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 regu- larizers, 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 regular- ization using the split Bregman iterative method. The resulting algorithm only requires the ability to e±ciently compute the denoising operator associated to each in- volved 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.