Array imaging using intensity-only measurements

We introduce a new approach for narrow band array imaging of localized scatterers from intensity-only measurements by considering the possibility of reconstructing the positions and reflectivities of the scatterers exactly from only partial knowledge of the array data, since we assume that phase information is not available. We reformulate this intensity-only imaging problem as a non-convex optimization problem and show that we can have exact recovery by minimizing the rank of a positive semidefinite matrix associated with the unknown reflectivities. Since this optimization problem is NP-hard and is computationally intractable, we replace the rank of the matrix by its nuclear norm, the sum of its singular values, which is a convex programming problem that can be solved in polynomial time. Numerical experiments explore the robustness of this approach, which recovers sparse reflectivity vectors exactly as solutions of a convex optimization problem.