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ЖУРНАЛЫ // Записки научных семинаров ПОМИ

Зап. научн. сем. ПОМИ, 2016, том 444, страницы 47–88 (Mi znsl6268)

An alternative approach towards the higher order denoising of images. Analytical aspects
M. Bildhauer, M. Fuchs, J. Weickert

Литература

1. R. Acar, C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems”, Inverse Problems, 10 (1994), 1217–1229  crossref  mathscinet  zmath  isi
2. R. A. Adams, Sobolev Spaces, Academic Press, San Diego, 1975  mathscinet  zmath
3. L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Oxford Science Publications, Clarendon, Oxford, 2000  mathscinet  zmath
4. G. Aubert, L. Vese, “A variational method in image recovery”, SIAM J. Numer. Anal., 34:5 (1997), 1948–1979  crossref  mathscinet  zmath  isi
5. G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer, New York, 2006  mathscinet  zmath
6. P. N. Belhumeur, “A binocular stereo algorithm for reconstructing sloping, creased, and broken surfaces in the presence of half-occlusion”, Proc. Fourth International Conference on Computer Vision (Berlin, May 1993), IEEE Computer Society Press, 431–438
7. M. Bildhauer, Convex variational problems: linear, nearly linear and anisotropic growth conditions, Lect. Notes Math., 1818, Springer, Berlin–Heidelberg–New York, 2003  crossref  mathscinet  zmath
8. M. Bildhauer, M. Fuchs, “Twodimensional anisotropic variational problems.”, Calc. Variations, 16 (2003), 177–186  mathscinet  zmath
9. M. Bildhauer, M. Fuchs, “A variational approach to the denoising of images based on different variants of the TV-regularization”, Appl. Math. Optim., 66:3 (2012), 331–361  crossref  mathscinet  zmath  isi  elib
10. M. Bildhauer, M. Fuchs, “On some pertubations of the total variation image inpainting method. Part I: regularity theory”, J. Math. Sciences, 202:2 (2013), 154–169  crossref  mathscinet
11. M. Bildhauer, M. Fuchs, C. Tietz, “$C^{1,\alpha}$-interior regularity for minimizers of a class of variational problems related to image inpainting”, Algebra Analiz, 27:3 (2015), 51–65  mathnet
12. M. Bildhauer, M. Fuchs, X. Zhong, “A lemma on the higher integrability of functions with applications to the regularity theory of two dimensional generalized Newtonian fluids”, Manus. Math., 116:2 (2005), 135–156  crossref  mathscinet  zmath
13. P. Blomgren, T. F. Chan, P. Mulet, L. Vese, W. L. Wan, “Variational PDE models and methods for image processing”, Numerical Analysis 1999 (Dundee), Chapman & Hall/CRC Res. Notes Math., 420, Chapman & Hall/CRC, Boca Raton, FL, 2000, 43–67  mathscinet  zmath
14. K. Bredies, K. Kunisch, T. Pock, “Total generalized variation”, SIAM J. Imaging Sci., 3 (2010), 492–526  crossref  mathscinet  zmath  isi
15. K. Bredies, K. Kunisch, T. Valkonen, “Properties of $L^1$-TVG$^2$: The one-dimensional case”, J. Math. Analysis Appl., 398 (2013), 438–454  crossref  mathscinet  zmath
16. K. Bredies, T. Valkonen, “Inverse problems with second-order total generalized variation constraints”, Proc. 9th International Conference on Sampling Theory and Applications, Singapore, 2011
17. C. Brito-Loeza, K. Chen, “On high-order denoising models and fast algorithms for vector-valued images”, IEEE Transactions on Image Processing, 19 (2010), 1518–1526  crossref  mathscinet  isi
18. M. Burger, K. Papafitsoros, E. Papoutsellis, C.-B. Schönlieb, Infimal convolution regularisation functionals of $BV$ and $L^p$ spaces. Part I: The finite $p$ case, April 2015, arXiv: 1504.01956[math.NA]; Journal of Mathematical Imaging and Vision (to appear)  mathscinet
19. M. Burger, K. Papafitsoros, E. Papoutsellis, C.-B. Schönlieb, Infimal convolution regularisation functionals of $BV$ and $L^p$ spaces. The case $p=\infty$, Oct. 2015, arXiv: 1510.09032[math.NA]  mathscinet
20. V. Caselles, A. Chambolle, M. Novaga, “Regularity for solutions of the total variation denoising problem”, Rev. Mat. Iberoam., 27 (2011), 233–252  crossref  mathscinet  zmath  isi
21. A. Chambolle, P.-L. Lions, “Image recovery via total variation minimization and related problems”, Numer. Math., 76 (1997), 167–188  crossref  mathscinet  zmath  isi
22. T. F. Chan, S. Esedoglu, F. E. Park, A fourth order dual method for staircase reduction in texture extraction and image restoration problems, Technical Report CAM-05-28, Dept. of Mathematics, University of California at Los Angeles, LA, 2005
23. T. Chan, J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, Philadelphia, 2005  mathscinet  zmath
24. Y. Chen, S. Levine, M. Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM J. Appl. Math., 66 (2006), 1383–1406  crossref  mathscinet  zmath  isi
25. E. Di Benedetto, “$C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations”, Nonlinear Anal., 7 (1983), 827–850  crossref  mathscinet
26. S. Didas, J. Weickert, B. Burgeth, “Properties of higher order nonlinear diffusion filtering”, J. Math. Imaging Vision, 35 (2009), 208–226  crossref  mathscinet  isi
27. D. Ferstl, C. Reinbacher, R. Ranftl, M. Rüther, H. Bischof, “Image guided depth upsampling using anisotropic total generalized variation”, Proc. International Conference on Computer Vision, Sydney, Australia, Dec. 2013, 993–1000
