A method based on the wavelet transform is used to localize the causative sources of potential field anomalies. In previous studies we introduced a particular class of analyzing wavelets belonging to the Poisson semigroup and such that the analyzed anomaly has a conical signature in the wavelet domain with its apex pointing at the location of the causative homogeneous source. In the present paper we apply this formalism to the special case of anomalies produced by elongated sources like faults and dikes. We show that, for this particular type of anomalies, the two-dimensional (2-D) wavelet transform corresponds to the ridgelet analysis and reduces to the 1-D wavelet transform applied in the Radon domain. A complete synthetic example is used to illustrate all steps of the analysis method: Radon transform of the anomaly map, selection of the Radon signature of elongated anomalies, complex wavelet transform, and source localization with the conical signature in the wavelet domain. The azimuthal filtering performed in the Radon domain leads to high signal-to-noise ratio and good localization of the sources both horizontally and vertically. The synthetic example is completed by an application of the method to a real aeromagnetic survey acquired in Britanny (France) and the results are compared with source depth determinations made with the Euler deconvolution method.