Wang Jian-Li, Huang Ting-Zhu, Zhao Xi-Le, Jiang Tai-Xiang, Ng Michael K
IEEE Trans Image Process. 2021;30:3581-3596. doi: 10.1109/TIP.2021.3062995. Epub 2021 Mar 11.
This paper addresses the tensor completion problem, which aims to recover missing information of multi-dimensional images. How to represent a low-rank structure embedded in the underlying data is the key issue in tensor completion. In this work, we suggest a novel low-rank tensor representation based on coupled transform, which fully exploits the spatial multi-scale nature and redundancy in spatial and spectral/temporal dimensions, leading to a better low tensor multi-rank approximation. More precisely, this representation is achieved by using two-dimensional framelet transform for the two spatial dimensions, one/two-dimensional Fourier transform for the temporal/spectral dimension, and then Karhunen-Loéve transform (via singular value decomposition) for the transformed tensor. Based on this low-rank tensor representation, we formulate a novel low-rank tensor completion model for recovering missing information in multi-dimensional visual data, which leads to a convex optimization problem. To tackle the proposed model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the structured optimization problem. Numerical examples on color images, multispectral images, and videos illustrate that the proposed method outperforms many state-of-the-art methods in qualitative and quantitative aspects.
本文研究张量补全问题，其旨在恢复多维图像的缺失信息。如何表示底层数据中嵌入的低秩结构是张量补全中的关键问题。在这项工作中，我们提出了一种基于耦合变换的新型低秩张量表示方法，该方法充分利用了空间多尺度特性以及空间和光谱/时间维度上的冗余性，从而实现更好的低张量多秩逼近。更确切地说，这种表示是通过对两个空间维度使用二维小波框架变换、对时间/光谱维度使用一维/二维傅里叶变换，然后对变换后的张量使用卡尔胡宁 - 洛伊夫变换（通过奇异值分解）来实现的。基于这种低秩张量表示，我们制定了一种新型的低秩张量补全模型，用于恢复多维视觉数据中的缺失信息，这导致了一个凸优化问题。为了解决所提出的模型，我们开发了针对结构化优化问题量身定制的交替方向乘子法（ADMM）算法。彩色图像、多光谱图像和视频的数值示例表明，所提出的方法在定性和定量方面均优于许多现有方法。
