A variational approach to the denoising of images based on different variants of the TV-regularization.(English)Zbl 1260.49074

The purpose of this short paper is to investigate the existence and the regularity of solutions for some variational problems related to varational partial differential methods used in image recovery. Loosely speaking, one has to reconstruct an original image from an observed pattern, and the technique to this end is to minimize a functional in the form of an integral which defines the quality of the data fitting. The authors consider successively nearly linear growth and linear growth and in each case they state some convergence properties for the approximation of the solution.

MSC:

 49N90 Applications of optimal control and differential games 49N60 Regularity of solutions in optimal control 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text:

References:

 [1] Adams, R.A.: Sobolev Spaces. Academic Press, San Diego (1975) · Zbl 0314.46030 [2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford Science Publications. Clarendon, Oxford (2000) · Zbl 0957.49001 [3] Apushkinskaya, D., Bildhauer, M., Fuchs, M.: Steady states of anisotropic generalized Newtonian fluids. J. Math. Fluid Mech. 7, 261–297 (2005) · Zbl 1162.76375 [4] Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217–1229 (1994) · Zbl 0809.35151 [5] Aubert, G., Vese, L.: A variational method in image recovery. SIAM J. Numer. Anal. 34(5), 1948–1979 (1997) · Zbl 0890.35033 [6] Blomgren, P., Chan, T.F., Mulet, P., Vese, L., Wan, W.L.: Variational PDE models and methods for image processing. In: Numerical Analysis 1999, Dundee. Chapman & Hall/CRC Res. Notes Math., vol. 420, pp. 43–67. Chapman & Hall/CRC Press, Boca Raton (2000) · Zbl 0953.68621 [7] Bildhauer, M.: Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions. Lecture Notes in Mathematics, vol. 1818. Springer, Berlin (2003) · Zbl 1033.49001 [8] Bildhauer, M.: Two dimensional variational problems with linear growth. Manuscr. Math. 110, 325–342 (2003) · Zbl 1026.49028 [9] Bildhauer, M., Fuchs, M.: On a class of variational integrals with linear growth satisfying the condition of {$$\mu$$}-ellipticity. Rend. Mat. Appl. 22, 249–274 (2002) · Zbl 1047.49029 [10] Bildhauer, M., Fuchs, M.: Relaxation of convex variational problems with linear growth defined on classes of vector-valued functions. Algebra Anal. 14(1), 26–45 (2002) · Zbl 1029.49013 [11] Bildhauer, M., Fuchs, M.: Partial regularity for a class of anisotropic variational integrals with convex hull property. Asymptot. Anal. 32, 293–315 (2002) · Zbl 1076.49018 [12] Bildhauer, M., Fuchs, M.: Two-dimensional anisotropic variational problems. Calc. Var. 16, 177–186 (2003) · Zbl 1026.49027 [13] Bildhauer, M., Fuchs, M.: A regularity result for stationary electrorheological fluids in two dimensions. Math. Methods Appl. Sci. 27, 1607–1617 (2004) · Zbl 1058.76073 [14] Bildhauer, M., Fuchs, M.: A geometric maximum principle for variational problems in spaces of vector valued functions of bounded variation. Zap. Naučn. Semin. POMI 385, 5–17 (2010) [15] Caselles, V., Chambolle, A., Novaga, M.: Regularity for solutions of the total variation denoising problem. Rev. Mat. Iberoam. 27, 233–252 (2011) · Zbl 1228.94005 [16] Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997) · Zbl 0874.68299 [17] Chan, T.F., Esedoḡlu, S.: Aspects of total variation regularized L 1 function approximation. SIAM J. Appl. Math. 65, 1817–1837 (2005) · Zbl 1096.94004 [18] Chan, T., Shen, J., Vese, L.: Variational PDE models in image processing. Not. Am. Math. Soc. 50(1), 14–26 (2003) · Zbl 1168.94315 [19] Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006) · Zbl 1102.49010 [20] Demengel, F., Temam, R.: Convex functions of a measure and applications. Indiana Univ. Math. J. 33, 673–709 (1984) · Zbl 0581.46036 [21] Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976) · Zbl 0322.90046 [22] Frehse, J.: Two dimensional variational problems with thin obstacles. Math. Z. 143, 279–288 (1975) · Zbl 0302.49002 [23] Frehse, J., Seregin, G.: Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening. Transl. Am. Math. Soc. 193, 127–152 (1999) [24] Fuchs, M., Seregin, G.: Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids. Lecture Notes in Mathematics, vol. 1749. Springer, Berlin (2000) · Zbl 0964.76003 [25] Fuchs, M., Seregin, G.: A regularity theory for variational integrals with LlogL-growth. Calc. Var. 6, 171–187 (1998) · Zbl 0929.49022 [26] Fuchs, M., Osmolovskii, V.: Variational integrals on Orlicz-Sobolev spaces. Z. Anal. Anwend. 17, 393–415 (1998) · Zbl 0913.49023 [27] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der Math. Wiss., vol. 224. Springer, Berlin (1998) · Zbl 1042.35002 [28] Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser, Basel (1984) · Zbl 0545.49018 [29] Kawohl, B.: Variational Versus PDE-Based Approaches in Mathematical Image Processing. CRM Proceedings and Lecture Notes, vol. 44, pp. 113–126 (2008) · Zbl 1144.35401 [30] Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Nauka, Moskow (1964). English translation. Academic Press, New York (1968) [31] Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992) · Zbl 0780.49028 [32] Vese, L.: A study in the BV space of a denoising-deblurring variational problem. Appl. Math. Optim. 44, 131–161 (2001) · Zbl 1003.35009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.