This paper considers the denoising and reconstruction of images corrupted by Poisson noise. Poisson noise arises in the context of counting the emission or scattering of photons. In various application domains, such as astronomy and medical imaging, photons counts are low resulting in very low signal-to-noise ratio images. Recently, Azzari and Foi investigated using BM3D for Poisson image denoising in a coarse-to-fine image resolution framework. Specifically, the denoised result at a coarse resolution is used to improve the denoising of the next finer resolution, resulting in state-of-the-art denoising results. This paper presents an alternative regularized maximum likelihood formulation of the reconstruction problem, and explains how it can be solved using a coarse-to-fine proximal gradient optimization algorithm. The proposed methods of this paper are compared to the methods of Azzari and Foi, highlighting their strong similarities. The advantage of the proposed method of this paper is that it easily generalizes to inverse problem settings, which is demonstrated in the context of denoising a Poisson noisy image with missing pixels (i.e. image inpainting); in contrast there is no known generalization of the coarse-to-fine BM3D denoising method that was proposed by Azzari and Foi.