  Recovering color images and videos from highly undersampled data is a
fundamental and challenging task in face recognition and computer vision. By
the multi-dimensional nature of color images and videos, in this paper, we
propose a novel tensor completion approach, which is able to efficiently
explore the sparsity of tensor data under the discrete cosine transform (DCT).
Specifically, we introduce two ``sparse + low-rank'' tensor completion models
as well as two implementable algorithms for finding their solutions. The first
one is a DCT-based sparse plus weighted nuclear norm induced low-rank
minimization model. The second one is a DCT-based sparse plus $p$-shrinking
mapping induced low-rank optimization model. Moreover, we accordingly propose
two implementable augmented Lagrangian-based algorithms for solving the
underlying optimization models. A series of numerical experiments including
color image inpainting and video data recovery demonstrate that our proposed
approach performs better than many existing state-of-the-art tensor completion
methods, especially for the case when the ratio of missing data is high.
