  In recent years, quaternion matrix completion (QMC) based on low-rank
regularization has been gradually used in image de-noising and
de-blurring.Unlike low-rank matrix completion (LRMC) which handles RGB images
by recovering each color channel separately, the QMC models utilize the
connection of three channels by processing them as a whole. Most of the
existing quaternion-based methods formulate low-rank QMC (LRQMC) as a
quaternion nuclear norm (a convex relaxation of the rank) minimization
problem.The main limitation of these approaches is that the singular values
being minimized simultaneously so that the low-rank property could not be
approximated well and efficiently. To achieve a more accurate low-rank
approximation, the matrix-based truncated nuclear norm has been proposed and
also been proved to have the superiority. In this paper, we introduce a
quaternion truncated nuclear norm (QTNN) for LRQMC and utilize the alternating
direction method of multipliers (ADMM) to get the optimization.We further
propose weights to the residual error quaternion matrix during the update
process for accelerating the convergence of the QTNN method with admissible
performance. The weighted method utilizes a concise gradient descent strategy
which has a theoretical guarantee in optimization. The effectiveness of our
method is illustrated by experiments on real visual datasets.
