Image Restoration with Compound Regularization Using a Bregman Iterative Algorithm

Some imaging inverse problems may require the solution to simultaneously exhibit properties that are not enforceable by a single regularizer. One way to attain this goal is to use a linear combinations of regularizers, thus encouraging the solution to simultaneously exhibit the characteristics enforced by each individual regularizer. In this paper, we address the optimization problem resulting from this type of compound regularization using the split Bregman iterative method. The resulting algorithm only requires the ability to efficiently compute the denoising operator associated to each involved regularizer. Convergence is guaranteed by the theory behind the Bregman iterative approach to solving constrained optimization problems. In experiments with images that are simultaneously sparse and piece-wise smooth, the proposed algorithm successfully solves the deconvolution problem with a compound regularizer that is the linear combination of the `1 and total variation (TV) regularizers. The lowest MSE obtained with the (`1+TV) regularizer is lower than that obtained with TV or `1 alone, for any value of the corresponding regularization parameters.