## 1 Introduction  
Image denoising is one of the most important research areas in the field of image processing, and it has great value in both theoretical studies and engineering applications. Its usage spans the broad fields of image restoration [1], detection [2], photoelectric detection [3], geological exploration [4], remote sensing [5], and medical image analysis [6], among others [7; 8]. With the development of compressed sensing theory, image processing algorithms based on sparse representation and constrained regularization have evolved into promising methods of image restoration [9]. Models based on total variation (TV) regularization [10; 11; 12] are found to be effective in removing random noise. The TV model has been successfully used in image restoration tasks such as denoising [13], deblurring [14], and super-resolution [15]. Although TV regularization can recover sharp edges of a degraded image, it also leads to some undesired effects and transforms the smooth signal into piecewise constants, the so-called staircase effect. Several models have been proposed by scholars to make an improvement on the TV model [16; 17; 18; 19; 20; 21]. One usual method is to replace the original TV norm by a high-order TV norm. The high-order TV overcomes the staircase effect while preserving the edges in the restored image. However, the high-order TV based methods may transform the smooth signal to over-smoothing and take more time to compute. More details can be referred to in References [22]. In 2010, [PERSON] et al. proposed the total generalized variation (TGV) model [19]. The TGV model puts constraints on both the first- and second-order gradients of an image, thus effectively attenuating the staircase effect of the TV model. Still, it is difficult to both preserve the image details and suppress the noise simultaneously in TGV. Furthermore, some scholars pay attention to fractional-order gradients replacing integer-order gradients [23]. Their research shows that using a fractional differential operator with 0 \(<\) v \(<\) 1 can appropriately process the noise and edge information, but it also has un-denoised \"spots\" in the image. Although these improved methods can alleviate the staircase artifacts, they might lead to \"spots\" effects on the processed image. How to choose a good regularization functional is a key point in imaging science, in order to balance the staircase artifacts and \"spots\" effects. Recently, [PERSON] and [PERSON] proposed the total variation with overlapping group sparsity (OGS-TV) [24; 25; 26; 27; 28], which introduces the concept of the group gradient into the TV model and takes into full consideration the dissimilarity between smooth and edge regions. The OGS-TV model can distinguish the individual noise point and image edge point, so it greatly alleviates the staircase effect. Based on this work, [PERSON] et al. applied this method in the removal of speckle noise. [PERSON] and [PERSON] applied the OGS model in the field of Magnetic Resonance (MR) image reconstruction [27]. In this paper, we introduce the OGS model into the denoising of impulse noises. In the typical denoising method, the L1-norm is commonly used as the fidelity term of impulse noise. However, the solution to the L1-norm usually involves the soft-thresholding function, which reduces large values by a constant amount. As a result, noise signals are estimated systematically underestimated for large signal values [26]. To improve this shortcoming, many non-convex reconstruction methods are proposed. Non-convex regularizers have also shown to exhibit a sparser solution than the L1 regularizer [24; 29; 30]. Inspired by their research, we propose a total variation model based on overlapping group sparsity and Lp-pseudo-norm shrinkage (called OGS-Lp for short). Compared with the L1-norm, the Lp-pseudo-norm adds another degree of freedom to the model, which better characterizes the sparsity features of the image [31].