The study aimed to solve the common problem that hardware limitations and degradation make the data obtained in reality usually incomplete and improve the quality of communication transmission. In this paper, we propose a new low-rank tensor complementation model LRTC-CSC, which is based on tensor kernel parametrization (TNN), preserves the low-rank structure of information while restoring the detail features, and finally solves the problem using the efficient alternating direction multiplier method (ADMM). Based on the low-rank nature of the tensor, adding convolutional sparse coding (CSC) can well represent the characteristics of the high-frequency part of the information to handle the details while recovering the global information. The experimental results show that the training set of this paper saves much time compared with other models in several metrics by using only ten images of similar color for each data. At the same time, the data recovery effect is much higher than the novel TV canonical prior. In particular, the LRTC-CSC model is 5.18 dB higher than the LRTC-TV model in terms of PSNR value for image recovery at a 70% missing rate. The LRTC-CSC model proposed in this paper is more accurate and efficient for communication data restoration.