We propose and analyze a data completion algorithm based on the representation of the solution in terms of surface integral operators to solve the Cauchy problem for the Helmholtz or the Laplace equations. The proposed method is non iterative and intrinsically handle the case of noisy and incompatible data. In order to cope with the ill-posedness of the problem, our formulation is compatible with standard regularization methods associated with linear ill posed inverse problems and leads to convergent scheme. We numerically validate our method with different synthetic examples using a Tikhonov regularization.