A Nonnegatively Constrained Convex Programming Method for Image Reconstruction

We consider a large-scale convex minimization problem with nonnegativity constraints that arises in astronomical imaging. We develop a cost functional which incorporates the statistics of the noise in the image data and Tikhonov regularization to induce stability. We introduce an efficient hybrid gradient projection-reduced Newton (active set) method. By “reduced Newton” we mean taking Newton steps only in the inactive variables. Due to the large size of our problem, we compute approximate reduced Newton steps using conjugate gradient (CG) iteration. We also introduce a highly effective sparse preconditioner that dramatically speeds up CG convergence. A numerical comparison between our method and other standard large-scale constrained minimization algorithms is presented.