Abstract: The tensor completion problem aims at recovering the complete structure from partially observed tensor data and is centred on the use of a priori properties of the data for missing information completion. Low rank and local smoothness as two typical a priori characteristics of tensor data that is often utilised separately as independent constraints in traditional recovery methods. Paper innovatively introduce the modified correlated total variation with a joint low-rank and local smoothness. Compared with traditional recovery models, this method can more fully exploit the intrinsically coupled low-rank global structure and local smoothing features in tensor data. Demonstrate significant advantages in hyperspectral image, multispectral data and color image recovery tasks. Especially in color image processing, the proposed model has a higher fidelity of edge details as well as a higher degree of color reproduction.