Low-rank tensor decomposition (LRTD) has demonstrated significant efficacy in multidimensional image reconstruction. Indeed, LRTD driven by nonlinear relationship can capture the underlying low-rank structure more accurately, since real-world data often exhibits complex nonlinear interactions. However, the existing nonlinear LRTD methods do not to investigate the inherent nonlinear interactions in spatial neighborhoods and spectral or temporal models. To address these challenges, we propose a novel deep nonlinear low-rank tensor decomposition (DNLRTD). Specifically, we design a deep nonlinear transform network (DNTN) using multiple convolutional layers and channel attention modules to form a deep nonlinear transform (DNT). The custom-designed DNT effectively captures nonlinear interactions within spatial neighborhoods while paying attention to the nonlinear interactions of spectral or temporal dimensions, consequently achieving a lower-rank representation. By integrating DNT into the low-tubal-rank decomposition framework, we induce the deep tubal-rank and form the DNLRTD. Also, we design a customized DNLRTD optimization strategy to make it flexible for different multidimensional image reconstruction tasks. Based on DNLRTD, we construct two multidimensional image reconstruction models and develop corresponding algorithms based on the alternating direction method of multipliers (ADMM) to solve them. Extensive experimental results on spectral compressive imaging and dynamic magnetic resonance image (MRI) reconstruction verify the superior performance of the proposed method.

Tensor decompositions are powerful tools for capturing the low-rank structure of dynamic videos. However, existing tensor decompositions primarily consider pixel-wise interactions, thus capturing solely global spatio-temporal correlations and struggling to handle the complex patterns that are inherent to dynamic videos in real-world applications. To overcome this limitation, we propose a dynamic Bhattacharya-Mesner (DyBM) decomposition, which represents the dynamic video as a sum of terms, with each term being the convolution of a BM-rank 1 tensor and a learnable three-dimensional filter. The newly constructed filters enable DyBM decomposition to establish patch-wise interactions in BM-rank 1 tensors, effectively capturing both global and local spatio-temporal correlations in dynamic videos. We further provide a physical interpretation of the factors in DyBM decomposition and offer an in-depth discussion of its relationship to the original BM decomposition. To evaluate the effectiveness of DyBM decomposition, we build a dynamic video recovery model. To solve the model, we develop a corresponding optimization algorithm with a theoretical convergence guarantee. Extensive experiments verify that DyBM decomposition-based method performs more favorably than the state-of-the-art tensor decomposition-based methods especially for dynamic videos.

Recently, the block-term decomposition with rank- $(L_{r}, L_{r}, 1)$ (termed as LL1 decomposition), which decomposes a third-order tensor into the sum of the outer products between vector and matrix factors, has received increasing attention for high-dimensional image reconstruction. However, the fixed low-rank matrix decomposition in LL1 is restricted to third-order tensors, which hinders its development for higher-order tensor data (i.e., order $N \gt 3$ ). To address this, we propose a Block Customized Topology Term Decomposition (BCTD), which represents an $N$ th-order tensor as a sum of outer products of basis vectors and customized $(N-1)$ th-order coefficient tensors with flexible internal topological structures. The proposed BCTD enjoys two advantages: Firstly, it allows tackling higher-order tensors beyond the third-order tensor setting of LL1, which can better preserve the high-dimensional structure of the tensor. Secondly, it allows each term to have a customized topological structure beyond the fixed topological structure (i.e., low-rank matrix decomposition) in LL1, which can better explore the intrinsic high-dimensional low-rank structures of the tensor. To evaluate the performance of the proposed BCTD, we build the corresponding high-dimensional image reconstruction model and provide a theoretical generalization error bound between the recovered tensor of the proposed model and the underlying tensor. To solve the resulting optimization problem, we apply a proximal alternating minimization (PAM)-based algorithm with a theoretical convergence guarantee. Extensive experimental results on high-dimensional image completion and compression tasks using real-world datasets (color videos and light field images) demonstrate the superiority of the proposed model over other baseline models.

