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The linear sampling method (LSM) is a simple and effective approach to image the shape of an unknown target through the solution of a linear equation whose known term is the field radiated by an elementary (i.e., monopolar) source located in a set of test points. In this paper, we extend the LSM formulation by generalizing the right-hand side of the equation to higher order multipoles. In particular, we discuss which is the expected method's outcome after such a change and propose a post-processing scheme to combine the results obtained for different multipoles. As shown with simulated and experimental data, the proper combination of higher order multipoles allows to achieve better images of complex shaped targets as compared to the standard LSM and to overcome some of its limitations.