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In recent years, quaternion matrix completion (QMC) based on low-rankregularization has been gradually used in image de-noising andde-blurring.Unlike low-rank matrix completion (LRMC) which handles RGB imagesby recovering each color channel separately, the QMC models utilize theconnection of three channels by processing them as a whole. Most of theexisting quaternion-based methods formulate low-rank QMC (LRQMC) as aquaternion nuclear norm (a convex relaxation of the rank) minimizationproblem.The main limitation of these approaches is that the singular valuesbeing minimized simultaneously so that the low-rank property could not beapproximated well and efficiently. To achieve a more accurate low-rankapproximation, the matrix-based truncated nuclear norm has been proposed andalso been proved to have the superiority. In this paper, we introduce aquaternion truncated nuclear norm (QTNN) for LRQMC and utilize the alternatingdirection method of multipliers (ADMM) to get the optimization.We furtherpropose weights to the residual error quaternion matrix during the updateprocess for accelerating the convergence of the QTNN method with admissibleperformance. The weighted method utilizes a concise gradient descent strategywhich has a theoretical guarantee in optimization. The effectiveness of ourmethod is illustrated by experiments on real visual datasets.

Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion