The traditional Total Variation (TV) minimization algorithm is a classical iterative reconstruction algorithm based on Compressed Sensing (CS), and can accurately reconstruct images from sparse and noisy data. However, the block artifacts may be brought by the algorithm during the reconstruction of image having not obvious piecewise constant feature. Researches show that the use of High Order Total Variation (HOTV) in the image denoising can effectively suppress the block artifacts brought by the TV model. Therefore, a HOTV image reconstruction model and its Chambolle-Pock (CP) solving algorithm were proposed. Specifically, the second order TV norm was constructed by using the second order gradient, then a data fidelity constrained second order TV minimization model was designed, and the corresponding CP algorithm was derived. The Shepp-Logan phantom in wave background, grayscale gradual changing phantom and real CT phantom were used to perform the image reconstruction experiments and qualitative and quantitative analysis under ideal data projection and noisy data projection conditions. The reconstruction results of ideal data projection show that compared to the traditional TV algorithm, the HOTV algorithm can effectively suppress the block artifacts and improve the reconstruction accuracy. The reconstruction results of noisy data projection show that both the traditional TV algorithm and the HOTV algorithm have good denoising effect but the HOTV algorithm is able to protect the image edge information better and has higher anti-noise performance. The HOTV algorithm is a better reconstruction algorithm than the TV algorithm in the reconstruction of image having not obvious piecewise constant feature and obvious grayscale fluctuation feature. The proposed HOTV algorithm can be extended to CT reconstruction under different scanning modes and other imaging modalities.

In order to solve the problem that the Total Variation (TV) iterative constraint model is easy to cause staircase artifact and cannot save the details in Computer Tomography (CT) images, an adaptive step size Non-Local Total Variation (NLTV) constraint iterative reconstruction algorithm was proposed. Considering the NLTV model is able to preserve and restore the details and textures of image, firstly, the CT model was regarded as a constraint optimization model for searching the solutions satisfying specific regular term, which means the NLTV minimization, in the solution set that satisfies the fidelity term of projection data. Then, the Algebraic Reconstruction Technique (ART) and the Split Bregman (SB) algorithm were used to ensure that the reconstructed results were constrained by the data fidelity term and regularization term. Finally, the Adaptive Steepest Descent-Projection Onto Convex Sets (ASD-POCS) algorithm was used as basic iterative framework to reconstruct images. The experimental results show that the proposed algorithm can achieve accurate results by using the projection data of 30 views under the noise-free sparse reconstruction condition. In the noise-added sparse data reconstruction experiment, the algorithm obtains the result similar to final convergence and has the Root Mean Squared Error (RMSE) as large as 2.5 times of that of ASD-POCS algorithm. The proposed algorithm can reconstruct the accurate result image under the sparse projection data and suppress the noise while improving the details reconstruction ability of TV iterative model.