The effective integration and classification of hyperspectral images (HSIs) and light detection and ranging (LiDAR) data is of great significance in Earth observation missions, which are confronted with challenges such as insufficient information utilization and feature heterogeneity. This article proposes a multimodal quaternion representation network (MMQRN) for multisource remote sensing (RS) data classification. Specifically, we first propose the multimodal quaternion representation (MMQR), which employs the orthogonal imaginary components of quaternions to model the complex nonlinear interactions among complementary features, thereby enabling their comprehensive fusion and utilization. Subsequently, we design a multimodal feature cross-fusion (MFCF) framework to integrate multisource, multimodal, and multilevel features adequately. Finally, we leverage the ability to capture long-term dependencies of transformers to design a quaternion convolutional transformer network (QCTN) for modeling global and local spatial-spectral information, respectively. Experiments conducted on three multisource RS datasets demonstrate the superior performance of the proposed MMQRN relative to other state-of-the-art classification methods.

Recently, the fully-connected tensor network (FCTN) decomposition has shown a powerful capability of depicting intrinsic correlations between any pair of tensor modes. But there exists a challenging question in FCTN decomposition-based methods, i.e., the accurate determination of the complicated FCTN-rank, which contains${N(N-1)}/{2}$elements for$N$th-order tensors. In this paper, we design a structural sparsity regularization for the FCTN decomposition, which estimates the complicated FCTN-rank by adaptively pruning near-zero groups in FCTN factor. Based on this regularization, we propose a noisy tensor completion (NTC) model, aiming at the recovery of a tensor from its partial and noisy observation. Besides, we design a proximal alternating minimization (PAM)-based algorithm to solve the model. In theorem, we prove a guarantee for the global convergence of the developed algorithm. To further accelerate our method for large-scale data sets, we customize the randomized block sampling strategy for general tensor network decomposition methods by updating factors from small samples. Experiments demonstrate that our strategy can accurately estimate the FCTN-rank and achieve better reconstruction performances, and our methods outperform the state-of-the-art methods in the reconstruction of different types of real-world tensors.

Hyperspectral image (HSI) denoising is a critical preprocessing step in remote sensing. Recently, the denoising diffusion probabilistic models (DDPMs) have emerged as the powerful generative models. However, applying DDPM to HSI denoising task remains challenging owing to the scarcity and acquisition difficulty of HSIs. Thus, how to effectively incorporate physical priors into the DDPM to improve denoising performance remains an underexplored issue. To this end, we propose an Unmixing-Guided Learnable Abundance-Latent Diffusion for HSI Denoising (ULADiff), which is a from-scratch, task-specific diffusion framework that incorporates physically interpretable priors and conditional information into the DDPM. ULADiff comprises three key components, including a Spectral Unmixing Transformer (SUT) network, an abundance-based diffusion model, and a reconstruction module. Specifically, we employ a learnable block-based SUT module in a self-supervised manner to decompose noisy HSIs into the abundance maps and endmembers. The SUT module enables the diffusion model to operate in a lower-dimensional abundance domain that better captures the underlying structure of HSIs. Then, we incorporate the first eigenimage, the reconstructed image via Singular Value Decomposition, as a physically meaningful condition to facilitate controllable generation. Furthermore, we propose a reconstruction module that enforces a spatial-spectral consistency prior by simultaneously imposing total variation regularization on the endmembers and a sparsity constraint on the abundance maps. This design preserves the intrinsic structures of the HSI and improves reconstruction quality. Comprehensive evaluations on synthetic and real-world datasets demonstrate that ULADiff outperforms state-of-the-art methods in both quantitative performance and visual fidelity.

