Modeling And Algorithm For Tensor Completion Based On Collaborative Sparse And Low-Rank Transforms

Multidimensional data/images usually can be incomplete and corrupted in practice,severely degrading data quality and limiting applications.Tensor completion(TC)aims to recover missing entries from an incomplete observation and has numerous applications in imaging processing.The transform-based tensor nuclear norm(TNN)methods have recently yielded promising results for TC.The primary goal of these methods is to exploit the low-rank structure of frontal slices under the transform along the third mode.However,these methods typically neglect that the third mode fiber of a tensor is a one-dimensional signal,which is sparse under some transforms.The main contributions of this thesis are summarized as follows:1.We suggest a collaborative sparse and low-rank transforms model called CSLRT for third-order TC,which simultaneously exploit the sparsity of third mode fibers and the low-rankness of frontal slices under the learned transforms.Here,nuclear norm and l₁norm are used to characterize the low-rankness and the sparsity of the transformed ten-sor under the learned transforms,respectively.In our work,the transformed sparsity is complementary to the transformed low-rankness,and they are organically combined and benefit from each other.Moreover,we suggest three-directional CSLRT(3DCSLRT)to fully explore the transformed sparsity of fibers and transformed low-rankness of slices along three modes.2.Due to the strong non-convexity,the proposed CSLRT and 3DCSLRT are chal-lenging to solve.We develop the multi-block proximal alternating minimization(PAM)algorithms to efficiently solve them.Under the PAM framework,the iterative sequence generated by our algorithm is bounded,and converges to a critical point.Extensive nu-merical experiments on multispectral images,videos,and color images for TC validate that the proposed CSLRT and 3DCSLRT are competitive compared with state-of-the-art methods.