Abstract Tensor completion has attracted attention because of its promising ability and generality. However, there are few studies on noisy scenarios which directly solve an optimization problem consisting of a “noise inequality constraint”. In this paper, we propose a new tensor completion and denoising model including tensor total variation and tensor nuclear norm minimization with a range of values and noise inequalities. Furthermore, we developed its solution algorithm based on a primal-dual splitting method, which is computationally efficient as compared to tensor decomposition based non-convex optimization. Lastly, extensive experiments demonstrated the advantages of the proposed method for visual data retrieval such as for color images, movies, and 3D-volumetric data