Thick cloud removal for multitemporal remote sensing images (MTRSIs) is a necessary preprocessing step for subsequent applications. Existing methods for cloud removal ignore spectral–temporal consistency prior (STCP), such as smooth regions of different time and bands existing in the same spatial location. To address this problem, we propose a factor-based group sparsity regularization within the low-rank tensor factorization (LRTF) framework and theoretically prove that it can characterize the STCP in MTRSIs. Based on this regularization, we construct a cloud removal model for MTRSIs. On one hand, the introduction of STCP enables the model to achieve superior cloud removal performance. On the other hand, regularization on small-sized factors rather than on the original data enables the model to have extremely low computational complexity. To solve this model, we develop a proximal alternating minimization (PAM)-based algorithm, in which we integrate a mask acquisition method based on separated cloud and shadow components. Comparative experiments using both simulated and real data demonstrate that the proposed method outperforms recent mask-unknown and mask-known methods in terms of performance and efficiency.

Cloud removal is crucial for enhancing the quality of remote sensing images (RSIs) and broadening their applicability. Tensor decomposition, which extracts latent correlations across multidimensions, has driven the development of various methods for thick cloud removal in multitemporal RSI (MTRSI). However, current tensor decomposition methods are not specifically tailored for MTRSIs, resulting in insufficient correlation representation and requiring high computational costs in processing MTRSIs. In this article, we construct a novel bilateral tensor ring (BTR) decomposition, the first method specifically designed for MTRSIs, which enables a customized representation of spatial, spectral, and temporal correlations with lower computational complexity. The fundamental idea behind BTR decomposition is to effectively distinguish between the weak spatial correlation and the strong spectral–temporal correlation while simultaneously capturing the interaction between these two components. With the support of BTR decomposition, we propose an MTRSI cloud removal model and develop an efficient proximal alternating minimization (PAM)-based algorithm to solve it. In theory, we prove a convergence guarantee for the algorithm. Extensive experimental results verify that our method offers superior cloud removal performance and delivers a 10- to 100-fold acceleration in computational efficiency compared to the state-of-the-art tensor-based methods. The code is available at:https://yubangzheng.github.io.

Convolutional long short-term memory (ConvLSTM) possesses a remarkable capability of encoding spatial information and capturing long-range dependencies in sequential data. As a result, ConvLSTM has garnered success in hyperspectral image (HSI) classification. Nonetheless, the design of the special gate structures and convolution operations contributes to a high model complexity, making it challenging to deploy in resource-constrained environments. In this article, we propose a fully tensorized ConvLSTM model for HSI spatial-spectral classification under the premise of low complexity. First, we devise a novel and efficient tensor-sequenced convolution in the tensor train (TT) format, called ETTConv. ETTConv can reduce the number of parameters and computations in the standard convolutional layer by tensorizing the convolution kernels and mapping them to a series of smaller ones. Building upon this innovation, we present a novel ETTConvLSTM unit, formed by jointly compressing all weight tensors within the recurrent units. Using it as the fundamental unit, we construct the lightweight a efficient tensor train ConvLSTM 2-D neural network (ETTCL2DNN) model, characterized by reduced complexity without compromised classification performance. Furthermore, to better preserve the joint spatial-spectral structure of HSI data, we extend the ETTConv layer and the ETTConvLSTM unit to their 3-D versions, resulting in a new lightweight a efficient tensor train ConvLSTM 3-D neural network (ETTCL3DNN) model. Extensive quantitative experimental results on three widely used HSI datasets demonstrate the superiority of the proposed methods, exhibiting enhanced classification performance with reduced model complexity.

