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August  2015, 9(3): 853-874. doi: 10.3934/ipi.2015.9.853

The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting

1. 

Laboratoire LAMSIN, ENIT, University of Tunis El Manar, B.P. 37, 1002 Tunis-Belvédère, Tunisia, Tunisia, Tunisia

Received  June 2014 Revised  March 2015 Published  July 2015

Image inpainting or disocclusion, which refers to the process of restoring a damaged image with missing information, has many applications in different fields. Different techniques can be applied to solve this problem. In particular, many variational models have appeared in the literature. These models give rise to partial differential equations for which Dirichlet boundary conditions are usually used. The basic idea of the algorithms that have been proposed in the literature is to fill-in damaged regions with available information from their surroundings. The aim of this work is to treat the case where this information is not available in a part of the boundary of the damaged region. We formulate the image inpainting problem as a nonlinear Cauchy problem. Then, we give a Nash-game formulation of this Cauchy problem and we present different numerical experiments using the finite-element method for solving the image inpainting problem.
Citation: Moez Kallel, Maher Moakher, Anis Theljani. The Cauchy problem for a nonlinear elliptic equation: Nash-game approach and application to image inpainting. Inverse Problems and Imaging, 2015, 9 (3) : 853-874. doi: 10.3934/ipi.2015.9.853
References:
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R. Aboulaïch, A. Ben Abda and M. Kallel, Missing boundary data reconstruction via an approximate optimal control, Inverse Probl. Imaging, 2 (2008), 411-426. doi: 10.3934/ipi.2008.2.411.

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H. Attouch, P. Redont and A. Soubeyran, A new class of alternating proximal minimization algorithms with costs-to-move, SIAM J. Optim., 18 (2007), 1061-1081 (electronic). doi: 10.1137/060657248.

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S. Avdonin, et al., Iterative methods for solving a nonlinear boundary inverse problem in glaciology, J. Inverse Ill-Posed Probl., 17 (2009), 239-258. doi: 10.1515/JIIP.2009.018.

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M. Azaïez, F. Ben Belgacem and H. El Fekih, On Cauchy's problem. II. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336. doi: 10.1088/0266-5611/22/4/012.

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C. Ballester, et al., Filling-in by joint interpolation of vector fields and gray levels, IEEE Trans. Image Process, 10 (2001), 1200-1211. doi: 10.1109/83.935036.

[7]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104. doi: 10.1088/0266-5611/21/3/018.

[8]

A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22 (2006), 1191-1206. doi: 10.1088/0266-5611/22/4/005.

[9]

T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088.

[10]

T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62 (2001/02), 1019-1043. doi: 10.1137/S0036139900368844.

[11]

T. Chan and J. Shen, Non-texture inpainting by curvature-driven diffusions, Journal of Visual Communication and Image Representation, 12 (2001), 436-449.

[12]

J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9 (1951), 225-236.

[13]

J. X. Cruz Neto, et al., Learning how to play Nash, potential games and alternating minimization method for structured nonconvex problems on Riemannian manifolds, J. Convex Anal., 20 (2013), 395-438.

[14]

H. Egger and A. Leitao, Efficient reconstruction methods for nonlinear elliptic Cauchy problems with piecewise constant solutions, Adv. Appl. Math. Mech., 1 (2009), 729-749.

[15]

S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13 (2002), 353-370. doi: 10.1017/S0956792502004904.

[16]

H. Grossauer and O. Scherzer, Using the complex Ginzburg-Landau equation for digital inpainting in 2D and 3D, in Scale Space Methods in Computer Vision (eds. L. D. Griffin and M. Lillholm), Lecture Notes in Computer Science, 2695, Springer Berlin Heidelberg, 2003, 225-236. doi: 10.1007/3-540-44935-3_16.

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A. Habbal, A topology Nash game for tumoral antiangiogenesis, Struct. Multidiscip. Optim., 30 (2005), 404-412. doi: 10.1007/s00158-005-0525-1.

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A. Habbal and M. Kallel, Data completion problems solved as Nash games, Journal of Physics, Conference Series, 386 (2012), 012004. doi: 10.1088/1742-6596/386/1/012004.

[19]

A. Habbal and M. Kallel, Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems, SIAM J. Control Optim., 51 (2013), 4066-4083. doi: 10.1137/120869808.

[20]

A. Habbal, J. Petersson and M. Thellner, Multidisciplinary topology optimization solved as a Nash game, Internat. J. Numer. Methods Engrg., 61 (2004), 949-963. doi: 10.1002/nme.1093.

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J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953.

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D. N. Hào and D. Lesnic, The Cauchy problem for Laplace's equation via the conjugate gradient method, IMA J. Appl. Math., 65 (2000), 199-217. doi: 10.1093/imamat/65.2.199.

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A. Hasanov, Inverse coefficient problems for monotone potential operators, Inverse Problems, 13 (1997), 1265-1278. doi: 10.1088/0266-5611/13/5/011.

[24]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.

[25]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Grundlehren der Mathematischen Wissenschaften, 274, Springer, Berlin, 1985.

