Image deblurring has been widely studied in image science. Blurred images corrupted by Poisson noise appear in various applications such as astronomical imaging, electronic microscopy, positron emission, tomography, etc. The standard criteria for deblurring Poissonian images aim to solve a minimization problem. Recently, various algorithms were proposed for it. In all these approaches, a hard problem is the selection of suitable value of the regularization parameter. In the existing literature, the main approaches for automatic parameter selection include the generalized cross validation, the Stein unbiased risk estimator and the discrepancy principle. Very recently, based on the statistical characteristic of the generalized Kullback-Leibler divergence, a new discrepancy principle was proposed for selecting the regularization parameter of the TV-KL model. In these works the regularization parameter selection is not space dependent. In order to improve the restored image quality, spatially adapted regularization parameter selection is considered in this paper. The numerical results indicate that the proposed algorithm outperforms the other approaches in both image detail preservation and Poisson noise removal. The paper is well written. The theorems are rigourously proved. However, it is hard to read for readers without a strong mathematical background.