Tensor network (TN) representation is a powerful technique for computer vision and machine learning. TN structure search (TN-SS) aims to search for a customized structure to achieve a compact representation, which is a chal-lenging NP-hard problem. Recent “sampling-evaluation”-based methods require sampling an extensive collection of structures and evaluating them one by one, resulting in pro-hibitively high computational costs. To address this issue, we propose a novel TN paradigm, named SVD-inspired TN decomposition (SVDinsTN), which allows us to efficiently solve the TN-SS problem from a regularized modeling per-spective, eliminating the repeated structure evaluations. To be specific, by inserting a diagonal factor for each edge of the fully-connected TN, SVDinsTN allows us to calculate TN cores and diagonal factors simultaneously, with the factor sparsity revealing a compact TN structure. In theory, we prove a convergence guarantee for the proposed method. Experimental results demonstrate that the proposed method achieves approximately 100 ~ 1000 times acceleration compared to the state-of-the-art TN-SS methods while maintaining a comparable level of representation ability.

To address the problems of poor robustness and target over-shrinkage in complex backgrounds of infrared small target detection (ISTD) algorithms, we propose a novel model by two steps. Firstly, we introduce a weighted tensor average nuclear norm with lpfunction (WTANN-lp) as the constraint term of the infrared background patch tensor to eliminate the issue of information’s underutilization due to transposition variability. This improves the robustness of the algorithm under complex background. Secondly, we design a directional structure tensor (DST) for extracting the target’s local prior information, which can distinguish the target from sparse residuals in multiple directions and overcome the challenge of target over-shrinkage. We use the alternating direction multiplier method (ADMM) to solve the proposed model, and extensive experiments demonstrate the superior target detection and shape reproduction performance of the proposed model compared to seven baseline detection methods.

The pixel intensity of remote sensing images at different time nodes exhibits significant differences due to factors such as changes in solar illumination angles. Consequently, the previous cloud removal methods, primarily based on the original pixel domain, yield unsatisfactory results. In this paper, considering the sharing of similar features among remote sensing images at different time nodes, we first design a novel feature-domain fidelity that leverages the feature extraction capability of convolution operator, allowing for the precise preservation of intricate details and textures inherent in multi-temporal remote sensing images. Building upon the feature-domain fidelity, we propose a cloud removal model that organically integrates the low fully-connected tensor network rank regularization, which comprehensively captures the spatial-spectral-temporal correlations of multi-temporal remote sensing images. Moreover, we develop an effective algorithm based on proximal alternating minimization to solve the proposed model. Numerical experiments conducted on both simulated and real-world data validate that the proposed method outperforms the compared ones.

The existence of thick clouds covers the comprehensive Earth observation of optical remote sensing images (RSIs). Cloud removal is an effective and economical preprocessing step to improve the subsequent applications of RSIs. Deep learning (DL)-based methods have attracted much attention and achieved state-of-the-art results. However, most of these methods suffer from the following issues: 1) ignore the physical characteristics of RSIs; 2) require paired images with/without cloud or extra auxiliary images (such as SAR); and 3) demand the cloud mask. These issues might have limited the flexibility of existing networks. In this paper, we propose a novel low-rank regularized self-supervised network (LRRSSN) that couples model-driven and data-driven methods to remove the thick cloud from multitemporal remote sensing images (MRSIs). First, motivated by the equal importance of image and cloud components as well as their intrinsic characteristics, we decompose the observed image into low-rank image and structural sparse cloud components. In this way, we obtain a model-driven thick cloud removal method where the spectral-temporal low-rank correlation of the image component and the spectral structural sparsity of the cloud component are effectively exploited. Second, to capture the complex nonlinear features of different scenarios, the data-driven self-supervised network that does not require external training datasets is designed to explore the deep prior of the image component. Third, the coupled model-driven and data-driven LRRSSN is optimized by an efficient half-quadratic splitting algorithm. Finally, without knowing the exact cloud mask, we estimate the cloud mask to preserve information in cloud-free areas as much as possible. Experiments conducted in synthetic and real-world scenarios demonstrate the effectiveness of the proposed approach.

