Weighted Order-p Tensor Nuclear Norm Minimization and Its Application to Hyperspectral Image Mixed Denoising

Chengxun He, Qiujie Cao, Yang Xu, Le Sun, Zebin Wu, Zhihui Wei

Abstract: Recently, tensor singular value decomposition (t-SVD) has demonstrated excellent performance in various high-dimensional information processing applications. However, in adapting t-SVD to handle the typical tensor data restoration tasks, such as hyperspectral image (HSI) denoising, the following questions remain inadequately addressed: 1) the existing tensor nuclear norm minimization (TNN) regime treats all tensor singular values alike; thus, it lacks flexibility and dominance in dealing with the sophisticated HSI tensor; 2) the existing t-SVD-based denoising methods cannot directly process order- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p&gt;3$ </tex-math></inline-formula> ) tensors; thus, they fail to comprehensively exploit the high-dimensional structural correlation of the HSI tensor along different modes. To address the above challenges, in this study, we first generalize a novel weighted order- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> TNN minimization regime, which integrates the adaptively reweighting strategy for matrix, third-order, and order- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> tensors in a unified architecture. Subsequently, an efficient subspace low-rank learning model is established, using HSI denoising tasks as an application example to corroborate the superiority of the proposed regime in approximating the high-dimensional low-rank structure of natural tensor data. Extensive experimental results substantiate that our effort surpasses existing state-of-the-art low-rank tensor recovery methods in both restoration accuracy and efficiency. The source code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/CX-He/WTNN.git</uri> .