In many applications such as Magnetic Resonance Imaging, images are acquired using Fourier transform measurements. Such measurements can be expensive, and it is of interest to exploit the wavelet domain sparsity of natural images to reduce the number of measurements without destroying reconstruction quality. Much work in compressed sensing has been devoted to this problem in recent years. However, a rigorous theory for sampling with compressive frequency measurements has to date only been developed for bases that, unlike wavelet bases, are incoherent to the Fourier basis. Nevertheless, it has been shown empirically that variable density sampling strategies seem to overcome this obstacle. We introduce a theory which reveals suitable variable density sampling strategies and provides the first theoretical reconstruction results for compressive imaging via frequency measurements.