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The authors present a gradient-based optimisation algorithm for the reconstruction of the shape of a metal protrusion on a flat ground plane based on the scattering matrix of an array antenna. The sensitivities are based on the continuum form of Maxwell's equations, which offers good flexibility with respect to the choice of field solver. The computational cost for the gradient is independent of the number of degrees of freedom that are used to represent the shape of the protrusion. Initial tests show that the sensitivities yield good accuracy in the 3D setting and that relatively good reconstructions of simple surfaces are feasible with a computational cost that amounts to about some ten evaluations of the scattering matrix.