Due to the confusion of target-like sparse structures and the interference of linear features in complex scenarios, many infrared small target detection methods struggle to effectively detect dim and small targets. In response to this challenge, we propose a new 3-D paradigm framework, which combines spatial-temporal weighting and regularization within a low-rank sparse tensor decomposition model. First, we design a novel spatial-temporal local prior structure tensor, named 3DST, which can significantly distinguish between targets and target-like sparse structures. Second, we introduce a three-directional log-based tensor nuclear norm (3DLogTNN) to provide a full characterization of the low-rankness of the background tensor. Third, we suggest a weighted three-directional total variation (3DTV) regularization to constrain smoothness features in background images. Finally, we develop an efficient alternating direction method of multipliers (ADMMs) to solve the proposed model. In particular, we devise a fast and accurate Sylvester tensor equation for accelerated subproblem solving. Extensive experimental results demonstrate that the proposed model has superior target detection and background suppression performance in complex scenarios compared with other detection methods.

The fusion and classification of hyperspectral images (HSIs) and light detection and ranging (LiDAR) data have been extensively studied using deep learning. However, traditional real-valued deep learning methods have limitations in distinguishing internal and external relations and capturing fine spatial characteristics. To break through these limitations, this letter proposes a quaternion convolutional neural network (QCNN) with extended morphological attribute profile (EMAP) quaternion representation (called EQR) for multisource remote sensing (RS) data classification by utilizing quaternion properties. Specifically, we first propose the EQR for each single-source data, which encodes the multi-attribute features in a compact yet comprehensive manner, highlighting the internal relations. Secondly, we embed EQR into QCNN to preserve the internal relations and enable the interaction of multi-attribute features. Then we develop the 3-D quaternion convolution to better exploit the 3-D characteristic of HSI. Finally, we design different attention mechanisms and a two-level fusion strategy for multisource data to learn enhanced features. Experiments on two multisource RS data sets show that the proposed method achieved better performance than other state-of-the-art classification methods.

Thick clouds in remote sensing (RS) images deteriorate the visual quality and hinder subsequent applications. The emerging multitemporal RS images with rich temporal information bring the opportunity for cloud removal. How to effectively exploit the rich temporal information of the multitemporal RS images remains challenging. As multitemporal RS images with the same geographic scene, the spatial gradient of RS images at different time nodes has a resemblance, which can guide the reconstruction of the cloudy region. Motivated by this, we suggest a gradient domain fidelity with respect to the guided gradient for thick cloud removal in multitemporal RS images, which faithfully preserves the fine edges and textures compared to the original pixel domain fidelity. Armed with the gradient domain fidelity, we propose a low-rank tensor ring decomposition model (TRGFid) for the thick cloud removal problem. In the proposed model, the guided gradient of the cloudy region is availably estimated by using the regression method from the cloud-free region of different time nodes. Moreover, we develop an efficient proximal alternating minimization-based algorithm for solving the proposed nonconvex model. Extensive simulated and real experiments show that the proposed method outperforms its competitors, and preserves fine edges and textures.

Recently, the transform-based tensor nuclear norm (TNN) framework has yielded promising results for hyperspectral image (HSI) denoising as compared with previous original-domain tensor-based models. However, the TNN framework only exploits the low-rankness of each band of HSIs (tensors) under a single spectral transform. The correlation between all bands under the transform (i.e., the global low-rankness of the transformed tensor) and the sparsity of the transformed HSI, which are beneficial for HSI denoising, is usually neglected in the TNN framework. In this article, we propose to reconcile sparse and low-tensor-ring (TR)-rank priors in the learned transformed domain (called T-RSTR model) for HSI denoising. In T-RSTR, the transform-based low-TR-rank and sparse regularizers are designed to characterize the global low-rankness and sparsity of the transformed tensors, respectively, and then the transform-based low-TR-rank and sparse regularizers are organically integrated and benefit from each other for substantially boosting denoising performance. To tackle the T-RSTR model, we elaborately design a proximal alternating minimization-based algorithm with the theoretical convergence. Extensive numerical results demonstrate that T-RSTR is superior to the competing methods.

