On the localization of fractal discontinuity lines from noisy data

Abstract

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.