## 1 Introduction
In order to balance the staircase artifacts and \"spots\" effects, [PERSON] and [PERSON] proposed a method based on the fractional-order gradient; in particular, in their method, they replaced the integer-order gradient with a fractional-order gradient [22]. Their research shows that using a fractional differential operator with \(0<v<1\) order to process the image not only appropriately processes the noise and edge information, but also enhances the texture and preserves the information related to the smooth region of the image. In fact, for noisy images, the conventional super-resolution way is to denoise the image as a pre-processing step and then super-resolve the de-noised image. Some new methods have been proposed [23; 24; 25], such as the median filter transform (MFT) with parallelogram shaped windows [23], which could be effectively used to integrate SR and image denoising to provide improved results in comparison to the conventional way. In the case of SRR, another primary problem that needs to be addressed is the blocking artifact. To reduce the blocking artifact in a reconstructed image, [PERSON] et al. proposed a fractional TV image SRR method [26; 27]; their method involves using three parts: the integer-order TV item, the fractional-order TV (FTV) item, and the data fidelity item. The integer-order TV item can maintain the discontinuity and structure of the image, while the FTV item can suitably handle non-local information, such as the texture of the image. Thus, their method can slightly reduce the staircase artifacts in an image. To overcome the insufficiency of dealing with detail and texture information in the typical TV approach, [PERSON], [PERSON], and [PERSON] proposed a two-dimensional (2D) compressive sensing sparse method for image reconstruction [28]. Their method introduces the FTV regularization term. In addition, to achieve elastic sparse representation, the discrete wavelet and curvelet transform regularization terms are combined into the cost function. Furthermore, [PERSON] and [PERSON] proposed an SRR method based on the PDE of fractional Brownian motion to enhance the random texture information in an image [29]; this method is global and does not require image patches. Fractional Brownian motion is a stochastic process with self-similarity, which can significantly characterize natural texture information; consequently, it can effectively enhance the texture information of an image while enlarging it. Recently, [PERSON] et al. proposed a non-convex fractional order variational model for image super-resolution; the use of the non-convex data fitting term can efficiently reduce complex noise such as impulse noise, while the fractional order regularization term can preserve image features like edges and texture much better [30]. In contrast, the TV method can only minimize horizontal and vertical gradients, reducing image noise in the horizontal and vertical directions, whereas the four-direction gradient regularization approach can reduce noise in the 45\({}^{\circ}\) and 135\({}^{\circ}\) directions, thereby improving the quality of image reconstruction [31; 32]. Based on the four-direction TV (TV4) proposed by [PERSON] et al. [31], [PERSON] et al. presented a four-directional fractional-order TV (FTV4) [33], which applies the fractional-order difference operator to alleviate the staircase artifacts in TV4 and improve its denoising ability. In the typical reconstruction method, the TV is based on the \(L_{1}\) norm; however, in practice, many non-convex reconstruction methods outperform the \(L_{1}\) norm sparse constrained reconstruction method. [PERSON] first proposed the non-convex optimization problem with the \(L_{p}\) quasinorm minimization (\(0<p<1\)) as the objective function [34; 35]. Further, [PERSON] et al. replaced the \(L_{1}\) norm with the \(L_{p}\) norm in the minimization function and proposed a total \(p\)-variation (TpV) minimization algorithm [36]. Recently, [PERSON] et al.