He Chengxun, Xu Yang, Wu Zebin, Zheng Shangdong, Wei Zhihui
IEEE Trans Image Process. 2024;33:6409-6424. doi: 10.1109/TIP.2024.3475738. Epub 2024 Nov 12.
The recently proposed high-order tensor algebraic framework generalizes the tensor singular value decomposition (t-SVD) induced by the invertible linear transform from order-3 to order-d ( ). However, the derived order-d t-SVD rank essentially ignores the implicit global discrepancy in the quantity distribution of non-zero transformed high-order singular values across the higher modes of tensors. This oversight leads to suboptimal restoration in processing real-world multi-dimensional visual datasets. To address this challenge, in this study, we look in-depth at the intrinsic properties of practical visual data tensors, and put our efforts into faithfully measuring their high-order low-rank nature. Technically, we first present a novel order-d tensor rank definition. This rank function effectively captures the aforementioned discrepancy property observed in real visual data tensors and is thus called the discrepant t-SVD rank. Subsequently, we introduce a nonconvex regularizer to facilitate the construction of the corresponding discrepant t-SVD rank minimization regime. The results show that the investigated low-rank approximation has the closed-form solution and avoids dilemmas caused by the previous convex optimization approach. Based on this new regime, we meticulously develop two models for typical restoration tasks: high-order tensor completion and high-order tensor robust principal component analysis. Numerical examples on order-4 hyperspectral videos, order-4 color videos, and order-5 light field images substantiate that our methods outperform state-of-the-art tensor-represented competitors. Finally, taking a fundamental order-3 hyperspectral tensor restoration task as an example, we further demonstrate the effectiveness of our new rank minimization regime for more practical applications. The source codes of the proposed methods are available at https://github.com/CX-He/DTSVD.git.
最近提出的高阶张量代数框架将由可逆线性变换诱导的张量奇异值分解（t-SVD）从三阶推广到d阶（ ）。然而，推导得到的d阶t-SVD秩基本上忽略了张量高阶模式中非零变换高阶奇异值数量分布中的隐含全局差异。这种疏忽导致在处理实际多维视觉数据集时恢复效果次优。为应对这一挑战，在本研究中，我们深入研究了实际视觉数据张量的内在特性，并致力于忠实地测量它们的高阶低秩性质。从技术上讲，我们首先提出了一种新颖的d阶张量秩定义。这个秩函数有效地捕捉了在实际视觉数据张量中观察到的上述差异特性，因此被称为差异t-SVD秩。随后，我们引入了一个非凸正则化器，以促进相应的差异t-SVD秩最小化机制的构建。结果表明，所研究的低秩逼近具有闭式解，避免了先前凸优化方法所导致的困境。基于这一新机制，我们精心开发了两个用于典型恢复任务的模型：高阶张量补全和高阶张量鲁棒主成分分析。关于四阶高光谱视频、四阶彩色视频和五阶光场图像的数值示例证实，我们的方法优于以张量表示的现有竞争对手。最后，以一个基本的三阶高光谱张量恢复任务为例，我们进一步证明了我们新的秩最小化机制在更实际应用中的有效性。所提方法的源代码可在https://github.com/CX-He/DTSVD.git获取。
