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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 |
References:
[1] |
R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
L. Ambrosio, Compactness for a special class of functions of bounded variation, Boll. Un. Mat. Ital. B, 3 (1989), 857-881. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49. |
[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. |
[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. |
[15] |
G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, {233}, Birkhäuser Verlag, Basel, 2005. |
[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. |
[17] |
L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. |
[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. |
[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. |
[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. |
[21] |
V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997. |
[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. |
[23] |
M. Rosati, Asymptotic behavior of a Geman and McClure discrete model, Appl. Math. Optim., 41 (2000), 51-85.
doi: 10.1007/s002459911004. |
[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. |
[25] |
J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals, Preprint, submitted, 2014. |
show all references
References:
[1] |
R. Adams and J. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
L. Ambrosio, Compactness for a special class of functions of bounded variation, Boll. Un. Mat. Ital. B, 3 (1989), 857-881. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
G. Cortesani, Strong approximation of GSBV functions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.), 43 (1997), 27-49. |
[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. |
[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. |
[15] |
G. David, Singular Sets of Minimizers for the Mumford-Shah Functional, Progress in Mathematics, {233}, Birkhäuser Verlag, Basel, 2005. |
[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. |
[17] |
L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. |
[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. |
[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. |
[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. |
[21] |
V. Maz'ya nd S. Poborschi, Differentiable Functions on Bad Domains, World Scientific, Singapore, 1997. |
[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. |
[23] |
M. Rosati, Asymptotic behavior of a Geman and McClure discrete model, Appl. Math. Optim., 41 (2000), 51-85.
doi: 10.1007/s002459911004. |
[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. |
[25] |
J. Tiirola, Gamma-convergence of certain modified Perona-Malik functionals, Preprint, submitted, 2014. |
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