TRANSACTIONS ON IMAGE PROCESSING . 1 Modular Solvers for Constrained Image

|Many problems in image restoration cam be formulated as either an unconstrained nonlinear optimization problem: min u R(u) + 2 kKu ? zk 2 which is the Tikhonov 1] approach, where the regularization parameter is to be determined; or as a noise constrained problem: min u R(u); subject to 1 2 kKu ? zk 2 = 1 2 jj 2 ; where is the (estimated) variance of the the noise. In both formulations z is the measured (noisy, and blurred) image, and K a convolution operator. 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 nonlinear equations, which couples the constraint and the regularized problem. The existing solver is used in a block elimination algorithm. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identiies 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.