A robust tensor recovery model based on second-order difference-induced adaptive tensor nuclear norm regularization that consistently improves PSNR and ERGAS under all tested noise settings while achieving competitive SSIM values is proposed.
Abstract
Tensor data, such as hyperspectral images and videos, are often degraded by mixed noise, including Gaussian noise, sparse corruption, and outliers. In this paper, we propose a robust tensor recovery model based on second-order difference-induced adaptive tensor nuclear norm regularization. The underlying clean tensor is represented by a representative coefficient tensor and a learned orthogonal basis along the third mode, so that global low-rank correlations can be characterized in a compact and data-adaptive coefficient domain rather than in a fixed transform space. To incorporate local smoothness into the same representation, tensor nuclear norm penalties are imposed on the spatial second-order difference tensors of the representative coefficients. Compared with conventional first-order total variation, the proposed regularizer models local curvature variations and the correlations among second-order difference patterns, which helps reduce staircase artifacts while preserving structural details. A Hybrid Ordinary–Welsch fidelity term and an $\ell _{1}$ -norm sparse error term are further incorporated to improve robustness against mixed noise. The resulting optimization problem is solved by an ADMM-based algorithm. Experiments on hyperspectral image and video denoising demonstrate that the proposed method consistently improves PSNR and ERGAS under all tested noise settings while achieving competitive SSIM values.
Image Sparse Noise Removal constitutes a core issue within the domain of image processing. The tensor nuclear norm (TNN) has achieved great success in color image sparse denoising, but it suffers from the problem of direction sensitivity. In order to address this problem, a color image sparse noise removal scheme via the rotational tensor nuclear norm (RTNN) is proposed. First, to characterize the correlation across each dimension of the image, the RTNN is introduced. Then, on the basis of the RTNN, a model for image sparse noise removal is established. Next, an effective algorithm for the image sparse noise removal model is designed by adopting the ADMM. Finally, numerical experiments demonstrate that our scheme surpasses other compared methods with respect to evaluation metrics.
Li Guo, Pei-Yu Qin· Journal of Physics, Conferen...· 0 citations
This work provides a theoretical analysis which establishes a non-asymptotic reconstruction error bound that characterizes the effects of sampling complexity, optimization convergence, and model mismatch in a structured tensor approximation problem.
Jeongmin Chae, Usama Saleem, Selin Bac et al.· 0 citations
Principal component analysis is a second-order method, it selects covariance dominant directions. In local data models, however, a structural component may be rare or intermittent and therefore have modest variance but large fourth-order response. This note demonstrates a straightforward fourth-order enhancement based on Z-eigenvectors of fourth-order tensors. We prove a separation result showing that, in a fourth-order-dominant regime, rank-k PCA selects nuisance directions, while successively selected fourth-order maximizing directions recover the structural subspace. In the noiseless case, the resulting structural projection has strictly smaller squared reconstruction error for every nonzero structural vector. For arbitrary deterministic errors, we give an explicit sufficient condition under which the same improvement holds. We also establish the stability of our residual recovery scheme under tensor perturbation. Numerical experiments include a controlled additive-noise study and an application-motivated synthetic baseball-swing example, illustrating the distinction between PCA and residual fourth-order recovery under perturbation and in mixed feature coordinates.
Hyperspectral image (HSI) denoising remains a critical challenge due to noise corruption during acquisition. While nonlocal low-rank (LR) tensor methods leverage spatial–spectral correlations, they usually fail under heavy or complex noise, as directly estimating LR tensors from noisy observations usually leads to residual noise accumulation. To address this limitation, we propose a novel nonlocal low-rank residual (NLRR) approach, which reformulates LR tensor recovery as a progressive residual minimization problem. Unlike conventional methods that exclusively approximate LR tensors directly from degraded observations, the proposed NLRR approach iteratively refines the latent LR tensor structure by minimizing the rank residual, thereby decoupling noise suppression from tensor approximation. This residual-driven framework uniquely integrates two complementary priors: (1) a nonlocal LR residual prior that exploits spatial self-similarity, and (2) a global spectral LR prior that suppresses spectral redundancy. To generalize the proposed NLRR approach to real-world scenarios with mixed noise, we develop the NLRR-robust principal component analysis (NLRR-RPCA) framework, which incorporates the LR residual along with global spectral LR and sparse tensor priors for mixed noise removal. Additionally, to ensure both numerical stability and computational tractability, we develop an adaptive rank-adjusted alternating minimization algorithm, which dynamically adjusts the ranks of the estimated tensors to better handle different noise scenarios. Extensive experiments on both simulated and real HSI datasets demonstrate that our proposed NLRR approach outperforms numerous popular or state-of-the-art methods in both quantitative evaluation and visual perception.
Lixia Xia, Youqun Chen, Xin Wang et al.· Mathematics· 0 citations
Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor. We prove that squared relative reconstruction error orthogonally decomposes into interaction loss and out-of-subspace energy, and derive a Wedin-type bound establishing the stability of a plug-in Direction Loss estimator. Experiments on synthetic and hyperspectral datasets demonstrate that nearly identical reconstruction errors can yield markedly different structural-loss profiles; hyperspectral patches with comparable reconstruction errors exhibit up to a 4.6-fold difference in Direction Loss, correlating with severe visual blurring.
Multilinear principal component analysis (MPCA) reduces the dimension of tensor-valued data while preserving their mode-specific structure, but its quadratic scatter criterion can be unstable under heavy-tailed distributions and contamination. We propose spatial-sign-based multilinear principal component analysis (SMPCA), a robust dimension-reduction method that centers the observations by their spatial median, removes radial magnitude through spatial-sign normalization, and estimates the mode-wise loading spaces by alternating eigendecompositions. Under a separable tensor elliptical model, we show that the target mode-wise loading spaces uniquely maximize the population criterion and that one complete sweep of exact population block updates recovers them from any initialization. We also characterize exactly when their tensor-product subspace coincides with a leading unrestricted subspace of vectorized spatial-sign PCA and, when finite second moments exist, ordinary vectorized PCA. At the sample level, we derive explicit statistical rates for the mode-wise subspaces and the joint multilinear projector, obtain corresponding reconstruction guarantees, establish consistency of the cumulative-contribution dimension selector, and prove that the objective values generated by exact cyclic updates are nondecreasing and convergent. Simulations and an empirical application show that SMPCA is more accurate and stable than competitors under heavy-tailed distributions and outlier contamination, while retaining competitive performance under light-tailed settings.
Dong-Xu Yang, Wanfeng Liang, Le Zhou et al.· 0 citations
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