×

zbMATH — the first resource for mathematics

An alternative approach towards the higher order denoising of images. analytical aspects. (English) Zbl 1384.35019
J. Math. Sci., New York 224, No. 3, 414-441 (2017) and Zap. Nauchn. Semin. POMI 444, 47-88 (2016).
Usually, pattern recognition systems work via the minimization of a cost function with a weighted sum of a quadratic error term together with the variation of the error term. Here one proposes to rather replace the last term by the Hessian matrix of the control function. It is shown that this new problem has one solution.

MSC:
35J20 Variational methods for second-order elliptic equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] Acar, R; Vogel, CR, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10, 1217-1229, (1994) · Zbl 0809.35151
[2] R. A. Adams, Sobolev Spaces, Academic Press, San Diego (1975). · Zbl 0314.46030
[3] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, Clarendon, Oxford (2000). · Zbl 0957.49001
[4] Aubert, G; Vese, L, A variational method in image recovery, SIAM J. Numer. Anal., 34, 1948-1979, (1997) · Zbl 0890.35033
[5] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer, New York (2006). · Zbl 1110.35001
[6] P. N. Belhumeur, “A binocular stereo algorithm for reconstructing sloping, creased, and broken surfaces in the presence of half-occlusion,” in: Proc. Fourth International Conference on Computer Vision (1993), pp. 431-438. · Zbl 1003.35009
[7] M. Bildhauer, “Convex variational problems: linear, nearly linear and anisotropic growth conditions,” Lect. Notes Math., 1818 (2003). · Zbl 1033.49001
[8] Bildhauer, M; Fuchs, M, Two-dimensional anisotropic variational problems, Calc. Variations, 16, 177-186, (2003) · Zbl 1026.49027
[9] Bildhauer, M; Fuchs, M, A variational approach to the denoising of images based on different variants of the TV-regularization, Appl. Math. Optim., 66, 331-361, (2012) · Zbl 1260.49074
[10] Bildhauer, M; Fuchs, M, On some pertubations of the total variation image inpainting method. part I: regularity theory, J. Math. Sci., 202, 154-169, (2013) · Zbl 1321.49060
[11] Bildhauer, M; Fuchs, M; Tietz, C, \(C\)\^{}{1}-interior regularity for minimizers of a class of variational problems related to image inpainting, Algebra Analiz, 27, 51-65, (2015) · Zbl 1335.49058
[12] Bildhauer, M; Fuchs, M; Zhong, X, A lemma on the higher integrability of functions with applications to the regularity theory of two dimensional generalized Newtonian fluids, Manus. Math., 116, 135-156, (2005) · Zbl 1116.49018
[13] P. Blomgren, T. F. Chan, P. Mulet, L. Vese, and W. L. Wan, “Variational PDE models and methods for image processing,” in: Res. Notes Math. (2000), pp. 43-67. · Zbl 0953.68621
[14] Bredies, K; Kunisch, K; Pock, T, Total generalized variation, SIAM J. Imaging Sci., 3, 492-526, (2010) · Zbl 1195.49025
[15] Bredies, K; Kunisch, K; Valkonen, T, Properties of \(L\)\^{}{1}-TVG\^{}{2}: the one-dimensional case, J. Math. Analysis Appl., 398, 438-454, (2013) · Zbl 1253.49024
[16] K. Bredies and T. Valkonen, “Inverse problems with second-order total generalized variation constraints,” in: Proc. 9th International Conference on Sampling Theory and Applications, Singapore (2011).
[17] Brito-Loeza, C; Chen, K, On high-order denoising models and fast algorithms for vector-valued images, IEEE Trans. Image Processing, 19, 1518-1526, (2010) · Zbl 1371.94060
[18] M. Burger, K. Papafitsoros, E. Papoutsellis, and C.-B. Schönlieb, “Infimal convolution regularisation functionals of BV and \(L\)\^{}{\(p\)} spaces. Part I: The finite \(p\) case,” arXiv:1504.01956 [math.NA], 1-32 (2015). · Zbl 1116.49018
[19] M. Burger, K. Papafitsoros, E. Papoutsellis, and C.-B. Schönlieb, “Infimal convolution regularisation functionals of BV and \(L\)\^{}{\(p\)} spaces. The case \(p\) = \(∞\),” arXiv:1510.09032 [math.NA], 1-10 (2015). · Zbl 0539.35027
[20] 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
[21] Chambolle, A; Lions, P-L, Image recovery via total variation minimization and related problems, Numer. Math., 76, 167-188, (1997) · Zbl 0874.68299
[22] T. F. Chan, S. Esedoglu, and F. E. Park, “A fourth order dual method for staircase reduction in texture extraction and image restoration problems,” Technical Report CAM-05-28, University of California at Los Angeles (2005). · Zbl 1144.35401
[23] T. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia (2005). · Zbl 1095.68127
[24] 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
[25] Benedetto, E, \(C\)\^{}{1+\(α\)} local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 827-850, (1983) · Zbl 0539.35027
