Regular algorithms for the localization of discontinuity lines based on a separation of perturbed function values
A.L. Ageev, T.V. Antonova
N.N. Krasovsky Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia
Keywords: ill-posed problems, regularization method, discontinuity lines, global localization, discretization, separability threshold, image separation
Abstract
We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables, provided that outside the discontinuity lines the function satisfies a Lipschitz condition, and at each point on the lines there is a discontinuity of the first kind. For a uniform grid with step τ, it is assumed that at each node the mean values of the perturbed function on a square with side τ are known, and the perturbed function approximates the exact function in L2(ℝ2). The level of perturbation δ is assumed to be known. We propose a new approach to construct regularizing algorithms for localizing the discontinuity lines based on a separation of the original noisy data. New algorithms are constructed for a class of functions with piecewise linear discontinuity lines and a convergence theorem with estimates of approximation accuracy is proved.
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