## 1 Introduction  
Images are a primary source of information because they contain large volumes of information and are well-aligned with the cognitive functions of the brain. In the absence of visible light, infrared (IR) radiation generated by an object can be converted into visible IR images using an IR thermal imaging system. The brightness of each pixel in the image corresponds to the change in the intensity of the object's radiation energy [1]. IR thermal-imaging systems are widely used in biomedicine, military, industrial, and agricultural applications, owing to their strong environmental adaptability, concealment, anti-interference, and identification abilities. However, they have disadvantages such as high noise, low contrast, and blurring. Among the various types of noise, the salt-and-pepper and the Gaussian noises have the greatest impact on IR images [2]. Salt-and-pepper noise is a common random noise exhibiting sparsity in mathematics and statistics. The common methods for removing salt-and-pepper noise include median filter, partial differential equation (PDE) model-based methods, and total variation (TV) model-based methods. Although the median filter method can effectively remove salt-and-pepper noise, it produces deblurred images with incomplete image details [3]. In the last decade, PDE-based models have been developed for various physical applications in image restoration [4]. TV-based models can answer fundamental questions related to image restoration better than other models [5]. The main advantage of this method is that it preserves the image edge. Traditional TV models, including anisotropic TV (ATV) [6] and isotropic TV (ITV) [7], can only approximate the piecewise constant function, causing staircase effects in the restored images [8, 9]. Scholars have proposed several variants of the TV model to alleviate the staircase effect. For example, [PERSON] et al. used the L1 norm to describe the statistical characteristics of noise and introduced overlapping group sparsity TV into the salt-and-pepper noise model, achieving good results [10]. However, many non-convex models have achieved better sparse constraints in practical applications than those based on the L1 norm at low sampling rates. Therefore, [PERSON] and [PERSON] proposed a new sparse optimization model (L0 TVPADMM) that used an L0 norm as the fidelity term to solve the reconstruction problem based on the TV model [11]. [PERSON] et al. proposed the HNHOTV-OGS method, which combined non-convex high-order total variation and overlapping group sparse regularization [12]. [PERSON] proposed a non-convex optimization problem with the minimization of the Lp-pseudo-norm as the objective function [13; 14]. Subsequently, [PERSON] and [PERSON] provided a theoretical condition of the Lp-pseudo-norm to recover an arbitrary sparse signal [15]. [PERSON] and [PERSON] [16] and [PERSON] et al. [17] theoretically demonstrated the superiority of methods based on the Lp-pseudo-norm. The Lp-pseudo-norm exhibited a stronger sparse representation ability than the L1 norm. Therefore, it has garnered extensive research interest in recent years [18; 19; 20; 21; 22; 23]. For example, [PERSON] et al. imposed sparse constraints on the high-order gradients of the image, combined the Lp-pseudo-norm with the total generalized variation model, and proposed an image restoration algorithm that achieved good performance [22]. Based on the mathematical model in [22], [PERSON] et al. replaced the L1 norm with the Lp-pseudo-norm to describe the statistical characteristics of salt-and-pepper noise and proposed an image denoising method based on the Lp-pseudo-norm with low-order overlapping group sparsity (Lp_LOGS) [23]. Among these algorithms, Lp_LOGS performed the best in removing salt-and-pepper noise. However, this method only considers the prior information of the low-order gradient of an image. This study proposes an image-deblurring model based on the Lp_LOGS model to mine the high-order prior information of an IR image containing salt-and-pepper noise, called Lp_HOGS.