[26]

P. Houston, J. Robson and E. Sauli, Discontinuous Galerkin fnite element approximation of quasilinear elliptic boundary value problems I: The scalar case, IMA J. Numer., 50 (2005), 726-749.

[27]

C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface, Math. Comp., 29 (1975), 343-349. doi: 10.1090/S0025-5718-1975-0400741-X.

[28]

S. Korotov and M. Křížek, Finite element analysis of variational crimes for a quasilinear elliptic problem in 3D, Numer. Math., 84 (2000), 549-576. doi: 10.1007/s002110050010.

[29]

S. Korotov and M. Křížek, Finite element analysis of variational crimes for a nonlinear heat conduction problem in three-dimensional space, in ENUMATH 97 (Heidelberg), World Sci. Publ., River Edge, NJ, 1998, 421-428.

[30]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, (Russian) Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64-74; translation in U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52 (1992).

[31]

P. Kügler and A. Leitão, Mean value iterations for nonlinear elliptic Cauchy problems, Numer. Math., 96 (2003), 269-293. doi: 10.1007/s00211-003-0477-6.

[32]

M. A. Lavrentev, On the Cauchy problem for the Laplace equation, Izv. Akd. Nauk SSSR. Ser. Mat., 50 (1956), 819-842.

[33]

M. M. Lavrentev, On the problem of Cauchy for linear elliptic equations of the second order, Dokl. Akad. Nauk SSSR (N.S.), 112 (1957), 195-197.

[34]

S. Li and T. Başar, Distributed algorithms for the computation of noncooperative equilibria, Automatica J. IFAC, 23 (1987), 523-533. doi: 10.1016/0005-1098(87)90081-1.

[35]

I. Ly, An iterative method for solving Cauchy problems for the $p$-Laplace operator, Complex Var. Elliptic Equ., 55 (2010), 1079-1088. doi: 10.1080/17476931003628257.

[36]

I. Ly and N. Tarkhanov, A variational approach to the Cauchy problem for nonlinear elliptic differential equations, J. Inverse Ill-Posed Probl., 17 (2009), 595-610. doi: 10.1515/JIIP.2009.037.

[37]

S. Masnou, Disocclusion: A variational approach using level lines, IEEE Trans. Image Process., 11 (2002), 68-76. doi: 10.1109/83.982815.

[38]

V. G. Maz'ya and V. P. Havin, The solutions of the Cauchy problem for the Laplace equation (uniqueness, normality, approximation), Trudy Moskov. Mat. Obšč., 30 (1974), 61-114.

[39]

L. I. 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.

[40]

N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Mathematical Topics, 7, Akademie Verlag, Berlin, 1995.

[41]

R. Temam, Applications de l'analyse convexe au calcul des variations, in Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lecture Notes in Math., 543, Springer, Berlin, 1976, 208-237.

[42]

S. Uryas'ev and R. Y. Rubinstein, On relaxation algorithms in computation of noncooperative equilibria, IEEE Trans. Automat. Control, 39 (1994), 1263-1267. doi: 10.1109/9.293193.

show all references

References:
[1]

R. Aboulaïch, A. Ben Abda and M. Kallel, Missing boundary data reconstruction via an approximate optimal control, Inverse Probl. Imaging, 2 (2008), 411-426. doi: 10.3934/ipi.2008.2.411.

[2]

R. Aboulaich, S. Boujena and E. El Guarmah, A nonlinear parabolic model in processing of medical image, Math. Model. Nat. Phenom., 3 (2008), 131-145. doi: 10.1051/mmnp:2008084.

[3]

H. Attouch, P. Redont and A. Soubeyran, A new class of alternating proximal minimization algorithms with costs-to-move, SIAM J. Optim., 18 (2007), 1061-1081 (electronic). doi: 10.1137/060657248.

[4]

S. Avdonin, et al., Iterative methods for solving a nonlinear boundary inverse problem in glaciology, J. Inverse Ill-Posed Probl., 17 (2009), 239-258. doi: 10.1515/JIIP.2009.018.

[5]

M. Azaïez, F. Ben Belgacem and H. El Fekih, On Cauchy's problem. II. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336. doi: 10.1088/0266-5611/22/4/012.

[6]

C. Ballester, et al., Filling-in by joint interpolation of vector fields and gray levels, IEEE Trans. Image Process, 10 (2001), 1200-1211. doi: 10.1109/83.935036.

[7]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104. doi: 10.1088/0266-5611/21/3/018.

[8]

A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22 (2006), 1191-1206. doi: 10.1088/0266-5611/22/4/005.

[9]

T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592. doi: 10.1137/S0036139901390088.

[10]

T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62 (2001/02), 1019-1043. doi: 10.1137/S0036139900368844.

[11]

T. Chan and J. Shen, Non-texture inpainting by curvature-driven diffusions, Journal of Visual Communication and Image Representation, 12 (2001), 436-449.

[12]

J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9 (1951), 225-236.

[13]

J. X. Cruz Neto, et al., Learning how to play Nash, potential games and alternating minimization method for structured nonconvex problems on Riemannian manifolds, J. Convex Anal., 20 (2013), 395-438.

