Geodesic
Research of a threshold (correlation) method and application for localization of singularities
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 829-848.

Voir la notice de l'article provenant de la source Math-Net.Ru

The localization of singularities for functions of one ($\delta$-functions and discontinuities of the first kind) and two (line of discontinuity) dimensions is discussed. General scheme of the study of this ill-posed problems is presented. Using this scheme new problems of localization of singularities are investigated.
Keywords: ill-posed problems, discontinuities of the first kind, line of discontinuity, localization of singularities, regularizing method, separation threshold. 
@article{SEMR_2016_13_a104,
     author = {D. V. Kurlikovskii and A. L. Ageev and T. V. Antonova},
     title = {Research of a threshold (correlation) method and application for localization of singularities},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {829--848},
     publisher = {mathdoc},
     volume = {13},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2016_13_a104/}
}
TY  - JOUR
AU  - D. V. Kurlikovskii
AU  - A. L. Ageev
AU  - T. V. Antonova
TI  - Research of a threshold (correlation) method and application for localization of singularities
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2016
SP  - 829
EP  - 848
VL  - 13
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2016_13_a104/
LA  - ru
ID  - SEMR_2016_13_a104
ER  - 
%0 Journal Article
%A D. V. Kurlikovskii
%A A. L. Ageev
%A T. V. Antonova
%T Research of a threshold (correlation) method and application for localization of singularities
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2016
%P 829-848
%V 13
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2016_13_a104/
%G ru
%F SEMR_2016_13_a104
D. V. Kurlikovskii; A. L. Ageev; T. V. Antonova. Research of a threshold (correlation) method and application for localization of singularities. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 829-848. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a104/

[1] V. Yu. Terebizh, “Image restoration with minimum a priori information”, Phys. Usp., 38:2 (1995), 137–167 | DOI

[2] V. Yu. Terebizh, Introduction to statistical theory of inverse problems, Phizmatlit, M., 2005 (in Russian)

[3] A. V. Goncharskii, A. M. Cherepaschuk, A. G. Yagola, Numerical methods of solving inverse problems of astrophysics, Nauka, M., 1978 (in Russian) | Zbl

[4] G. Winkler, O. Wittich, V. Liebscher, A. Kempe, “Don't shed tears over breaks”, Jahresber. Deutsch. Math.-Verein, 107:2 (2005), 57–87 | MR | Zbl

[5] C. G. M. Oudshoorn, “Asymptotically minimax estimation of a function with jumps”, Bernoulli, 4:1 (1998), 15–33 | DOI | MR | Zbl

[6] V. S. Sizikov, Mathematical methods for processing the results of measurements, Politekhnika, St. Petersburg, 2001 (in Russian) | MR

[7] E. Yu. Derevtsov, V. V. Pickalov, “Reconstruction of vector fields and their singularities from ray transform”, Numerical Analysis and Applications, 4:1 (2011), 21–35 | DOI | Zbl

[8] S. Mallat, A wavelet tour of signal processing: the sparse way, Academic Press, San Diego, CA, 1999 | MR | Zbl

[9] Ya. A. Furman et al. (eds.), Introduction to contour analysis and its applications to image and signal processing, Fizmatlit, M., 2002 (in Russian)

[10] A. N. Tikhonov, V. Y. Arsenin, Solutions of ill-posed problems, Wiley, New York, 1977 | MR | Zbl

[11] V. K. Ivanov, V. V. Vasin, V. P. Tanana, Theory of linear ill-posed problems and its applications, VSP, Utrecht, 2002 | MR | Zbl

[12] V. V. Vasin, A. L. Ageev, Ill-posed problems with a priori information, VSP, Utrecht, 1995 | MR | Zbl

[13] A. L. Ageev, T. V. Antonova, “On a new class of ill-posed problems”, Izv. Ural'sk. Gos. Univ., 58 (2008), 24–42 (in Russian) | MR | Zbl

[14] A. L. Ageev, T. V. Antonova, “On ill-posed problems of localization of singularities”, Tr. Inst. Mat. Mech. UrO RAN, 17, no. 3, 2011, 30–45 (in Russian)

[15] A. L. Ageev, T. V. Antonova, “Regularizing algorithms for detecting discontinuities in ill-posed problems”, Comput. Math. and Math. Phys., 48:8 (2008), 1284–1292 | DOI | MR | Zbl

[16] T. V. Antonova, “Regularizing algorithms for localization of breakpoints of noisy functions convection”, Proceedings of the Steklov Institute of Mathematics, 2, 2009, 24–39 | DOI | MR | Zbl

[17] T. V. Antonova, “New methods for localizing discontinuities of a noisy function”, Numerical Analysis and Application, 3:4 (2010), 306–316 | DOI | Zbl

[18] T. V. Antonova, “A method for localization of discontinuity lines of an approximately defined function of two variables”, Numerical Analysis and Application, 5:4 (2012), 285–296 | DOI | Zbl

[19] A. L. Ageev, T. V. Antonova, “Approximation of discontinuity lines of a noisy function of two variables”, Journal of Applied and Industrial Mathematics, 6:3 (2012) | DOI | MR

[20] A. L. Ageev, T. V. Antonova, “Localization algorithms for singularities of solutions to convolution equation of the first kind”, J. Inverse and Ill-Posed Problems, 16:7 (2008), 639–650 | DOI | MR | Zbl

[21] D. V. Kurlikovskii, “Methods of localization of discontinuities in solution of first kind equation of convolution type”, Russian Mathematics, 58:3 (2014), 60–63 | DOI | MR | Zbl

[22] V. I. Bogachev, Fundamentals of measure theory, v. I, Regular and Chaotic Dynamics, M.–Izhevsk, 2006 (in Russian)

[23] E. M. Stein, G. L. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971 | MR | Zbl