In this paper we develop a wavelet-based statistical method for solving linear inverse problems. The Bayesian framework developed here is general enough to treat a wide class of linear inverse problems involving (white or colored) Gaussian observation noise. In this approach, a signal prior is developed by modeling the signal/imgage wavelet coefficients as independent Gaussian mixture random variabls. We first specify a uniform (non-informative) distribution on the mixing parameters, which leads to a simple and efficient iterative algorithm for MAP estimation. This algorithm is similar to the EM algorithm in that it alternates between a state estimation step and a maximization step. Moreover, we show that our algorithm converges monotonically to a local maximum of the posterior distribution. We next generalize the result to non-uniform priors and develop an efficient integer programming algorithm that enables a similar alternating optimization procedure. Experimental reults show that this new method outperforms recent results, including multiscale Kalman filtering and wavelet-vaguelette type methods based on linear inverse filtering followed by wavelet coefficient denoising.