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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 & Imaging, 2015, 9 (3) : 853-874. doi: 10.3934/ipi.2015.9.853
##### 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.  doi: 10.3934/ipi.2008.2.411.  Google Scholar [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.  doi: 10.1051/mmnp:2008084.  Google Scholar [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.  doi: 10.1137/060657248.  Google Scholar [4] S. Avdonin, et al., Iterative methods for solving a nonlinear boundary inverse problem in glaciology,, J. Inverse Ill-Posed Probl., 17 (2009), 239.  doi: 10.1515/JIIP.2009.018.  Google Scholar [5] M. Azaïez, F. Ben Belgacem and H. El Fekih, On Cauchy's problem. II. Completion, regularization and approximation,, Inverse Problems, 22 (2006), 1307.  doi: 10.1088/0266-5611/22/4/012.  Google Scholar [6] C. Ballester, et al., Filling-in by joint interpolation of vector fields and gray levels,, IEEE Trans. Image Process, 10 (2001), 1200.  doi: 10.1109/83.935036.  Google Scholar [7] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 21 (2005), 1087.  doi: 10.1088/0266-5611/21/3/018.  Google Scholar [8] A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem,, Inverse Problems, 22 (2006), 1191.  doi: 10.1088/0266-5611/22/4/005.  Google Scholar [9] T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting,, SIAM J. Appl. Math., 63 (2002), 564.  doi: 10.1137/S0036139901390088.  Google Scholar [10] T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings,, SIAM J. Appl. Math., 62 (): 1019.  doi: 10.1137/S0036139900368844.  Google Scholar [11] T. Chan and J. Shen, Non-texture inpainting by curvature-driven diffusions,, Journal of Visual Communication and Image Representation, 12 (2001), 436.   Google Scholar [12] J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics,, Quart. Appl. Math., 9 (1951), 225.   Google Scholar [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.   Google Scholar [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.   Google Scholar [15] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model,, European J. Appl. Math., 13 (2002), 353.  doi: 10.1017/S0956792502004904.  Google Scholar [16] H. Grossauer and O. 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Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations,, Dover Publications, (1953).   Google Scholar [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.  doi: 10.1093/imamat/65.2.199.  Google Scholar [23] A. Hasanov, Inverse coefficient problems for monotone potential operators,, Inverse Problems, 13 (1997), 1265.  doi: 10.1088/0266-5611/13/5/011.  Google Scholar [24] F. Hecht, New development in freefem++,, J. Numer. Math., 20 (2012), 251.   Google Scholar [25] L. Hörmander, The Analysis of Linear Partial Differential Operators. III,, Grundlehren der Mathematischen Wissenschaften, (1985).   Google Scholar [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.   Google Scholar [27] C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface,, Math. Comp., 29 (1975), 343.  doi: 10.1090/S0025-5718-1975-0400741-X.  Google Scholar [28] S. Korotov and M. Křížek, Finite element analysis of variational crimes for a quasilinear elliptic problem in 3D,, Numer. Math., 84 (2000), 549.  doi: 10.1007/s002110050010.  Google Scholar [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), (1998), 421.   Google Scholar [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.   Google Scholar [31] P. Kügler and A. Leitão, Mean value iterations for nonlinear elliptic Cauchy problems,, Numer. Math., 96 (2003), 269.  doi: 10.1007/s00211-003-0477-6.  Google Scholar [32] M. A. 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Image Process., 11 (2002), 68.  doi: 10.1109/83.982815.  Google Scholar [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.   Google Scholar [39] L. I. 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 [40] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Mathematical Topics, (1995).   Google Scholar [41] R. Temam, Applications de l'analyse convexe au calcul des variations,, in Nonlinear Operators and the Calculus of Variations (Summer School, (1975), 208.   Google Scholar [42] S. Uryas'ev and R. Y. Rubinstein, On relaxation algorithms in computation of noncooperative equilibria,, IEEE Trans. Automat. Control, 39 (1994), 1263.  doi: 10.1109/9.293193.  Google Scholar

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.  doi: 10.3934/ipi.2008.2.411.  Google Scholar [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.  doi: 10.1051/mmnp:2008084.  Google Scholar [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.  doi: 10.1137/060657248.  Google Scholar [4] S. Avdonin, et al., Iterative methods for solving a nonlinear boundary inverse problem in glaciology,, J. Inverse Ill-Posed Probl., 17 (2009), 239.  doi: 10.1515/JIIP.2009.018.  Google Scholar [5] M. Azaïez, F. Ben Belgacem and H. El Fekih, On Cauchy's problem. II. Completion, regularization and approximation,, Inverse Problems, 22 (2006), 1307.  doi: 10.1088/0266-5611/22/4/012.  