## 1 Introduction  
In recent years, X-ray computed tomography (CT) has been able to capture images of internal organ tissues and structures by a non-destructive technique, which has been successfully used in various fields such as archaeology, medical diagnosis, and industrial inspection [1, 2]. But there has been growing concern that the human body will absorb some X-rays, leading to a risk of cancer [3, 4]. To avoid excessive radiation doses, the well-known ALARA (as low as reasonably achievable) principle is extensively applied in clinical practice. Commonly, there are two methods to reduce the radiation dose, for example, reducing the detector unit X-ray flux and decreasing the scanned views [5, 6, 7, 8]. However, reducing the radiation dose and the scanned views usually cause high noise and artifacts in the reconstructed images by conventional filtered back projection (FBP) [9]. Therefore, how to reconstruct a satisfactory image from noisy projection data or under-sampled projection data has become a hot research issue. This work focuses on the under-sampled projection data case. In order to obtain finely detailed reconstructed images from under-sampled projection data, researchers have proposed some iterative reconstruction methods. Typical iterative reconstruction algorithms such as the algebraic reconstruction technique (ART) [10], the simultaneous algebraic reconstruction technique (SART) [11], and expectation maximization (EM) [12] combined with prior regularization constraint could improve image quality to a certain extent. Total variation (TV), as a simple and effective regularization, has been widely applied in the CT field [13; 14]. But the disadvantage of TV is the assumption that the reconstructed image is constructed of piecewise constants [15], leading to serious staircase artifacts in the reconstructed image. Therefore, variant methods have been presented to improve the image quality. [PERSON] et al. proposed an edge-preserving total variation (EPTV) method, which allocated different weights to penalize the pixels at the non-edge or edge position in the image [16]. [PERSON] et al. developed a non-local total-variation (NLTV) minimization method, which combined with reweighted \(L_{1}\)-norm to process image [17]. [PERSON] et al. presented an adaptive-weighted total variation (AWTV) regularization, which considered the weights depending on the anisotropic edge properties between adjacent image voxels [18]. [PERSON] et al. used a penalized weighted least-squares (PWLS) method with a total generalized variation (TGV), which showed the advantage in reconstruction accuracy [19]. [PERSON] et al. presented a novel regularization combined constraint structural group sparsity with TV prior sparsity, which effectively suppressed the noise and artifacts [20]. Although the above-mentioned methods have advantages in suppressing staircase artifacts effects, they are still based on TV and may lose subtle details and blur the image edge. Later in the CT reconstruction field, dictionary learning (DL) was proposed and had demonstrated great potential. DL methods were firstly developed by [PERSON] et al. [21] for low-dose CT (LDCT) image reconstruction via combining the PWLS method with regularization involving redundant pre-learned dictionaries. [PERSON] et al. designed a new discriminative dictionary for both the reconstruction and segmentation, providing better results than common dictionary-based methods [22]. [PERSON] et al. proposed an iterative reconstruction algorithm for LDCT images based on image block classification and dictionary learning to address the limitation that all image blocks are represented by the same dictionary, which greatly improved the image quality [23]. [PERSON] et al. combined the \(L_{1}\)-norm dictionary learning penalty with a statistical iterative reconstruction framework for LDCT image reconstruction, which accelerated the global convergence rate [24]. [PERSON] et al.