Approximation of discontinuity lines of a noisy function of two variables

We construct and study methods for localizing (determining the location) a line in whose neighborhood the function of two variables in question is smooth while having discontinuity of the first kind along the line. Instead of the exact function, an approximation in $L_2$ is available with a known noise level. This problem belongs to the class of ill-posed nonlinear problems and, in order to solve it, we have to construct regularizing algorithms. We propose a simplified theoretical approach to the problem of discontinuity line localization for a noisy function with conditions on the exact function imposed in an arbirarily thin strip crossing the discontinuity line. We construct averaging methods and estimate the precision of localization for them.