L ¹-Regularized inverse problems for image deblurring via bound- and equality-constrained optimization

Image deblurring is typically modeled as an ill-posed, linear inverse problem. By adding an L¹-penalty to the negative-log likelihood function, the resulting minimization problem becomes well-posed. Moreover, the penalty enforces sparsity. The difficulty with L¹-penalties, however, is that they are non-differentiable. Here we replace the L¹-penalty by a linear penalty together with bound and equality constraints. We consider two statistical models for measurement error: Gaussian and Poisson. In either case, we obtain a bound- and equality-constrained minimization problem, which we solve using an iterative augmented Lagrangian (AL) method. Each iteration of the AL method requires the solution of a bound-constrained minimization problem, which is convex-quadratic in the Gaussian case and convex in the Poisson case. We recommend two highly efficient methods for the solution of these subproblems that allows us to apply the AL method to large-scale imaging examples. Results are shown on synthetic data in one and two dimensions, as well as on a radiograph used to calibrate the transmission curve of a pulsed-power X-ray source at a US Department of Energy radiography facility.