Poissonian image deconvolution is a key issue in various applications, such as astronomical imaging, medical imaging, and electronic microscope imaging. A large amount of literature on this subject is analysis-based methods. These methods assign various forward measurements of the image. Meanwhile, synthesis-based methods are another well-known class of methods. These methods seek a reconstruction of the image. In this paper, we propose an approach that combines analysis with synthesis methods. The method is proposed to address Poissonian image deconvolution problem by minimizing the energy functional, which is composed of a sparse representation prior over a learned dictionary, the data fidelity term, and framelet based analysis prior constraint as the regularization term. The minimization problem can be efficiently solved by the split Bregman technique. Experiments demonstrate that our approach achieves better results than many state-of-the-art methods, in terms of both restoration accuracy and visual perception. 1. Introduction Poissonian image deconvolution appears in various applications such as astronomical imaging , medical imaging , and electronic microscope imaging . It aims to reconstruct a high quality image from the degraded image . Mathematically, the process of Poissonian image deconvolution can be generally modeled by where denotes the point spread function (PSF), denotes the unknown image to be estimated, denotes the Poisson noise process, denotes the blurred noisy image, and is the length of image vector which is stacked by columns. It is known that Poissonian image deconvolution is a typical ill-posed inverse problem. In general, the solution of (1) is not unique. Prior knowledge of image, including analysis-based and synthesis-based priors, can be used to address this problem. For an overview of the two classes of priors, we refer to . Analysis-based priors are frequently used as regularization term in energy functional where is the regularization constraint term. Two main analysis-based methods have been proposed to solve problem (2): the total variation (TV) [5, 6] based methods and wavelet frames-based methods. TV-based methods have shown good performance on blurred images for discontinuous solution and edge-preserving advantages. Many authors have combined the TV regularization term with variant methods to solve problem (2). For example, Sawatzky et al. combined expectation-maximization algorithm with TV regularization  in positron emission tomography. Setzer et al. considered using TV regularization term with split
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