To suppress 3D point cloud noise, a feature-preserving reconstruction method using sparse optimization is proposed, which can restore sharp features while suppressing noise. First, the curvature of the underlying manifold surface is estimated using the eigenvalues of the local tensor matrix, which is constructed by using the neighboring points. To avoid the influence of outliers on normal estimation, pair consistency voting is used to realize robust statistical identification of feature points in the neighborhood. In the L0 minimization framework, an adaptive differential operator, based on feature identification, is introduced to avoid generation of artifacts in the alternating optimization process, and the projection regularization term is used to alleviate curved surface degradation. According to the optimized normal field, the sharp features are restored by projection optimization. The experimental results show that the reconstructed point cloud error is reduced by 10.2% on average, and the normal error is reduced by 29.7% on average. In addition, the subjective visual effect is better than several state-of-the-art algorithms. The introduced method can effectively improve the point cloud quality and provide technical support for 3D measurement and reverse modeling based on the point cloud.