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The equivalent magnetic currents method can be used to characterize radiating structures and to obtain information about the current distribution on the radiating device under test (DUT). In order to do so, the system is assumed to be linear and inversion from near-field measurements can be carried out using numerical techniques such as the singular value decomposition. However this approach is very sensitive to modeling and measurement errors, which is a consequence of the ill-conditioned nature of the inversion process. Here, we propose to account for the ill-conditioning by using a Tikhonov regularization technique to perform the inversion. This technique requires a regularization parameter to be set to an appropriate value, and we show that a generalized cross-validation approach yields adequate determination of the regularization parameter directly from the measured data. The proposed method can be tailored to account for specific characteristics of the DUT. This point is illustrated by the estimation of the current distribution of a bidimensional radiating structure for which prior information about the orientation of radiating edges is incorporated into the inversion method.