Methods for the approximating the discontinuity lines of a noisy function of two variables with countably many singularities

We study the localization methods for the discontinuity lines of a noisy function of two variables. The function is assumed to have countably many discontinuity lines: finitely many discontinuity lines have “large” jump, and the jumps at the remaining discontinuity lines satisfy some smallness condition. It is required, from the noisy function and the error in $L_2$, to determine the number and localize the position of the discontinuity lines that form the first set for the exact function. The problem under consideration belongs to the class of nonlinear ill-posed problems, and for solution we have to construct regularizing algorithms. We propose a simplified theoretical approach when conditions on the exact function are imposed in a narrow strip intersecting the discontinuity lines. We construct methods for the averaging and localization of discontinuity lines and obtain estimates for the accuracy of the localization.