The color video inpainting problem is one of the most challenging problem in the modern imaging science. It aims to recover a color video from a small part of pixels that may contain noise. However, there are less of robust models that can simultaneously preserve the coupling of color channels and the evolution of color video frames. In this paper, we present a new robust quaternion tensor completion (RQTC) model to solve this challenging problem and derive the exact recovery theory. The main idea is to build a quaternion tensor optimization model to recover a low-rank quaternion tensor that represents the targeted color video and a sparse quaternion tensor that represents noise. This new model is very efficient to recover high dimensional data that satisfies the prior low-rank assumption. To solve the case without low-rank property, we introduce a new low-rank learning RQTC model, which rearranges similar patches classified by a quaternion learning method into smaller tensors satisfying the prior low-rank assumption. We also propose fast algorithms with global convergence guarantees. In numerical experiments, the proposed methods successfully recover color videos with eliminating color contamination and keeping the continuity of video scenery, and their solutions are of higher quality in terms of PSNR and SSIM values than the state-of-the-art algorithms.

翻译：彩色视频修复问题是现代成像科学中最具挑战性的问题之一，其目标是从可能含有噪声的少量像素中恢复彩色视频。然而，目前缺乏能够同时保持颜色通道耦合与视频帧演变的鲁棒模型。本文提出一种新的鲁棒四元数张量补全(RQTC)模型来解决这一难题，并推导了精确恢复理论。核心思想是构建一个四元数张量优化模型，恢复代表目标彩色视频的低秩四元数张量和代表噪声的稀疏四元数张量。该模型在恢复满足低秩先验假设的高维数据方面非常高效。针对不满足低秩特性的情况，我们引入一种新的低秩学习RQTC模型，通过四元数学习方法将相似块重排为满足低秩先验假设的小型张量。我们还提出了具有全局收敛保证的快速算法。数值实验中，所提方法成功恢复了彩色视频，消除了颜色污染并保持了视频场景的连续性，其解在PSNR和SSIM值上均优于现有最优算法。