Localization method for lines of discontinuity of approximately defined function of two variables

A function of two variables with the lines of discontinuity of the first kind is considered. It is assumed that outside discontinuity lines the function to be measured is smooth and has a limited partial derivative. Instead of the accurate function its approximation in $L_2$ and perturbation level are known. The problem in question belongs to the class of nonlinear ill-posed problems, for whose solution it is required to construct regularizing algorithms. We propose a reduced theoretical approach to solving the problem of localizing the discontinuity lines of the function that is noisy in the space $L_2$. This is done in the case when conditions of an exact function are imposed “in the small”. Methods of averaging have been constructed, the estimations of localizing the line (in the small) have been obtained.