The moment approach to solve the Cauchy problems is investigated. First, we consider the Cauchy problem for the Laplace equation, and we present a moment method for solving it in the case of a flat boundary. Second, we consider the reciprocity gap concept used to solve the problem of crack detection, as a stopping criterion and we study the case of noise data. Finally, we propose an application to the Cauchy problem for the Laplace equation, for the inpainting problem. Some numerical results showing the efficiency of the method proposed are also given.