Geodesic
Stable methods for reconstruction of noisy images
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 9 (2011), pp. 32-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the numerical reconstruction of noisy images by using two regularization algorithms. The basis of algorithms is Tikhonov regularization with two special nondifferentiable stabilizers. To solve the problem of a nonsmooth minimizing the proximal method and the subgradient process are involved. The results of calculations on the supercomputer «Uranus» are presented.
Keywords: numerical methods, algorithms, ill-posed problems, inverse problems, iterative regularization, nonsmooth optimization. 
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T. I. Serezhnikova. Stable methods for reconstruction of noisy images. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 9 (2011), pp. 32-42. http://geodesic.mathdoc.fr/item/VYURU_2011_9_a3/

[1] V. V. Vasin, “Regularization and iterative approximation for linear ill-posed problems in the space of functions of bounded variation”, Proc. Steclov Inst. Math., 1, 2002, S225–S229 | MR

[2] V. V. Vasin, “Approksimatsiya negladkikh reshenii lineinykh nekorrektnykh zadach”, Tr. In-ta matematiki i mekhaniki UrO RAN, 12, no. 1, 2006, 64–77 | Zbl

[3] V. V. Vasin, T. I. Serezhnikova, “Dvukhetapnyi metod approksimatsii negladkikh reshenii i vosstanovlenie zashumlennogo izobrazheniya”, Avtomatika i telemekhanika, 2004, no. 2, 126–135 | MR | Zbl

[4] V. V. Vasin, T. I. Serezhnikova, “Regulyarnyi algoritm approksimatsii negladkikh reshenii dlya integralnykh uravnenii Fredgolma pervogo roda”, Vychislitelnye tekhnologii, 15:2 (2010), 15–23 | Zbl

[5] V. V. Vasin, Proksimalnyi algoritm s proektirovaniem v zadachakh vypuklogo programmirovaniya, Preprint, In–t matematiki i mekhaniki UNTs AN SSSR, Sverdlovsk, 1982 | MR

[6] A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnykh zadach, Nauka, M., 1976

[7] V. S. Sizikov, Matematicheskie metody obrabotki rezultatov izmerenii, Politekhnika, SPb., 2001, 240 pp. | MR

[8] A. S. Leonov, Reshenie nekorrektno postavlennykh obratnykh zadach: ocherk teorii, prakticheskie algoritmy i demonstratsii v MATLAB, Knizh. dom «Librokom», M., 2010, 336 pp.

[9] V. A. Belokurov, E. V. Shimanovskaya, M. V. Sazhin i dr., “Vosstanovlenie izobrazheniya gravitatsionnoi QSO linzy 2237+0305 «Krest Einshteina»”, Astronom. zhurnal, 78:10 (2001), 1–11

[10] A. B. Bakushinskii, V. S. Sizikov, “Nekotorye nestandartnye regulyarizuyuschie algoritmy i ikh chislennaya realizatsiya”, Zhurn. vychisl. matem. i matem. fiziki, 22:3 (1982), 532–539 | MR

[11] C. R. Vogel, Computational methods for inverse problems, SIAM, Philadelphia, 2002 | MR | Zbl

[12] S. Uebb (red.), Fizika vizualizatsii izobrazhenii v meditsine, v 2-kh t., per. s angl., v. 2, Mir, M., 1991

[13] E. Dzh. Cheisson, “Pervye rezultaty s kosmicheskogo teleskopa «Khabbl»”, V mire nauki, 1992, no. 8, 6–14

[14] G. Endryus, Primenenie vychislitelnykh mashin dlya obrabotki izobrazhenii, Energiya, M., 1977

[15] R. Ernst, Dzh. Bodenkhauzen, A. Vokaun, YaMR v odnom i dvukh izmereniyakh, Mir, M., 1990

[16] A. Kawanaka, M. Takagi, “Estimation of static magnetic field and gradient fields from NMR image”, J. Phys. Sci. Instrum., 19 (1986), 871–875 | DOI

[17] R. Beits, M. Mak-Donnell, Vosstanovlenie i rekonstruktsiya izobrazhenii, Mir, M., 1989

[18] R. T. Rockafellar, “Monotone operators and the proximal point algorithm”, SIAM J. Control and Optimization, 14:5 (1976), 871–898 | DOI | MR