This paper examines the challenge of Computed Tomography (CT) image reconstruction problem from incomplete projection data. Based on the underdetermined system of equations, we incorporate a Total Variation (TV) regularization term and data fidelity term to formulate a TV-CT model for image reconstruction. We present a comprehensive derivation of the primal-dual method to solve the corresponding saddle point problem, as well as the primal-dual algorithms. Particularly, we introduce adaptive stepsize strategies for the proposed primal-dual algorithm to enhance the reconstruction performance. Finally, numerical experiments are conducted to verify the proposed method, including comparisons with state-of-the-art methods.

Convex clustering, turning clustering into a convex optimization problem, has drawn wide attention. It overcomes the shortcomings of traditional clustering methods such as K-means, Density-Based Spatial Clustring of Applications with Noise (DBSCAN) and hierarchical clustering that can easily fall into the local optimal solution. However, convex clustering is vulnerable to the occurrence of outlier features, as it uses the Frobenius norm to measure the distance between data points and their corresponding cluster centers and evaluate clusters. To accurately identify outlier features, this paper decomposes data into a clustering structure component and a normalized component that captures outlier features. Different from existing convex clustering evaluating features with the exact measurement, the proposed model can overcome the vast difference in the magnitude of different features and the outlier features can be efficiently identified and removed. To solve the proposed model, we design an efficient algorithm and prove the global convergence of the algorithm. Experiments on both synthetic datasets and UCI datasets demonstrate that the proposed method outperforms the compared approaches in convex clustering.