Geodesic
On the localization of fractal discontinuity lines from noisy data
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2023), pp. 27-44.

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We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables: the function is smooth outside the discontinuity lines, and at each point on the line it has a discontinuity of the first kind. We construct averaging procedures and study global discrete regularizing algorithms for approximating discontinuity lines. Lipschitz conditions are imposed on the discontinuity lines. A parametric family of fractal lines is constructed, for which all conditions can be checked analytically. A fractal is indicated that has a large fractal dimension, for which the efficiency of the constructed methods can be guaranteed.
Keywords: ill-posed problems, regularization method, discontinuity lines, global localization, discretization, Lipschitz condition.
Mots-clés : fractal 
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A. L. Ageev; T. V. Antonova. On the localization of fractal discontinuity lines from noisy data. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2023), pp. 27-44. http://geodesic.mathdoc.fr/item/IVM_2023_9_a2/

[1] Tikhonov A.N., Arsenin V.Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979

[2] Ivanov V.K., Vasin V.V., Tanana V.P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978

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

[4] Malla S., Veivlety v obrabotke signalov, Mir, M., 2005

[5] Vvedenie v konturnyi analiz i ego prilozheniya k obrabotke izobrazhenii i signalov, Fizmatlit, M., 2002

[6] Gonsales R., Vuds R., Tsifrovaya obrabotka izobrazhenii, Izd. $3$-e ispr. i dop., Tekhnosfera, M., 2012

[7] Antonova T.V., “Metod lokalizatsii linii razryva priblizhenno zadannoi funktsii dvukh peremennykh”, Sib. zhurn. vychisl. matem., 15:4 (2012), 345–357 | Zbl

[8] Ageev A.L., Antonova T.V., “Approksimatsiya linii razryva zashumlennoi funktsii dvukh peremennykh”, Sib. zhurn. industr. matem., 15:1(49) (2012), 3–13 | Zbl

[9] Ageev A.L., Antonova T.V., “Diskretnyi algoritm lokalizatsii linii razryva funktsii dvukh peremennykh”, Sib. zhurn. industr. matem., 20:4 (72) (2017), 3–12 | Zbl

[10] Ageev A.L., Antonova T.V., “K voprosu o globalnoi lokalizatsii linii razryva funktsii dvukh peremennykh”, Tr. IMM UrO RAN, 24, no. 2, 2018, 12–23

[11] Ageev A.L., Antonova T.V., “O lokalizatsii negladkikh linii razryva funktsii dvukh peremennykh”, Tr. IMM UrO RAN, 25, no. 3, 2019, 9–23

[12] Ageev A.L., Antonova T.V., “Novye otsenki tochnosti metodov lokalizatsii linii razryva zashumlennoi funktsii”, Sib. zhurn. vychisl. matem., 23:4 (2020), 351–364

[13] Kronover R.M., Fraktaly i khaos v dinamicheskikh sistemakh. Osnovy teorii, Postmarket, M., 2000

[14] Fikhtengolts G.M., Kurs differentsialnogo i integralnogo ischisleniya, v. 1, 8-e izd., Fizmatlit, M., 2003