Google Scholar [6] C. Ballester, et al., Filling-in by joint interpolation of vector fields and gray levels,, IEEE Trans. Image Process, 10 (2001), 1200.  doi: 10.1109/83.935036.  Google Scholar [7] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 21 (2005), 1087.  doi: 10.1088/0266-5611/21/3/018.  Google Scholar [8] A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem,, Inverse Problems, 22 (2006), 1191.  doi: 10.1088/0266-5611/22/4/005.  Google Scholar [9] T. F. Chan, S. H. Kang and J. Shen, Euler's elastica and curvature-based inpainting,, SIAM J. Appl. Math., 63 (2002), 564.  doi: 10.1137/S0036139901390088.  Google Scholar [10] T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings,, SIAM J. Appl. Math., 62 (): 1019.  doi: 10.1137/S0036139900368844.  Google Scholar [11] T. Chan and J. Shen, Non-texture inpainting by curvature-driven diffusions,, Journal of Visual Communication and Image Representation, 12 (2001), 436.   Google Scholar [12] J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics,, Quart. Appl. Math., 9 (1951), 225.   Google Scholar [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.   Google Scholar [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.   Google Scholar [15] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model,, European J. Appl. Math., 13 (2002), 353.  doi: 10.1017/S0956792502004904.  Google Scholar [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), (2695), 225.  doi: 10.1007/3-540-44935-3_16.  Google Scholar [17] A. Habbal, A topology Nash game for tumoral antiangiogenesis,, Struct. Multidiscip. Optim., 30 (2005), 404.  doi: 10.1007/s00158-005-0525-1.  Google Scholar [18] A. Habbal and M. Kallel, Data completion problems solved as Nash games,, Journal of Physics, 386 (2012).  doi: 10.1088/1742-6596/386/1/012004.  Google Scholar [19] A. Habbal and M. Kallel, Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems,, SIAM J. Control Optim., 51 (2013), 4066.  doi: 10.1137/120869808.  Google Scholar [20] A. Habbal, J. Petersson and M. Thellner, Multidisciplinary topology optimization solved as a Nash game,, Internat. J. Numer. Methods Engrg., 61 (2004), 949.  doi: 10.1002/nme.1093.  Google Scholar [21] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations,, Dover Publications, (1953).   Google Scholar [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.  doi: 10.1093/imamat/65.2.199.  Google Scholar [23] A. Hasanov, Inverse coefficient problems for monotone potential operators,, Inverse Problems, 13 (1997), 1265.  doi: 10.1088/0266-5611/13/5/011.  Google Scholar [24] F. Hecht, New development in freefem++,, J. Numer. Math., 20 (2012), 251.   Google Scholar [25] L. Hörmander, The Analysis of Linear Partial Differential Operators. III,, Grundlehren der Mathematischen Wissenschaften, (1985).   Google Scholar [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.   Google Scholar [27] C. Johnson and V. Thomée, Error estimates for a finite element approximation of a minimal surface,, Math. Comp., 29 (1975), 343.  doi: 10.1090/S0025-5718-1975-0400741-X.  Google Scholar [28] S. Korotov and M. Křížek, Finite element analysis of variational crimes for a quasilinear elliptic problem in 3D,, Numer. Math., 84 (2000), 549.  doi: 10.1007/s002110050010.  Google Scholar [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), (1998), 421.   Google Scholar [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.   Google Scholar [31] P. Kügler and A. Leitão, Mean value iterations for nonlinear elliptic Cauchy problems,, Numer. Math., 96 (2003), 269.  doi: 10.1007/s00211-003-0477-6.  Google Scholar [32] M. A. Lavrentev, On the Cauchy problem for the Laplace equation,, Izv. Akd. Nauk SSSR. Ser. Mat., 50 (1956), 819.   Google Scholar [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.   Google Scholar [34] S. Li and T. Başar, Distributed algorithms for the computation of noncooperative equilibria,, Automatica J. IFAC, 23 (1987), 523.  doi: 10.1016/0005-1098(87)90081-1.  Google Scholar [35] I. Ly, An iterative method for solving Cauchy problems for the $p$-Laplace operator,, Complex Var. Elliptic Equ., 55 (2010), 1079.  doi: 10.1080/17476931003628257.  Google Scholar [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.  doi: 10.1515/JIIP.2009.037.  Google Scholar [37] S. Masnou, Disocclusion: A variational approach using level lines,, IEEE Trans. Image Process., 11 (2002), 68.  doi: 10.1109/83.982815.  Google Scholar [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.   Google Scholar [39] L. I. 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 [40] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Mathematical Topics, (1995).   Google Scholar [41] R. Temam, Applications de l'analyse convexe au calcul des variations,, in Nonlinear Operators and the Calculus of Variations (Summer School, (1975), 208.   Google Scholar [42] S. Uryas'ev and R. Y. Rubinstein, On relaxation algorithms in computation of noncooperative equilibria,, IEEE Trans. Automat. Control, 39 (1994), 1263.  doi: 10.1109/9.293193.  Google Scholar
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