In the context of compressed sensing, we present a signal recovery framework based on the fact that the correlation matrix of a signal being recovered is available as side information to the receiver, and can therefore be exploited to improve the recovery performances of a standard BPDN formulation of the recovery problem. This is attained by quadratic, non-smooth convex optimization that can be solved through proximal methods, in a scheme that we dub C-BPDN. In order to show that the above information is correctly leveraged, we finally present evidence on a compressive imaging example, which highlights how the provided side information is properly leveraged by C-BPDN to yield a high-quality image with a smaller amount of measurements than BPDN.