Deconvolving Poissonian image has been a significant subject in various application areas such as astronomical, microscopic, and medical imaging. In this paper, a regularization-based approach is proposed to solve Poissonian image deconvolution by minimizing the regularization energy functional, which is composed of the generalized Kullback-Leibler divergence as the data-fidelity term and sparsity prior constraints as the regularization term, and a non-negativity constraint. We consider two sparsity prior constraints which include framelet-based analysis prior and combination of framelet and total variation analysis priors. Furthermore, we show that the resulting minimization problems can be efficiently solved by the split Bregman method. The comparative experimental results including quantitative and qualitative analysis manifest that our algorithm can effectively remove blur, suppress noise, and reduce artifacts.