Mathematical analysis of a inf-convolution model for image processing. (English) Zbl 1332.65030
Summary: We deal with a second-order image decomposition model to perform denoising and texture extraction that was previously presented. We look for the decomposition of an image as the summation of three different order terms. For highly textured images, the model gives a two-scale texture decomposition: The first-order term can be viewed as a macro-texture (larger scale) which oscillations are not too large, and the zero-order term is the micro-texture (very oscillating) that contains the noise. Here, we perform mathematical analysis of the model and give qualitative properties of solutions using the dual problem and inf-convolution formulation.
MSC:
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 68U10 | Computing methodologies for image processing |
| 65K10 | Numerical optimization and variational techniques |
References:
| [1] | Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F |
| [2] | Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000) · Zbl 0957.49001 |
| [3] | Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Applications to PDEs and optimization (2006) · Zbl 1095.49001 |
| [4] | Evans, L.C., Gariepy, R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992) · Zbl 0804.28001 |
| [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] | Aujol, J.-F., Aubert, G., Blanc-Féraud, L., Chambolle, A.: Image decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vision 22(1), 71-88 (2005) · Zbl 1371.94031 · doi:10.1007/s10851-005-4783-8 |
| [7] | Osher, S., Sole, A., Vese, L.: Image decomposition and restoration using total variation minimization and the h \[^11\] norm. SIAM J. Multiscale Model. Simul. 1(3), 349-370 (2003) · Zbl 1051.49026 · doi:10.1137/S1540345902416247 |
| [8] | 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 |
| [9] | Osher, S., Vese, L.: Image denoising and decomposition with total variation minimization and oscillatory functions. Special issue on mathematics and image analysis. J. Math. Imaging Vision 20(1-2), 7-18 (2004) · Zbl 1366.94072 |
| [10] | Yin, W., Goldfarb, D., Osher, S.: A comparison of three total variation based texture extraction models. J. Vis. Commun. Image 18, 240-252 (2007) · doi:10.1016/j.jvcir.2007.01.004 |
| [11] | Bergounioux, M., Piffet, L.: A second-order model for image denoising. Set-Valued Var. Anal. 18(3-4), 277-306 (2010) · Zbl 1203.94006 · doi:10.1007/s11228-010-0156-6 |
| [12] | Bergounioux, M., Piffet, L.: A full second order variational model for multiscale texture analysis. Comput. Optim. Appl. 54, 215-237 (2013) · Zbl 1291.90235 · doi:10.1007/s10589-012-9484-9 |
| [13] | Demengel, F.: Fonctions à hessien borné. Annales de l’institut Fourier 34(2), 155-190 (1984) · Zbl 0525.46020 · doi:10.5802/aif.969 |
| [14] | Bergounioux, M.: On poincaré-wirtinger inequalities in bv - spaces. Control Cybern 4(40), 921-929 (2011) · Zbl 1318.46019 |
| [15] | Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492-526 (2010) · Zbl 1195.49025 · doi:10.1137/090769521 |
| [16] | Bredies, K., Kunisch, K., Valkonen, T.: Properties of \[l^1\] l \[1-{\rm tgv}^2\] tgv2: the one-dimensional case. J. Math. Anal. Appl. 398(1), 438-454 (2013) · Zbl 1253.49024 · doi:10.1016/j.jmaa.2012.08.053 |
| [17] | Bredies, K., Holler, M.: Regularization of linear inverse problems with total generalized variation. J. Inverse Ill Probl. 22(6), 871-913 (2014) · Zbl 1302.65167 |
| [18] | Ambrosio, L., Faina, L., March, R.: Variational approximation of a second order free discontinuity problem in computer vision. SIAM J. Math. Anal. 32(6), 1171-1197 (2001) · Zbl 0996.46014 · doi:10.1137/S0036141000368326 |
| [19] | Carriero, M., Leaci, A., Tomarelli, F.: Uniform density estimates for the blake & zisserman functional. Discrete Contin. Dyn. Syst. 31(4), 1129-1150 (2011) · Zbl 1242.49043 · doi:10.3934/dcds.2011.31.1129 |
| [20] | Carriero, M., Leaci, A., Tomarelli, F.: Free gradient discontinuity and image inpainting. J. Math. Sci. 181(6), 805-819 (2012) · Zbl 1260.49025 |
| [21] | Meyer, Y.: Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, volume 22 of University Lecture Series. AMS, (2001) · Zbl 0987.35003 |
| [22] | Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167-188 (1997) · Zbl 0874.68299 · doi:10.1007/s002110050258 |
| [23] | Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, Partial Differential Equations and the Calculus of Variations, volume 147 of Applied Mathematical Sciences. Springer, Berlin (2006) · Zbl 1110.35001 |
| [24] | Ziemer, W.P.: Weakly Differentiable Functions - Sobolev Space and Functions of Bounded Variation. Springer, Berlin (1980) |
| [25] | Carriero, M., Leaci, A., Tomarelli, F.: Special bounded hessian and elastic-plastic plate. Rend. Ac. Naz. Sci. XV I(13), 223-258 (1992) · Zbl 0829.49014 |
| [26] | Moreau, J.-J.: Inf-convolution des fonctions numériques sur un espace vectoriel. C. R. Acad. Sci. Paris 256, 50475049 (1963) · Zbl 0137.31303 |
| [27] | Hiriart-Urruty, J.-B., Phelps, R.R.: Subdifferential calculus using \[\varepsilon\] ε -subdifferentials. J. Funct. Anal 118, 150-166 (1993) · Zbl 0791.49017 |
| [28] | Ekeland, I., Temam, R.: Convex analysis and variational problems. In: Classics in Applied Mathematics, vol. 1. Society for Industrial and Applied Mathematics (1999) · Zbl 0939.49002 |
| [29] | Chambolle, A.: An algorithm for total variation minimization and applications. J.Math. Imaging Vis. 20, 89-97 (2004) · Zbl 1366.94048 · doi:10.1023/B:JMIV.0000011320.81911.38 |
| [30] | Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the tv denoising problem and some extensions. SIAM Multiscale Model. Simul. 6(3), 879-894 (2007) · Zbl 1145.49024 · doi:10.1137/070683003 |
| [31] | Bergounioux, M.: Inf-convolution model : numerical experiment. Technical report, hal.archives-ouvertes.fr/hal-01002958v2, (2014) · Zbl 1018.49021 |
| [32] | Ring, W.: Structural properties of solutions of total variation regularization problems. ESAIM Math Model. Numer. Anal. 34, 799-840 (2000) · Zbl 1018.49021 · doi:10.1051/m2an:2000104 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
