Tensor data often suffer from missing value problem due to the complex high-dimensional structure while acquiring them. To complete the missing information, lots of Low-Rank Tensor Completion (LRTC) methods have been proposed, most of which depend on the low-rank property of tensor data. In this way, the low-rank component of the original data could be recovered roughly. However, the shortcoming is that the detail information can not be fully recovered. On the contrary, in the field of signal processing, Convolutional Sparse Coding (CSC) can provide a good representation of the high-frequency component of the image, which is generally associated with the detail component of the data. Nevertheless, CSC can not handle the low-frequency component well. To this end, we propose a novel method, LRTC-CSC, which adopts CSC as a supplementary regularization for LRTC to capture the high-frequency components. Therefore, LRTC-CSC can not only solve the missing value problem but also recover the details. Moreover, LRTC-CSC can be trained with small samples due to the sparsity characteristic of CSC. Extensive experiments show the effectiveness of LRTC-CSC, and quantitative evaluation indicates that the performance of our model is superior to state-of-the-art methods.