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, (2003).   Google Scholar

[2]

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

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford Mathematical Monographs, (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.  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, (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.  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.  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.  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.  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.  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.  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.   Google Scholar

[13]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Progress in Nonlinear Differential Equations and their Applications, (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.  doi: 10.1007/BF02392977.  Google Scholar

[15]

G. David, Singular Sets of Minimizers for the Mumford-Shah Functional,, Progress in Mathematics, (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.   Google Scholar

[17]

L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (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, (1985), 12.   Google Scholar

[19]

M. Gobbino, Finite difference approximation of the Mumford-Shah functional,, Comm. Pure Appl. Math., 51 (1998), 197.  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.   Google Scholar

[21]

V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains,, World Scientific, (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.  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.  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.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[25]

J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals,, Preprint, (2014).   Google Scholar

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev Spaces,, Second edition, (2003).   Google Scholar

[2]

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

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford Mathematical Monographs, (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.  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, (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.  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.  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.  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.  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.  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.  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.   Google Scholar

[13]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Progress in Nonlinear Differential Equations and their Applications, (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.  doi: 10.1007/BF02392977.  Google Scholar

[15]

G. David, Singular Sets of Minimizers for the Mumford-Shah Functional,, Progress in Mathematics, (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.   Google Scholar

[17]

L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (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, (1985), 12.   Google Scholar

[19]

M. Gobbino, Finite difference approximation of the Mumford-Shah functional,, Comm. Pure Appl. Math., 51 (1998), 197.  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.   Google Scholar

[21]

V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains,, World Scientific, (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.  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.  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.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[25]

J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals,, Preprint, (2014).   Google Scholar

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