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A time dependent variational approach to image restoration. (English) Zbl 1380.94012

Summary: In this paper we introduce a purely variational approach to the gradient flows, naturally arising in image denoising models, yielding the existence of global parabolic minimizers, in the sense that \[ \int_0^{T} \left[\int_\Omega u \partial_t\phi\, dx + \mathbf{F}(u)\right] \,dt \leq \int_0^{T}\mathbf{F}(u+\phi) \,dt, \] whenever \(T>0\) and \(\phi\in C_0^\infty(\Omega\times(0,T))\). Our method applies to a wide class of nonparametric regression models in image restoration analysis, such as quantile, robust, and logistic regression. A prototype functional \(\mathbf{F}\) is the by now classical \(\mathrm{TV}(L^2)\)-functional (i.e., the pure \(\mathrm{TV}\)-denoising case in image reconstruction) introduced by Rudin, Osher and Fatemi [L. I. Rudin et al., Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)]: \(\mathbf F(u):= \mathbf{TV}(u)+\frac{\kappa}{2}\int_\Omega |u-u_o|^2\, dx, \) where \(u_o:\Omega\to [0,1]\) is a noisy, monochromatic image and \(\kappa\gg 1\) a large penalization parameter. The evolutionary variational solutions are obtained as limits of maps, minimizing a convex variational functional in \(n+1\) dimensions with domain \(\Omega_T:=\Omega \times (0,T)\). Our approach yields a new way of proving the existence of global weak solutions to the associated Cauchy-Dirichlet problem, \(\partial _{t}u- \mathrm{div} (\frac{D u}{|D u|})=\kappa (u-u_o)\) in \(\Omega \times (0,\infty)\) and \(u=u_o\) on the parabolic boundary. Our approach also applies in situations where the considered functionals do not allow the derivation of the associated parabolic equation. We are able to deal with Dirichlet and Neumann type boundary conditions on the lateral boundary, and furthermore with the gradient flow associated to functionals modeling image deblurring, such as \(\mathbf{F}(u)=\mathbf{TV}(u)+\frac\kappa2\int_\Omega| \mathbf{K}[u]-u_o|^2\, dx, \) where \(\mathbf{K}: L^1(\Omega) \to L^2(\Omega)\) is a bounded, linear, injective operator satisfying the DC-condition \(\mathbf{K}[1]=1\).

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
68U10 Computing methodologies for image processing

Citations:

Zbl 0780.49028
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References:

