  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.
