): 835-851. doi: 10.3934/ipi.2015.9.835

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

 1 Department of Mathematics and Statistics, FI-20014 University of Turku, Finland 2 Department of Mathematical Sciences, FI-90014 University of Oulu, Finland, Finland

Received  October 2014 Revised  February 2015 Published  July 2015

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.
Citation: Petteri Harjulehto, Peter Hästö, Juha Tiirola. Point-wise behavior of the Geman--McClure and the Hebert--Leahy image restoration models. Inverse Problems & Imaging, 2015, 9 (3) : 835-851. doi: 10.3934/ipi.2015.9.835
##### References:
 [1] R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar [2] L. Ambrosio, Compactness for a special class of functions of bounded variation, Boll. Un. Mat. Ital. B, 3 (1989), 857-881.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [5] 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.  Google Scholar [6] 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.  Google Scholar [7] 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.  Google Scholar [8] 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.  Google Scholar [9] A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional, M2AN Math. Model. Numer. Anal., 33 (1999), 261-288. doi: 10.1051/m2an:1999115.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49.  Google Scholar [13] 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.  Google Scholar [14] 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.  Google Scholar [15] G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, {233}, Birkhäuser Verlag, Basel, 2005.  Google Scholar [16] 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.  Google Scholar [17] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar [18] 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. Google Scholar [19] 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.  Google Scholar [20] 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. Google Scholar [21] V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997.  Google Scholar [22] 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.  Google Scholar [23] M. Rosati, Asymptotic behavior of a Geman and McClure discrete model, Appl. Math. Optim., 41 (2000), 51-85. doi: 10.1007/s002459911004.  Google Scholar [24] 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.  Google Scholar [25] J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals, Preprint, submitted, 2014. Google Scholar

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##### References:
 [1] R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar [2] L. Ambrosio, Compactness for a special class of functions of bounded variation, Boll. Un. Mat. Ital. B, 3 (1989), 857-881.  Google Scholar [3] 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.  Google Scholar [4] 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.  Google Scholar [5] 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.  Google Scholar [6] 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.  Google Scholar [7] 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.  Google Scholar [8] 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.  Google Scholar [9] A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional, M2AN Math. Model. Numer. Anal., 33 (1999), 261-288. doi: 10.1051/m2an:1999115.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49.  Google Scholar [13] 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.  Google Scholar [14] 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.  Google Scholar [15] G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, {233}, Birkhäuser Verlag, Basel, 2005.  Google Scholar [16] 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.  Google Scholar [17] L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.  Google Scholar [18] 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. Google Scholar [19] 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.  Google Scholar [20] 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. Google Scholar [21] V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997.  Google Scholar [22] 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.  Google Scholar [23] M. Rosati, Asymptotic behavior of a Geman and McClure discrete model, Appl. Math. Optim., 41 (2000), 51-85. doi: 10.1007/s002459911004.  Google Scholar [24] 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.  Google Scholar [25] J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals, Preprint, submitted, 2014. Google Scholar
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