This paper addresses the problem of recovering an image degraded by a linear operator and corrupted with an additive Gaussian noise with a signal-dependent variance. The considered observation model arises in several digital imaging devices. To solve this problem, a variational approach is adopted relying on a weighted least squares criterion which is penalized by a non-smooth function. In this context, the choice of an efficient optimization algorithm remains a challenging task. We propose here to extend a recent primal-dual proximal splitting approach by introducing a preconditioning strategy that is shown to significantly speed up the algorithm convergence. The good performance of the proposed method is illustrated through image restoration examples.