Geodesic
On the localization of nonsmooth discontinuity lines of a function of two variables
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 9-23 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider ill-posed problems of localizing (finding the position of) the discontinuity lines of a perturbed function of two variables (an image). For each node of a uniform square grid with step $\tau$, the average values of the function over a square $\tau\times\tau$ are assumed to be known. The perturbed function approximates an exact function in the space $L_2(\mathbb{R}^2)$, and the perturbation level $\delta$ is known. Earlier, the authors studied the case of piecewise smooth discontinuity lines, which, as a rule, correspond to the borders of artificial objects in the corresponding image. In the present paper, an approach to the study of localization algorithms is developed, which makes it possible to weaken the conditions on the smoothness of discontinuity lines and consider, in particular, nonsmooth discontinuity lines, which can describe the boundaries of natural objects. To solve the problem under consideration, we construct and analyze global discrete algorithms for the approximation of discontinuity lines by sets of points of a uniform grid on the basis of averaging procedures. Conditions on the exact function are formulated and a correctness class is constructed, which includes functions with nonsmooth discontinuity lines. A theoretical analysis of the constructed algorithms is carried out on this class. It is established that the proposed algorithms make it possible to obtain a localization error of order $O(\delta)$. We also estimate other important parameters, which characterize the operation of the localization algorithm.
Keywords: ill-posed problem, regularization method, discontinuity lines, global localization, discretization, separability threshold. 
@article{TIMM_2019_25_3_a1,
     author = {A. L. Ageev and T. V. Antonova},
     title = {On the localization of nonsmooth discontinuity lines of a function of two variables},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {9--23},
     year = {2019},
     volume = {25},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a1/}
}
TY  - JOUR
AU  - A. L. Ageev
AU  - T. V. Antonova
TI  - On the localization of nonsmooth discontinuity lines of a function of two variables
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2019
SP  - 9
EP  - 23
VL  - 25
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a1/
LA  - ru
ID  - TIMM_2019_25_3_a1
ER  - 
%0 Journal Article
%A A. L. Ageev
%A T. V. Antonova
%T On the localization of nonsmooth discontinuity lines of a function of two variables
%J Trudy Instituta matematiki i mehaniki
%D 2019
%P 9-23
%V 25
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a1/
%G ru
%F TIMM_2019_25_3_a1
A. L. Ageev; T. V. Antonova. On the localization of nonsmooth discontinuity lines of a function of two variables. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 9-23. http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a1/

[1] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1974

[2] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978, 206 pp.

[3] Vasin V. V., Ageev A. L., Ill-posed problems with a priori information, VSP, Utrecht, 1995, 255 pp. | MR | Zbl

[4] Malla S., Veivlety v obrabotke signalov, Mir, M., 2005, 671 pp.

[5] Vvedenie v konturnyi analiz i ego prilozheniya k obrabotke izobrazhenii i signalov, ed. red. Ya. A. Furmana, Fizmatlit, M., 2002, 596 pp.

[6] Gonsales R., Vuds R., Tsifrovaya obrabotka izobrazhenii, izd. 3-e ispr. i dop., Tekhnosfera, M., 2012, 1104 pp.

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

[8] Ageev A.L., Antonova T.V., “Approksimatsiya linii razryva zashumlennoi funktsii dvukh peremennykh”, Sib. zhurn. industr. matematiki, 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. matematiki, 20:4(72) (2017), 3–12 | DOI | Zbl

[10] Ageev A.L., Antonova T.V., “K voprosu o globalnoi lokalizatsii linii razryva funktsii dvukh peremennykh”, Tr. In-ta matematiki i mekhaniki, 24:2 (2018), 12–23 | MR