Multispectral image (MSI) inpainting plays an important role in real applications. Recently, fully connected tensor network (FCTN) decomposition has been shown the remarkable ability to fully characterize global correlation. Considering global correlation and nonlocal self-similarity (NSS) of MSIs, this letter introduces FCTN decomposition to the whole MSI and its NSS groups and proposes a novel nonlocal patch-based FCTN (NL-FCTN) decomposition for MSI inpainting. More specially, the NL-FCTN decomposition-based method, which increases tensor order by stacking similar small-sized patches to NSS groups, cleverly leverages the remarkable ability of FCTN decomposition to deal with higher-order tensors. Besides, we propose an efficient proximal alternating minimization (PAM)-based algorithm to solve the proposed NL-FCTN decomposition-based model with a theoretical convergence guarantee. Extensive experiments on MSIs demonstrate that the proposed method achieves the state-of-the-art inpainting performance among all compared methods.

Hyperspectral images (HSIs) are often contaminated by several types of noise, which significantly limits the accuracy of subsequent applications. Recently, low-rank modeling based on tensor singular value decomposition (T-SVD) has achieved great success in HSI restoration. Most of them use the convex and nonconvex surrogates of the tensor rank, which cannot well approximate the tensor singular values and obtain suboptimal restored results. We suggest a novel HSI restoration model by introducing a fibered rank constrained tensor restoration framework with an embedded plug-and-play (PnP)-based regularization (FRCTR-PnP). More precisely, instead of using the convex and nonconvex surrogates to approximate the fibered rank, the proposed model directly constrains the tensor fibered rank of the solution, leading to a better approximation to the original image. Since exploiting the low-fibered-rankness of HSI is mainly to capture the global structure, we further employ an implicit PnP-based regularization to preserve the image details. Particularly, the above two building blocks are complementary to each other, rather than isolated and uncorrelated. Based on the alternating direction multiplier method (ADMM), we propose an efficient algorithm to tackle the proposed model. For robustness, we develop a three-directional randomized T-SVD (3DRT-SVD), which preserves the intrinsic structure of the clean HSI and removes partial noise by projecting the HSI onto a low-dimensional essential subspace. Extensive experimental results including simulated and real data demonstrate that the proposed method achieves superior performance over compared methods in terms of quantitative evaluation and visual inspection.

Image denoising is often empowered by accurate prior information. In recent years, data-driven neural network priors have shown promising performance for RGB natural image denoising. Compared to classic handcrafted priors (e.g., sparsity and total variation), the “deep priors” are learned using a large number of training samples, which can accurately model the complex image generating process. However, data-driven priors are hard to acquire for hyperspectral images (HSIs) due to the lack of training data. A remedy is to use the so-called unsupervised deep image prior (DIP). Under the unsupervised DIP framework, it is hypothesized and empirically demonstrated that proper neural network structures are reasonable priors of certain types of images, and the network weights can be learned without training data. Nonetheless, the most effective unsupervised DIP structures were proposed for natural images instead of HSIs. The performance of unsupervised DIP-based HSI denoising is limited by a couple of serious challenges, namely network structure design and network complexity. This work puts forth an unsupervised DIP framework that is based on the classic spatiospectral decomposition of HSIs. Utilizing the so-called linear mixture model of HSIs, two types of unsupervised DIPs, that is, U-Net-like network and fully connected networks, are employed to model the abundance maps and endmembers contained in the HSIs, respectively. This way, empirically validated unsupervised DIP structures for natural images can be easily incorporated for HSI denoising. Besides, the decomposition also substantially reduces network complexity. An efficient alternating optimization algorithm is proposed to handle the formulated denoising problem. Simulated and real data experiments are employed to showcase the effectiveness of the proposed approach.

