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The edge and curve of an image surface are crucial visual cues in vision psychology. Studies show that human beings can effectively process curvature information, such as distinguishing the concavity and convexity of an image. This finding indicates that curvature is essential for a desired image to be felt authentic and real. In this paper, a novel few-view computed tomography (CT) image reconstruction model is proposed based on mean curvature (MC). Similar to the total variation model, the MC employs the $L_{1}$ -norm to utilize the sparse prior information. Constructing efficient numerical algorithms for minimizing the MC model is significant due to the associated high-order Euler–Lagrange equations. A two-step numerical method, including curvature smoothing and surface fitting, is presented to solve the proposed model, which can be stably and efficiently solved by the alternating direction minimization. By applying the variable splitting method, the explicit solutions of the corresponding subproblems can be efficiently and quickly approximated by fast Fourier transform and the proximal point method. The accuracy and efficiency of the simulated and real data are qualitatively and quantitatively evaluated to verify the efficiency and feasibility of the proposed method. Comparisons with conventional algorithms demonstrate that the proposed approach has considerable advantages in few-view CT reconstruction problems.

The presence of metal objects remains a challenge in x-ray computed tomography (CT) imaging. Sinograms passing through metals, called metal trace, usually provide uncorrected information and are considered missing in CT image reconstruction. The sparse prior of an image in some appropriate transform domains, defined implicit sparsity, is often used in sinogram inpainting methods for metal trace recovery. However, conventional inpainting methods only employ the implicit sparsity of a sinogram and often result in several artifacts in the reconstructed images. In this paper, we propose a sinogram inpainting model with implicit sparsity exploitation for both sinogram and image. A newly added regularization term, which minimizes the sparse representation of image objects, is utilized to reduce unwanted artifacts and preserve fine structures. To solve the proposed model, we then present an efficient iterative algorithm based on the Chambolle-Pock optimization approach. The results for both digital phantom and actual CT data indicate that the new inpainting method exhibits reasonable performance and outperforms the conventional methods when applied to metal artifact reduction problems.