Approximation strategy based on the T-product for third-order quaternion tensors with application to color video compression

Following up the ideas of T-product algebra for third-order real tensors, this paper aims to establish an approximation strategy based on the T-product for third-order quaternion tensors. First, we constructively prove the existence of the tensor-quaternion singular value decomposition (t-QSVD) for third-mode symmetric quaternion tensors, and provide a way to compute the t-QSVD via fast Fourier transforms. Then further, we propose a T-product based compression strategy for any given third-order quaternion tensor. Last, the effectiveness of the T-product based compression strategy is demonstrated by numerical simulations on color videos.