On the localization of nonsmooth discontinuity lines of a function of two variables

A. L. Ageev ; T. V. Antonova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 9-23

Résumé

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.

Détail

Keywords: ill-posed problem, regularization method, discontinuity lines, global localization, discretization, separability threshold.