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A second-order model for image denoising. (English) Zbl 1203.94006
Summary: We present a variational model for image denoising and/or texture identification. Noise and textures may be modelled as oscillating components of images. The model involves a $L^{2}$-data fitting term and a Tychonov-like regularization term. We choose the $BV^{2}$ norm instead of the classical $BV$ norm. Here $BV^{2}$ is the bounded hessian function space that we define and describe. The main improvement is that we do not observe staircasing effects any longer, during denoising process. Moreover, texture extraction can be performed with the same method. We give existence results and present a discretized problem. An algorithm close to the one set by {\it A. Chambolle} [An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89--97 (2004; \url{doi:10.1023/B:JMIV.0000011320.81911.38})] is used: we prove convergence and present numerical tests.

94A08Image processing (compression, reconstruction, etc.)
65D18Computer graphics, image analysis, and computational geometry
68U10Image processing (computing aspects)
65K10Optimization techniques (numerical methods)
Full Text: DOI
[1] Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10(6), 1217--1229 (1994) · Zbl 0809.35151 · doi:10.1088/0266-5611/10/6/003
[2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford University Press (2000) · Zbl 0957.49001
[3] Attouch, H., Briceño-Arias, L.M., Combettes, P.L.: A parallel splitting method for coupled monotone inclusions. SIAM J. Control Optim. 48, 3246 (2010) · Zbl 1218.47089 · doi:10.1137/090754297
[4] Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in sobolev and BV spaces: applications to PDEs and optimization. MPS-SIAM series on optimization. Philadelphia, ISBN 0-89871-600-4 (2006) · Zbl 1095.49001
[5] Aubert, G., Aujol, J.F.: Modeling very oscillating signals, application to image processing. Appl. Math. Optim. 51(2), 163--182 (2005) · Zbl 1162.49306 · doi:10.1007/s00245-004-0812-z
[6] Aubert, G., Aujol, J.F., Blanc-Feraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vis. 22(1), 71--88 (2005) · Zbl 02225030 · doi:10.1007/s10851-005-4783-8
[7] Aubert, G., Kornprobst, P.: Mathematical problems in image processing, partial differential equations and the calculus of variations. Applied Mathematical Sciences, vol. 147. Springer Verlag (2006) · Zbl 1110.35001
[8] Aujol, J.F.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34, 307--327 (2009) · Zbl 1287.94012 · doi:10.1007/s10851-009-0149-y
[9] Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89--97 (2004) · Zbl 02060335 · doi:10.1023/B:JMIV.0000011320.81911.38
[10] Demengel, F.: Fonctions à hessien borné. Annales de l’institut Fourier, Tome 34(2), 155--190 (1984) · Zbl 0525.46020
[11] Echegut, R., Piffet, L.: A variational model for image texture identification (preprint). http://hal.archives-ouvertes.fr/hal-00439431/fr/
[12] Ekeland, I., Temam, R.: Convex Analysis and Variational problems. SIAM Classic in Applied Mathematics, vol. 28 (1999) · Zbl 0281.49001
[13] Fadili, J., Peyré, G.: Total variation projection with first order schemes (preprint). http://hal.archives-ouvertes.fr/hal-00380491/fr/
[14] Hinterberger, W., Scherzer, O.: Variational methods on the space of functions of bounded Hessian for convexification and denoising. Computing 76(1--2), 109--133 (2006) · Zbl 1098.49022 · doi:10.1007/s00607-005-0119-1
[15] Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. University Lecture Series, vol. 22. AMS (2002)
[16] Osher, S., Fatemi, E., Rudin L.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259--268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[17] Osher, S., Sole, A., Vese L.: Image decomposition and restoration using total variation minimization and the H 1 norm. SIAM J. Multiscale Model. Simul. 1(3), 349--370 (2003) · Zbl 1051.49026 · doi:10.1137/S1540345902416247
[18] Osher, S., Vese, L.: Modeling textures with total variation minimization and oscillating patterns in image processing. J. Sci. Comput. 19(1--3), 553--572 (2003) · Zbl 1034.49039 · doi:10.1023/A:1025384832106
[19] Osher, S.J., Vese, L.A.: Image denoising and decomposition with total variation minimization and oscillatory functions. Special issue on mathematics and image analysis. J. Math. Imaging Vis. 20(1--2), 7--18 (2004)
[20] Piffet, L.: Modèles variationnels pour l’extraction de textures 2D. Ph.D. Thesis, Orléans (2010)
[21] Weiss, P., Blanc-Féraud, L., Aubert, G.: Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31(n{$\deg$}3), 2047--2080 (2009) · Zbl 1191.94029 · doi:10.1137/070696143
[22] Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. J. Vis. Commun. Image 18, 240--252 (2007) · Zbl 05461642 · doi:10.1016/j.jvcir.2007.01.004