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In real applications where data acquisition is carried out under extreme conditions, post-processing techniques for systematic corrections are of critical importance. In micrometeorological studies, it is often the case that acquired data contains both missing information and impulse noise due to instrumentation failure, data transmission and data rejection for quality assurance. In this work, we propose a simple algorithm based on an $\ell_{1}-\ell_{1}$ variational formulation that simultaneously suppresses impulse noise and interpolates missing information. Our approach consists of relaxing the objective function in the variational formulation with a strictly convex and continuously differentiable function that depends on a regularization parameter. We solve a sequence of strictly convex optimization subproblems as the regularization parameter goes to zero, converging to the solution of the original problem. Numerical experiments on real micrometeorological datasets are conducted showing the effectiveness of the proposed approach. Furthermore, a convergence analysis is presented providing theoretical guaranties of our method.

Interpolating Sparsely Corrupted Signals in Micrometeorology