28. J. Frehse, “Two dimensional variational problems with thin obstacles”, Math. Z., 143 (1975), 279–288  crossref  mathscinet  zmath
29. J. Frehse, G. Seregin, “Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening”, Transl. Am. Math. Soc., 193 (1999), 127–152  mathscinet
30. M. Fuchs, “Computable upper bounds for the constants in Poincarè-type inequalities for fields of bounded deformation”, Math. Meth. Appl. Sciences, 34:15 (2011), 1920–1932  crossref  mathscinet  zmath  elib
31. M. Fuchs, S. Repin, “A posteriori error estimates for the approximations of the stresses in the Hencky plasticity problem”, Numer. Funct. Anal. Optim., 32:6 (2011), 610–640  crossref  mathscinet  zmath  isi  elib
32. M. Fuchs, G. Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids, Lect. Notes Math., 1749, Springer, Berlin–Heidelberg, 2000  crossref  mathscinet  zmath
33. E. Giusti, Minimal Surfaces and Functions of Bounded Variation., Monographs in Mathematics, 80, Birkhäuser, Basel, 1984  mathscinet  zmath
34. D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Math. Wiss., 224, 2nd edn., Springer, Berlin, 1989
35. J. B. Greer, A. L. Bertozzi, “Traveling wave solutions of fourth order PDEs for image processing”, SIAM J. Math. Analysis, 36 (2004), 38–68  crossref  mathscinet  zmath  isi
36. ter Haar Romeny B. M. (Ed.), Geometry-driven Diffusion in Computer Vision, Kluwer, Dordrecht, 1994  mathscinet  zmath
37. D. Hafner, C. Schroers, J. Weickert, “Introducing maximal anisotropy into second order coupling models”, Pattern Recognition, Lecture Notes in Computer Science, 9358, eds. Gall J., Gehler P., Leibe B., Springer, Berlin, 2015, 79–90  crossref  mathscinet
38. A. Hewer, J. Weickert, T. Scheffer, H. Seibert, S. Diebels, “Lagrangian strain tensor computation with higher order variational models”, Proc. 24th British Machine Vision Conference, eds. Burghardt T., Damen D., Mayol-Cuevas W., Mirmehdi M., BMVA Press, Bristol, UK, Sept. 2013
39. B. K. P. Horn, “Height and gradient from shading”, International J. Computer Vision, 5:1 (1990), 37–75  crossref  isi
40. B. Kawohl, “Variational versus PDE-based approaches in mathematical image processing”, CRM Proceedings and Lecture Notes, 44 (2008), 113–126  mathscinet  zmath  isi
41. O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Nauka, Moscow, 1964  zmath; English translation: Academic Press, New York, 1968  zmath
42. M. Lysaker, A. Lundervold, X.-C. Tai, “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time”, IEEE Transactions on Image Processing, 12 (2003), 1579–1590  crossref  zmath  isi
43. C. B. Morrey (Jr.), Multiple integrals in the calculus of variations, Reprint of the 1966 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2008  mathscinet  zmath
44. P. P. Mosolov, V. P. Mjasnikov, “On well-posedness of boundary value problems in the mechanics of continuous media”, Mat. Sbornik, 88(130):2 (1972), 256–267  mathnet  mathscinet  zmath; Engl. translation: Math. UssR Sbornik, 17:2 (1972), 257–268  crossref
45. K. Papafitsoros, K. Bredies, “A study of the one dimensional total generalised variation regularisation problem”, Inverse Problems and Imaging, 9 (2015), 511–550  crossref  mathscinet  zmath  isi
46. P. Perona, J. Malik, “Scale space and edge detection using anisotropic diffusion”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629–639  crossref  isi
47. L. Rudin, S. Osher, E. Fatemi, “Nonlinear total variation based noise removal algorithms”, Physica D, 60 (1992), 259–268  crossref  mathscinet  zmath  isi
48. O. Scherzer, “Denoising with higher order derivatives of bounded variation and an application to parameter estimation”, Computing, 60 (1998), 1–28  crossref  mathscinet  isi
49. S. Setzer, G. Steidl, T. Teuber, “Infimal convolution regularizations with discrete $\ell_1$-type functionals”, Communications in Mathematical Sciences, 9 (2011), 797–827  crossref  mathscinet  zmath  isi
50. G. Strang, R. Temam, “Functions of bounded deformation”, Arch. Rat. Mech. Anal., 75 (1981), 7–21  mathscinet
51. P. M. Suquet, “Sur une nouveau cadre fonctionnel pour les équations de la plasticité”, C. R. Acad. Sc. Paris (A), 286 (1978), 1129–1132  mathscinet  zmath
52. P. M. Suquet, “Un espace fonctionnel pour les équations de la plasticité”, Annales Faculté Sc. Toulouse $5^e$ sér., 1 (1979), 77–87  crossref  mathscinet  zmath
53. P. Tolksdorf, “Everywhere-regularity for some quasilinear systems with a lack of ellipticity”, Ann. Mat. Pura Appl., 134 (1983), 241–266  crossref  mathscinet  zmath  isi
54. L. Vese, “A study in the $\mathrm{BV}$ space of a denoising-deblurring variational problem”, Appl. Math. Optim., 44 (2001), 131–161  crossref  mathscinet  zmath  isi
55. J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart, 1998  mathscinet  zmath
56. Y.-L. You, M. Kaveh, “Fourth-order partial differential equations for noise removal.”, IEEE Transactions on Image Processing, 9 (2000), 1723–1730  crossref  mathscinet  zmath  isi


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