The problem of localization (determination of the position) of discontinuity lines for a function of two variables is considered. It is assumed that the function is smooth outside the discontinuity lines, and at each point on the lines it experiences a discontinuity of the first kind. The exact function is unknown, but information about the perturbed function is known. The case is considered where the perturbations are such that the problem is ill-posed. A new class of regularizing methods with an adaptive averaging domain is constructed. A well-posedness class containing fractal discontinuity lines is introduced, on which the problem is studied. Since the boundaries of natural objects in images are typically fractal in nature, studying methods on such lines is important. A geometric fractal with high Hausdorff dimension is constructed, for which all conditions on the discontinuity line can be verified. Guaranteed estimates of the accuracy of localization of discontinuity lines on the introduced correctness class are obtained.
Keywords: ill-posed problems, regularization method, discontinuity line, global localization, discretization, Lipschitz condition, fractal, Hausdorff dimension
Received January 19, 2026
Revised January 26, 2026
Accepted February 2, 2026
Alexander Leonidivich Ageev, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: ageev@imm.uran.ru
Tatiana Vladimirovna Antonova, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: tvantonova@imm.uran.ru
REFERENCES
1. Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach [Methods for the solution of ill-posed problems]. Moscow: Nauka Publ., 1979, 288 p.
2. Vasin V.V., Ageev A.L. Ill-posed problems with a priori information. Utrecht: VSP, 1995, 255 с. ISBN: 9789067641913 .
3. Mallat S. A wavelet tour of signal processing: the sparse way. NY: Acad. Press, 1999, 620 p. ISBN: 0-12-466606-X . Translated to Russian under the title Veivlety v obrabotke signalov, Moscow, Mir Publ., 2005, 671 p.
4. Gonzalez R.C., Woods R.E. Digital image processing (3rd Edition). Upper Saddle River, NJ: Pearson Prentice Hall, 2006, 976 p. ISBN: 978-0131687288 . Translated to Russian under the title Tsifrovaya obrabotka izobrazhenii. (Izdanie 3-e ispravlennoe i dopolnennoe), Moscow: Tekhnosfera Publ., 2012, 1104 p.
5. Mafi M., Rajaei H., Cabrerizo M., Adjouadi M. A robust edge detection approach un the presence of high impulse noise intensity through switching adaptive median and fixed weighted mean filtering. IEEE Trans. on image processing, 2018, vol. 27, no. 11, pp. 5475–5489. doi: 10.1109/TIP.2018.2857448
6. Mozerov M.G., van de Weijer J. Improved recursive geodesic distance computation for edge preserving filter. IEEE Trans. on image processing, 2017, vol. 26, no. 8, pp. 3696–3706. doi: 10.1109/TIP.2017.2705427
7. Antonova T.V. On the localization of nonsmooth discontinuity lines of a function of two variables. Trudy Inst. Mat. Mekh. UrO RAN, 2019, vol. 25, no. 3 (in Russian). doi: 10.21538/0134-4889-2019-25-3-9-23
8. Ageev A.L., Antonova T.V. On the localization of fractal lines of discontinuity from noisy data. Russian Math., 2023, vol. 67, no. 9, pp. 23–38. doi: 10.3103/S1066369X23090037
9. Crownover R M. Introduction to fractals and chaos. Boston, London: Jones and Bartlett Publ., 1995, 306 p. Translated to Russian under the title Fraktaly i khaos v dinamicheskikh sistemakh. Osnovy teorii, Moscow: Postmarket Publ., 2000, 352 p.
Cite this article as: A.L. Ageev, T.V. Antonova. Methods with adaptive averaging domain for localizing fractal break lines. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 44–60.