[26] Didas, S; Weickert, J; Burgeth, B, Properties of higher order nonlinear diffusion filtering, J. Math. Imaging Vision, 35, 208-226, (2009) · Zbl 1171.68788
[27] D. Ferstl, C. Reinbacher, R. Ranftl, M. Rüther, and H. Bischof, “Image guided depth up-sampling using anisotropic total generalized variation,” in: Proc. International Conference on Computer Vision, Sydney, Australia (2013), pp. 993-1000. · Zbl 0874.68299
[28] Frehse, J, Two dimensional variational problems with thin obstacles, Math. Z., 143, 279-288, (1975) · Zbl 0295.49003
[29] Frehse, J; Seregin, G, Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening, Transl. AMS, 193, 127-152, (1999)
[30] Fuchs, M, Computable upper bounds for the constants in Poincarè-type inequalities for fields of bounded deformation, Math. Meth. Appl. Sci., 34, 1920-1932, (2011) · Zbl 1403.74017
[31] Fuchs, M; Repin, S, A posteriori error estimates for the approximations of the stresses in the hencky plasticity problem, Numer. Funct. Anal. Optim., 32, 610-640, (2011) · Zbl 1419.74076
[32] M. Fuchs and G. Seregin, “Variational methods for problems from plasticity theory and for generalized Newtonian fluids,” Lect. Notes Math., 1749 (2000). · Zbl 0964.76003
[33] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel (1984). · Zbl 0545.49018
[34] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin (1989). · Zbl 0691.35001
[35] Greer, JB; Bertozzi, AL, Traveling wave solutions of fourth order PDE’s for image processing, SIAM J. Math. Anal., 36, 38-68, (2004) · Zbl 1082.35080
[36] B. M. ter Haar Romeny (ed.), Geometry-driven Diffusion in Computer Vision, Kluwer, Dordrecht (1994). · Zbl 0832.68111
[37] D. Hafner, C. Schroers, and J. Weickert, “Introducing maximal anisotropy into second order coupling models,” in: Lect. Notes Comp. Sci., Springer, Berlin (2015), pp. 79-90. · Zbl 1116.49018
[38] A. Hewer, J. Weickert, T. Scheffer, H. Seibert, and S. Diebels, “Lagrangian strain tensor computation with higher order and variational models,” in: Proc. 24th British Machine Vision Conference, BMVA Press, Bristol (2013). · Zbl 1371.94060
[39] Horn, BKP, Height and gradient from shading, Intern. J. Comput. Vision, 5, 37-75, (1990)
[40] Kawohl, B, Variational versus PDE-based approaches in mathematical image processing, CRM Proc. Lect. Notes, 44, 113-126, (2008) · Zbl 1144.35401
[41] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Nauka, Moskow (1964). · Zbl 0143.33602
[42] Lysaker, M; Lundervold, A; Tai, X-C, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Processing, 12, 1579-1590, (2003) · Zbl 1286.94020
[43] C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations. Reprint of the 1966 Edition, Springer-Verlag, Berlin (2008). · Zbl 1260.49074
[44] Mosolov, PP; Mjasnikov, VP, On well-posedness of boundary value problems in the mechanics of continuous media, Mat. Sb., 88, 256-284, (1972)
[45] Papafitsoros, K; Bredies, K, A study of the one dimensional total generalised variation regularisation problem, Inverse Probl. Imag., 9, 511-550, (2015) · Zbl 1336.49044
[46] Perona, P; Malik, J, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intel., 12, 629-639, (1990)
[47] Rudin, L; Osher, S; Fatemi, E, Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268, (1992) · Zbl 0780.49028
[48] Scherzer, O, Denoising with higher order derivatives of bounded variation and an application to parameter estimation, Computing, 60, 1-28, (1998) · Zbl 0891.65103
[49] Setzer, S; Steidl, G; Teuber, T, Infimal convolution regularizations with discrete \(ℓ\)_{1}-type functionals, Comm. Math. Sci., 9, 797-827, (2011) · Zbl 1269.49063
[50] Strang, G; Temam, R, Functions of bounded deformation, Arch. Rat. Mech. Anal., 75, 7-21, (1981) · Zbl 0472.73031
[51] Suquet, PM, Sur une nouveau cadre fonctionnel pour LES équations de la plasticité, C. R. A. S. Paris. Ser A, 286, 1129-1132, (1978) · Zbl 0378.35056
[52] P. M. Suquet, “Un espace fonctionnel pour leséquations de la plasticité,” Ann. Fac. Sci. Toulouse, Ser. 5, 1, 77-87 (1979). · Zbl 0405.46027
[53] Tolksdorf, P, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., 134, 241-266, (1983) · Zbl 0538.35034
[54] Vese, L, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim., 44, 131-161, (2001) · Zbl 1003.35009
[55] J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart (1998). · Zbl 0886.68131
[56] You, Y-L; Kaveh, M, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9, 1723-1730, (2000) · Zbl 0962.94011
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.