In this paper we consider narrow band, active array imaging of weak localized scatterers when only the intensities are recorded at an array with N transducers. We assume that the medium is homogeneous and, hence, wave propagation is fully coherent. This work is an extension of our previous paper [A. Novikov, M. Moscoso, and G. Papanicolaou, SIAM J. Imaging Sci., 8 (2015), pp. 1547-1573], where we showed that using linear combinations of intensity-only measurements, obtained from N-2 illuminations, imaging of localized scatterers can be carried out efficiently using imaging methods based on the singular value decomposition of the time-reversal matrix. Here we show that the same strategy can be accomplished with only 3N - 2 illuminations, therefore enormously reducing the data acquisition process. Furthermore, we show that in the paraxial regime one can form the images by using six illuminations only. In particular, this paraxial regime includes Fresnel and Fraunhofer diffraction. The key point of this work is that if one controls the illuminations, imaging with intensity only can be easily reduced to an imaging with phases and, therefore, one can apply standard imaging techniques. Detailed numerical simulations illustrate the performance of the proposed imaging strategy with and without data noise.