During the last five years, several convex optimization algorithms have been proposed for solving inverse problems. Most of the time, they allow us to minimize a criterion composed of two terms one of which permits to "stabilize" the solution. Different choices are possible for the so-called regularization term, which plays a prominent role for solving ill-posed problems. While a total variation regularization introduces staircase effects, a wavelet regularization may bring other kinds of visual artefacts. A compromise can be envisaged combining these regularization functions. In the context of Poisson data, we propose in this paper an algorithm to achieve the minimization of the associated (possibly constrained) convex optimization problem.