×

zbMATH — the first resource for mathematics

Convex regularization of multi-channel images based on variants of the TV-model. (English) Zbl 1391.49020
Summary: We discuss existence and regularity results for multi-channel images in the setting of isotropic and anisotropic variants of the TV-model.
MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
PDF BibTeX Cite
Full Text: DOI
References:
[1] Weickert, J; Schnörr, C, A theoretical framework for convex regularizers in PDE-based computation of image motion, Int J Comput Vision, 45, 3, 245-264, (2001) · Zbl 0987.68600
[2] Ball, JM, Convexity conditions and existence theorems in nonlinear elasticity, Arch Ration Mech Anal, 63, 4, 337-403, (197677) · Zbl 0368.73040
[3] Ball, JM, Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear Anal Mech Heriot-Watt Symp (Edinburgh, 1976), 1, 187-241, (1977)
[4] Ball, JM, Differentiability properties of symmetric and isotropic functions, Duke Math J, 51, 3, 699-728, (1984) · Zbl 0566.73001
[5] Tikhonov, AN; Arsenin, VY, Solutions of Ill-posed problems, (1977), Wiley, Washington (DC)
[6] Bertero, M; Poggio, TA; Torre, V, Ill-posed problems in early vision, Proc IEEE, 76, 8, 869-889, (1988)
[7] Schnörr, C, Unique reconstruction of piecewise smooth images by minimizing strictly convex non-quadratic functionals, J Math Imaging Vision, 4, 189-198, (1994)
[8] Nordström, N, Biased anisotropic diffusion – a unified regularization and diffusion approach to edge detection, Image Vision Comput, 8, 318-327, (1990)
[9] Perona, P; Malik, J, Scale space and edge detection using anisotropic diffusion, IEEE Trans Pattern Anal Mach Intell, 12, 629-639, (1990)
[10] Gerig, G; Kübler, O; Kikinis, R, Nonlinear anisotropic filtering of MRI data, IEEE Trans Med Imaging, 11, 221-232, (1992)
[11] Tschumperlé, D; Deriche, R, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1, Diffusion tensor regularization with constraints preservation, 948-953, (2001), IEEE Computer Society Press, Kauai (HI)
[12] Weickert, J; Brox, T; Nashed, MZ; Scherzer, O, Contemporary mathematics, 313, Diffusion and regularization of vector- and matrix-valued images, 251-268, (2002), AMS, Providence (RI) · Zbl 1047.68143
[13] Rudin, LI; Osher, S; Fatemi, E, Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268, (1992) · Zbl 0780.49028
[14] Chambolle, A, Proceeding of the 1994 IEEE International Conference on Image Processing, 1, Partial differential equations and image processing, 16-20, (1994), IEEE Computer Society Press, Austin (TX)
[15] Duran, J; Moeller, M; Sbert, C, Collaborative total variation: a general framework for vectorial TV models, SIAM J Imaging Sci, 9, 1, 116-151, (2000) · Zbl 1381.94016
[16] Christiansen, O; Lee, T-M; Lie, J, Total variation regularization of matrix-valued images, Int J Biomed Imaging, 11 pages, (2007)
[17] Berkels, B; Burger, M; Droske, M; Kobbelt, L; Kuhlen, T; Aach, T, Vision modelling, and visualization, Cartoon extraction based on anisotropic image classification, 293-300, (2006), AKA, Berlin
[18] Burger, M; Osher, S; Burger, M; Mennuci, ACG; Osher, S, Lecture notes in mathematics, 2090, A guide to the TV zoo, 1-70, (2013), Springer, Cham
[19] Ambrosio, L; Fusco, N; Pallara, D, Functions of bounded variation and free discontinuity problems, (2000), Clarendon Press, Oxford · Zbl 0957.49001
[20] Giusti, E, Monographs in mathematics, 80, (1984), Birkhäuser, Basel
[21] Bildhauer, M; Fuchs, M, A variational approach to the denoising of images based on different variants of the TV-regularization, Appl Math Optim, 66, 3, 331-361, (2012) · Zbl 1260.49074
[22] Bildhauer, M; Fuchs, M, On some perturbations of the total variation image inpainting method part I: regularity theory, J Math Sci, 202, 2, 154-169, (2014) · Zbl 1321.49060
[23] Bildhauer, M; Fuchs, M, On some perturbations of the total variation image inpainting method part II: relaxation and dual variational formulation, J Math Sci, 205, 2, 121-140, (2015) · Zbl 1321.49054
[24] Fuchs, M; Tietz, C, Existence of generalized minimizers and of dual solutions for a class of variational problems with linear growth related to image recovery, J Math Sci, 210, 4, 458-475, (2015) · Zbl 1331.49014
[25] Fuchs, M; Müller, J, A higher order TV-type variational problem related to the denoising and inpainting of images, Nonlinear Anal Theor Meth Appl, 154, 122-147, (2017) · Zbl 1358.49043
[26] Tietz, C, Existence and regularity theorems for variants of the TV-image inpainting method in higher dimensions and with vector-valued data [PhD thesis], (2016)
[27] Adams, RA, Pure and applied mathematics, 65, (1975), Academic Press, New York (NY)
[28] Fuchs, M; Müller, J; Tietz, C, Signal recovery via TV-type energies, Technical Report No. 381, Department of Mathematics, Saarland University · Zbl 1392.49003
[29] Bildhauer, M; Fuchs, M; Tietz, C, C1,α -interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting, Algebra i Anal, 27, 3, 51-65, (2015)
[30] Bildhauer, M; Fuchs, M, A geometric maximum principle for variational problems in spaces of vector-valued functions of bounded variation, J Math Sci, 178, 3, 235-242, (2011) · Zbl 1319.49074
[31] Bildhauer, M; Fuchs, M, Partial regularity for a class of anisotropic variational integrals with convex hull property, Asymp Anal, 32, 293-315, (2002) · Zbl 1076.49018
[32] Bildhauer, M; Fuchs, M; Weickert, J, Denoising and inpainting of images using TV-type energies: theoretical and computational aspects, J Math Sci, 219, 6, 899-910, (2016) · Zbl 1381.94011
[33] Demengel, F; Temam, R, Convex functions of a measure and applications, Indiana Univ Math J, 33, 5, 673-709, (1984) · Zbl 0581.46036
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.