## 1 Introduction  
Because of the imaging principle and other properties of infrared imaging systems, infrared images have low resolution with unclear edges; consequently, infrared images are not suitable for many applications. Considering this, it is considerably important to improve the quality of infrared images. There are currently two approaches to improve the resolution of infrared images: hardware and software approaches. For example, one hardware approach involves increasing the number of pixels per unit area, while another hardware approach involves increasing the size of the imaging system chip. These aforementioned hardware methods are not only difficult to implement, but also incur significant costs; thus, in recent years, various software approaches have been developed for the super-resolution reconstruction (SRR) of infrared images. The objective of these image SRR software approaches is to recover a high-resolution (HR) image using one or multiple low-resolution (LR) images. In general, the image SRR methods can be classified into three types, including interpolation-based [1], learning-based, and regularization-based methods. The interpolation-based methods involve the construction of an HR image by projecting LR images onto reference images. [PERSON] and [PERSON] [2] proposed a method based on a non-uniformity interpolation from a set of spatially shifted LR images by utilizing the generalized multichannel sampling theorem of [PERSON] [3] and [PERSON] [4]. [PERSON] et al. developed an SR algorithm based on a bivariate rational fractal interpolation model, which achieved competitive performance with finer detail and sharper edges [5]. However, in methods based on interpolation, the optimality of the entire reconstruction algorithm is not guaranteed. In recent years, learning-based SRR has become a hot topic for research; in learning-based methods, the relationship between HR and LR images is obtained by learning the training samples and reconstructing the HR images based on the obtained relationship. In another study, [PERSON] et al. obtained the relationship between HR and LR image blocks using the Markov random field [6]. [PERSON] proposed the concept of sparse representation, which is a new method of signal representation [7]. [PERSON] et al. proposed a super-resolution (SR) method based on this sparse representation [8; 9]. [PERSON] et al. proposed an SR model based on the iterative sparsity and locality-constrained representation [10]; [PERSON] et al. proposed a method which utilized an effective way to transform the large nonlinear feature space of LR images into a group of linear subspaces with mixture prior models in the training phase [11]. Though learning-based methods can achieve an excellent image visual effect, their high computational complexity limits its application. In contrast, methods based on regularization can capture the primary information and intrinsic geometry of an image with a small number of coefficients. In general, methods based on an integer-order partial differential equation (PDE) have obtained good results for image reconstruction; among these methods, a well-known method is total variation (TV) regularization [12; 13], which not only eliminates noise, but also preserves the edges of the image well. However, first-order TV assumes images to be piece-wise constant, which introduces staircase artifacts; to address this limitation, several high-order variational methods have been proposed [14; 15; 16; 17; 18; 19]. For example, [PERSON], [PERSON], and [PERSON] first replaced the first-order TV by a second-order TV [20]. In addition, [PERSON], [PERSON], and [PERSON] proposed the fourth-order TV combined with the Chambolle duality method to solve the issue of staircase artifacts [15]. [PERSON], [PERSON], and [PERSON] proposed total generalized variation (TGV) based on the combination of TV regularization with higher-order derivatives [21]; although these improved methods can alleviate the staircase artifacts, they might lead to \"spots\" effects on the processed image.