[14]

H. Egger and A. Leitao, Efficient reconstruction methods for nonlinear elliptic Cauchy problems with piecewise constant solutions, Adv. Appl. Math. Mech., 1 (2009), 729-749.

[15]

S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13 (2002), 353-370. doi: 10.1017/S0956792502004904.

[16]

H. Grossauer and O. Scherzer, Using the complex Ginzburg-Landau equation for digital inpainting in 2D and 3D, in Scale Space Methods in Computer Vision (eds. L. D. Griffin and M. Lillholm), Lecture Notes in Computer Science, 2695, Springer Berlin Heidelberg, 2003, 225-236. doi: 10.1007/3-540-44935-3_16.

[17]

A. Habbal, A topology Nash game for tumoral antiangiogenesis, Struct. Multidiscip. Optim., 30 (2005), 404-412. doi: 10.1007/s00158-005-0525-1.

[18]

A. Habbal and M. Kallel, Data completion problems solved as Nash games, Journal of Physics, Conference Series, 386 (2012), 012004. doi: 10.1088/1742-6596/386/1/012004.

[19]

A. Habbal and M. Kallel, Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems, SIAM J. Control Optim., 51 (2013), 4066-4083. doi: 10.1137/120869808.

[20]

A. Habbal, J. Petersson and M. Thellner, Multidisciplinary topology optimization solved as a Nash game, Internat. J. Numer. Methods Engrg., 61 (2004), 949-963. doi: 10.1002/nme.1093.

[21]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953.

[22]

D. N. Hào and D. Lesnic, The Cauchy problem for Laplace's equation via the conjugate gradient method, IMA J. Appl. Math., 65 (2000), 199-217. doi: 10.1093/imamat/65.2.199.

[23]

A. Hasanov, Inverse coefficient problems for monotone potential operators, Inverse Problems, 13 (1997), 1265-1278. doi: 10.1088/0266-5611/13/5/011.

[24]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.

[25]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Grundlehren der Mathematischen Wissenschaften, 274, Springer, Berlin, 1985.

[26]

P. Houston, J. Robson and E. Sauli, Discontinuous Galerkin fnite element approximation of quasilinear elliptic boundary value problems I: The scalar case, IMA J. Numer., 50 (2005), 726-749.

[27]

C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface, Math. Comp., 29 (1975), 343-349. doi: 10.1090/S0025-5718-1975-0400741-X.

[28]

S. Korotov and M. Křížek, Finite element analysis of variational crimes for a quasilinear elliptic problem in 3D, Numer. Math., 84 (2000), 549-576. doi: 10.1007/s002110050010.

[29]

S. Korotov and M. Křížek, Finite element analysis of variational crimes for a nonlinear heat conduction problem in three-dimensional space, in ENUMATH 97 (Heidelberg), World Sci. Publ., River Edge, NJ, 1998, 421-428.

[30]

V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, (Russian) Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64-74; translation in U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52 (1992).

[31]

P. Kügler and A. Leitão, Mean value iterations for nonlinear elliptic Cauchy problems, Numer. Math., 96 (2003), 269-293. doi: 10.1007/s00211-003-0477-6.

[32]

M. A. Lavrentev, On the Cauchy problem for the Laplace equation, Izv. Akd. Nauk SSSR. Ser. Mat., 50 (1956), 819-842.

[33]

M. M. Lavrentev, On the problem of Cauchy for linear elliptic equations of the second order, Dokl. Akad. Nauk SSSR (N.S.), 112 (1957), 195-197.

[34]

S. Li and T. Başar, Distributed algorithms for the computation of noncooperative equilibria, Automatica J. IFAC, 23 (1987), 523-533. doi: 10.1016/0005-1098(87)90081-1.

[35]

I. Ly, An iterative method for solving Cauchy problems for the $p$-Laplace operator, Complex Var. Elliptic Equ., 55 (2010), 1079-1088. doi: 10.1080/17476931003628257.

[36]

I. Ly and N. Tarkhanov, A variational approach to the Cauchy problem for nonlinear elliptic differential equations, J. Inverse Ill-Posed Probl., 17 (2009), 595-610. doi: 10.1515/JIIP.2009.037.

[37]

S. Masnou, Disocclusion: A variational approach using level lines, IEEE Trans. Image Process., 11 (2002), 68-76. doi: 10.1109/83.982815.

[38]

V. G. Maz'ya and V. P. Havin, The solutions of the Cauchy problem for the Laplace equation (uniqueness, normality, approximation), Trudy Moskov. Mat. Obšč., 30 (1974), 61-114.

[39]

L. I. 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.

[40]

N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Mathematical Topics, 7, Akademie Verlag, Berlin, 1995.

[41]

R. Temam, Applications de l'analyse convexe au calcul des variations, in Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lecture Notes in Math., 543, Springer, Berlin, 1976, 208-237.

[42]

S. Uryas'ev and R. Y. Rubinstein, On relaxation algorithms in computation of noncooperative equilibria, IEEE Trans. Automat. Control, 39 (1994), 1263-1267. doi: 10.1109/9.293193.

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