A DCT-based Tensor Completion Approach for Recovering Color Images and Videos from Highly Undersampled Data

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 DCT-based tensor completion models as well as two implementable algorithms for their solutions. The first one is a DCT-based weighted nuclear norm minimization model. The second one is called DCT-based $p$-shrinking tensor completion model, which is a nonconvex model utilizing $p$-shrinkage mapping for promoting the low-rankness of data. Moreover, we accordingly propose two implementable augmented Lagrangian-based algorithms for solving the underlying optimization models. A series of numerical experiments including color and MRI 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.