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

A flexible transform-based tensor product named $\star_{{\rm{QT}}}$-product for $L$th-order ($L\geq 3$) quaternion tensors is proposed. Based on the $\star_{{\rm{QT}}}$-product, we define the corresponding singular value decomposition named TQt-SVD and the rank named TQt-rank of the $L$th-order ($L\geq 3$) quaternion tensor. Furthermore, with orthogonal quaternion transformations, the TQt-SVD can provide the best TQt-rank-$s$ approximation of any $L$th-order ($L\geq 3$) 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.