# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 1 Introduction  
Image deblurring under impulse noise is important in digital image processing research. It is widely applied in many aspects of science and engineering such as space detection, photodetection, remote sensing technology, medical image analysis, and geological exploration. In the process of acquiring, storing, and transmitting images, owing to limitations of imaging equipment and conditions, images are prone to quality degradation, such as image blurring, noise pollution, contrast decline, and loss of details. Image restoration of degraded images is an inverse problem whose solution is often ill-posed [1]. For such types of problems, regularization proved to be an effective method. This involves the proper use of some prior information of the degraded image as a fidelity term and an image optimization model that contains an efficient solution algorithm as a regularization term, because recovering a clear image is of great importance for advanced image processing in the later stages. The total variation (TV) regularization proposed by [PERSON], [PERSON], and [PERSON] [2], also known as the ROF model, proved to be one of the most advanced regularizations in the field of image reconstruction. Based on the TV regularization, some extended models and their algorithms were widely applied to image restoration. For example, [PERSON] et al. [3] proposed a fast total variation deconvolution (FTvd) model for image deblurring. The model uses the total variation (TV) regularization based on the assumption that the image is piecewise smooth; the method can remove image noise sufficiently,but it is prone to producing a \"staircase effect\". Furthermore, the FTVd model adopted the L1 norm to express the sparsity of impulse noise. However, it cannot depict the statistic characteristics of impulse noise. Recently, some TV extensions emerged. [PERSON] et al. [4] proposed a four-directional total variation model to extend the vertical and horizontal gradient information in the TV regularization into four directions. [PERSON] et al. [5] proposed a fractional-order TV (FTV) model and tried to extend the integer-order TV model to the fractional order. [PERSON] and [PERSON] [6] proposed an overlapping group sparsity TV (OGSTV) model, in which the neighborhood information of the image gradient was taken into account and applied in conjunction with an L2-norm fidelity term to the Gaussian-noise denoising of one-dimensional signals. [PERSON] et al. [7] extended the model to a two-dimensional model based on an L1-norm fidelity term and applied the extended model to image deconvolution and impulse noise removal, achieving good results [8]. [PERSON] et al. [9] proposed a total generalized variation (TGV) model based on total variation, which simultaneously constrains the first-order and second-order gradients of the image and has many excellent mathematical properties such as convexity, lower semi-continuity, and rotation invariance; it is also capable of approximating any number of items. The TGV model proved--in theory and practice--to be able to remove the staircase effect effectively while preserving the edges and details of the image. TGV regularization terms attracted wide attention in the academic community and were applied to the reconstruction of nuclear magnetic resonance images [10] and seismic signal denoising [11; 12]. Inspired by the advantages of TGV regularization, we incorporate it here in the image deblurring process. The aforementioned image deblurring methods mostly employ the L1 norm to constrain the fidelity term. However, the L1 norm is a convex relaxation of the L0 norm, with limited ability to induce sparsity [13]. Recently, [PERSON] and [PERSON] [14] proved that Lp-pseudo-norm shrinkage is a method with better sparsity compared with soft-threshold shrinkage, which attracted extensive attention in academia. [PERSON] et al. [15] theoretically compared the advantages of Lp-minimization and L1-minimization, and tested them in a Gaussian noisy setting. [PERSON] et al.