Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non-linear minimization problem, usually with a Tikhonov-like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal-to-noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non-linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster.