We describe new wavelet-based techniques for removing noise from digital images. In the proposed approaches, the subband decompositions of images are modelled using alpha-stable prior models, which have been shown to be flexible enough in order to capture the heavy-tailed nature of wavelet coefficients. For improved denoising performance interscale dependencies of coefficients should also be taken into account and we achieve this by employing bivariate stable distributions. We restrict our study to the particular cases of the isotropic stable and the sub-Gaussian distributions. Using Bayesian estimation principles, we design both the bivariate minimum absolute error (MAE) and the bivariate maximum a posteriori (MAP) processors based on alpha-stable signal statistics. We also discuss methods of estimating stable distributions parameters from noisy observations. In implementing our algorithms, we make use of the dual-tree complex wavelet transform, which features near shift-invariance and improved directional selectivity compared to the standard wavelet transform. We test our algorithms in comparison with several recently published methods and show that our proposed techniques are competitive with the best wavelet-based denoising systems.