[1] R. Acar and C. R. Vogel, {\it Analysis of bounded variation penalty methods for ill-posed problems}, Inverse Problems, 10 (1994), pp. 1217-1229. · Zbl 0809.35151
[2] L. Ambrosio, N. Fusco, and D. Pallara, {\it Functions of Bounded Variation and Free Discontinuity Problems}, Oxford Math. Monogr., Oxford University Press, New York, 2000. · Zbl 0957.49001
[3] L. Ambrosio, N. Gigli, and G. Savaré, {\it Gradient Flows in Metric Spaces and in the Space of Probability Measures}, 2nd ed., Lectures Math. ETH Zürich, Birkhäuser Verlag, Basel, 2008. · Zbl 1145.35001
[4] F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, {\it The Dirichlet problem for the total variation flow}, J. Funct. Anal., 180 (2001), pp. 347-403. · Zbl 0973.35109
[5] F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, {\it Minimizing total variation flow}, Differential Integral Equations, 14 (2001), pp. 321-360. · Zbl 1020.35037
[6] F. Andreu, V. Caselles, J. I. Díaz, and J. I. J. M. Mazón, {\it Some qualitative properties for the total variation flow}, J. Funct. Anal., 188 (2002), pp. 516-547. · Zbl 1042.35018
[7] F. Andreu, V. Caselles, and J. M. Mazón, {\it Parabolic Quasilinear Equations Minimizing Linear Growth Functionals}, Progr. Math., 223, Birkhäuser Verlag, Basel, 2004. · Zbl 1053.35002
[8] F. Andreu, J. M. Mazón, and J. S. Moll, {\it The total variation flow with nonlinear boundary conditions}, Asymptot. Anal., 43 (2005), pp. 9-46. · Zbl 1082.35084
[9] F. Andreu, J. M. Mazón, J. S. Moll, and V. Caselles, {\it The minimizing total variation flow with measure initial conditions}, Commun. Contemp. Math., 6 (2004), pp. 431-494. · Zbl 1082.35090
[10] F. Andreu, V. Caselles, and J. M. Mazón, {\it A parabolic quasilinear problem for linear growth functionals}, Rev. Mat. Iberoam., 18 (2002), pp. 135-185. · Zbl 1010.35063
[11] F. Andreu, V. Caselles, and J. M. Mazón, {\it The Cauchy problem for a strongly degenerate quasilinear equation}, J. Eur. Math. Soc. (JEMS), 7 (2005), pp. 361-393. · Zbl 1082.35089
[12] F. Andreu, V. Caselles, and J. M. Mazón, {\it A strongly degenerate quasilinear equation: The parabolic case}, Arch. Ration. Mech. Anal., 176 (2005), pp. 415-453. · Zbl 1112.35111
[13] F. Andreu, V. Caselles, J. M. Mazón, and J. S. Moll, {\it Finite propagation speed for limited flux diffusion equations}, Arch. Ration. Mech. Anal., 182 (2006), pp. 269-297. · Zbl 1142.35455
[14] G. Bellettini, V. Caselles, and M. Novaga, {\it The total variation flow in \(R^N\)}, J. Differential Equations, 184 (2002), pp. 475-525. · Zbl 1036.35099
[15] V. Bögelein, F. Duzaar, and P. Marcellini, {\it Parabolic equations with p,q-growth}, J. Math. Pures Appl., 100 (2013), pp. 535-563. · Zbl 1288.35302
[16] V. Bögelein, F. Duzaar, and P. Marcellini, {\it Existence of evolutionary variational solutions via the calculus of variations}, J. Differential Equations, 256 (2014), pp. 3912-3942. · Zbl 1288.35007
[17] V. Bögelein, F. Duzaar, and P. Marcellini, {\it Parabolic systems with p,q-growth: A variational approach}, Arch. Ration. Mech. Anal., 210 (2013), pp. 219-267. · Zbl 1292.35146
[18] K. Bredies and D. Lorenz, {\it Mathematische Bildverarbeitung}, Vieweg + Teubner Verlag, Berlin, 2011.
[19] A. Chambolle and P. L. Lions, {\it Image recovery via total variation minimization and related problems}, Numer. Math., 76 (1997), pp. 167-188. · Zbl 0874.68299
[20] T. F. Chan and S. Esedoḡlu, {\it Aspects of total variation regularized \(L^1\) function approximation}, SIAM J. Appl. Math., 65 (2005), pp. 1817-1837. · Zbl 1096.94004
[21] T. F. Chan and J. Shen, {\it Image Processing and Analysis}, SIAM, Philadelphia, 2005. · Zbl 1095.68127
[22] E. De Giorgi, {\it Conjectures concerning some evolution problems} (in Italian), Duke Math. J., 81 (1996), pp. 255-268. · Zbl 0874.35027
[23] E. De Giorgi, {\it Selected Papers}, (L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, and S. Spagnolo eds., Springer-Verlag, Berlin, 2006.
[24] I. Ekeland and R. Teman, {\it Convex Analysis and Variational Problems}, North-Holland, Amsterdam, 1976.
[25] S. Esedoḡlu and S. Osher, {\it Decomposition of images by the anisotropic Rudin-Osher-Fatemi model}, Comm. Pure Appl. Math., 57 (2004), pp. 1609-1626. · Zbl 1083.49029
[26] E. Giusti, {\it Minimal Surfaces and Functions of Bounded Variation}, Monogr. Mathematics 80, Birkhäuser Verlag, Basel, 1984. · Zbl 0545.49018
[27] H. Hakkarainen and J. Kinnunen, {\it The BV-capacity in metric spaces}, Manuscripta Math., 132 (2010), pp. 51-73. · Zbl 1194.28001
[28] R. Hardt and X. Zhou, {\it An evolution problem for linear growth functionals}, Commun. Partial Differential Equations, 19 (1994), pp. 1879-1907. · Zbl 0811.35061
[29] Z. Jin and X. Yang, {\it Analysis of a new variational model for multiplicative noise removal}, J. Math. Anal. Appl., 362 (2010), pp. 415-426. · Zbl 1191.68788
[30] J. Kinnunen and P. Lindqvist, {\it Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation}, Ann. Mat. Pura Appl. (4), 185 (2006) pp. 411-435. · Zbl 1232.35080
[31] A. Lichnewsky and R. Temam, {\it Pseudosolutions of the time-dependent minimal surface problem}, J. Differential Equations, 30 (1978), pp. 340-364. · Zbl 0368.49016
[32] S. Osher and O. Scherzer, {\it G-norm properties of bounded variation regularization}, Commun. Math. Sci., 2 (2004), pp. 237-254. · Zbl 1082.49003
[33] L. Rudin, S. Osher, and E. Fatemi, {\it Nonlinear total variation based noise removal algorithms}, Phys. D, 268 (1992), pp. 259-268. · Zbl 0780.49028
[34] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, {\it Variational Methods in Imaging}, Appl. Math. Sci. 167, Springer, New York, 2009. · Zbl 1177.68245
[35] E. Serra and P. Tilli, {\it Nonlinear wave equations as limits of convex minimization problems: Proof of a conjecture by De Giorgi}, Ann. of Math. (2), 175 (2012), pp. 1551-1574. · Zbl 1251.49019
[36] J. Simon, {\it Compact sets in the space \(L^p(0,T;B)\)}, Ann. Mat. Pura Appl. (4), 146 (1987), pp. 65-96. · Zbl 0629.46031
[37] E. Spadaro and U. Stefanelli, {\it A variational view at the time-dependent minimal surface equation}, J. Evol. Equ., 11 (2011), pp. 793-809. · Zbl 1246.35117
[38] U. Stefanelli, {\it The De Giorgi conjecture on elliptic regularization}, Math. Models Methods Appl. Sci., 21 (2011), pp. 1377-1394. · Zbl 1228.35023
[39] W. Wieser, {\it Parabolic Q-minima and minimal solutions to variational flow}, Manuscripta Math., 59 (1987), pp. 63-107. · Zbl 0674.35042
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