  The tensor train rank (TT-rank) has achieved promising results in tensor
completion due to its ability to capture the global low-rankness of
higher-order (>3) tensors. On the other hand, recently, quaternions have proven
to be a very suitable framework for encoding color pixels, and have obtained
outstanding performance in various color image processing tasks. In this paper,
the quaternion tensor train (QTT) decomposition is presented, and based on that
the quaternion TT-rank (QTT-rank) is naturally defined, which are the
generalizations of their counterparts in the real number field. In addition, to
utilize the local sparse prior of the quaternion tensor, a general and flexible
transform framework is defined. Combining both the global low-rank and local
sparse priors of the quaternion tensor, we propose a novel quaternion tensor
completion model, i.e., QTT-rank minimization with sparse regularization in a
transformed domain. Specifically, we use the quaternion weighted nuclear norm
(QWNN) of mode-n canonical unfolding quaternion matrices to characterize the
global low-QTT-rankness, and the l1-norm of the quaternion tensor in a
transformed domain to characterize the local sparse property. Moreover, to
enable the QTT-rank minimization to handle color images and better handle color
videos, we generalize KA, a tensor augmentation method, to quaternion tensors
and define quaternion KA (QKA), which is a helpful pretreatment step for
QTT-rank based optimization problems. The numerical experiments on color images
and color videos inpainting tasks indicate the advantages of the proposed
method over the state-of-the-art ones.
