Quaternion tensor singular value decomposition using a flexible transform-based approach

A flexible transform-based tensor product named ?QT-product for Lth-order (L ≥ 3) quaternion tensors is proposed. Based on the ?QT-product, we define the corresponding singular value decomposition named TQt-SVD and the rank named TQt-rank of the Lth-order (L ≥ 3) quaternion tensor. Furthermore, with orthogonal quaternion transformations, the TQt-SVD can provide the best TQt-rank-s approximation of any Lth-order (L ≥ 3) quaternion tensor. In the experiments, we have verified the effectiveness of the proposed TQt-SVD in the application of the best TQt-rank-s approximation for color videos represented by third-order quaternion tensors.