Hyperspectral image (HSI) denoising is a fundamental task in remote sensing image processing, which is helpful for HSI subsequent applications, such as unmixing and classification. Thanks to the powerful representation ability of untrained deep neural networks, deep image prior (DIP)-based methods achieve tremendous successes in image processing (e.g., denoising and inpainting). However, DIP-based methods neglect the tensor low-rank prior of the underlying HSI which will be beneficial to capturing the global structure of the underlying HSI. To address this issue, we propose a novel model for HSI denoising, which can simultaneously take respective advantages of the tensor low-rank prior and the deep spatial-spectral prior. The tensor low-rank prior leads to a better global structure and the deep spatial-spectral prior is complementary to preserve better local details. On the one hand, we adopt low-rank tensor ring (TR) decomposition to characterize the tensor low-rank prior and capture the global structure of the underlying HSI. On the other hand, we employ untrained deep neural networks to flexibly represent the deep spatial-spectral prior and capture the local details of the underlying HSI. To solve the proposed model, we develop an efficient alternating minimization algorithm. Experimental results on simulated and real data validate the advantages of the proposed model in HSI denoising. Compared with state-of-the-art HSI denoising methods, the proposed method preserves better local details and the global structure of the underlying HSI.

The popular tensor train (TT) and tensor ring (TR) decompositions have achieved promising results in science and engineering. However, TT and TR decompositions only establish an operation between adjacent two factors and are highly sensitive to the permutation of tensor modes, leading to an inadequate and inflexible representation. In this paper, we propose a generalized tensor decomposition, which decomposes an Nth-order tensor into a set of Nth-order factors and establishes an operation between any two factors. Since it can be graphically interpreted as a fully-connected network, we named it fully-connected tensor network (FCTN) decomposition. The superiorities of the FCTN decomposition lie in the outstanding capability for characterizing adequately the intrinsic correlations between any two modes of tensors and the essential invariance for transposition. Furthermore, we employ the FCTN decomposition to one representative task, i.e., tensor completion, and develop an efficient solving algorithm based on proximal alternating minimization. Theoretically, we prove the convergence of the developed algorithm, i.e., the sequence obtained by it globally converges to a critical point. Experimental results substantiate that the proposed method compares favorably to the state-of-the-art methods based on other tensor decompositions.

As a preprocessing step, hyperspectral image (HSI) restoration plays a critical role in many subsequent applications. Recently, based on the framework of subspace representation and low-rank matrix/tensor factorization (LRMF/LRTF), many single-factor-regularized methods add various regularizations on the spatial factor to characterize its spatial prior knowledge. However, these methods neglect the common characteristics among different bands and the spectral continuity of HSIs. To tackle this issue, this article establishes a bridge between the factor-based regularization and the HSI priors and proposes a double-factor-regularized LRTF model for HSI mixed noise removal. The proposed model employs LRTF to characterize the spectral global low rankness, introduces a weighted group sparsity constraint on the spatial difference images (SpatDIs) of the spatial factor to promote the group sparsity in the SpatDIs of HSIs, and suggests a continuity constraint on the spectral factor to promote the spectral continuity of HSIs. Moreover, we develop a proximal alternating minimization-based algorithm to solve the proposed model. Extensive experiments conducted on the simulated and real HSIs demonstrate that the proposed method has superior performance on mixed noise removal compared with the state-of-the-art methods based on subspace representation, noise modeling, and LRMF/LRTF.

The tensor tubal rank, defined based on the tensor singular value decomposition (t-SVD), has obtained promising results in hyperspectral image (HSI) denoising. However, the framework of the t-SVD lacks flexibility for handling different correlations along different modes of HSIs, leading to suboptimal denoising performance. This article mainly makes three contributions. First, we introduce a new tensor rank named tensor fibered rank by generalizing the t-SVD to the mode-k t-SVD, to achieve a more flexible and accurate HSI characterization. Since directly minimizing the fibered rank is NP-hard, we suggest a three-directional tensor nuclear norm (3DTNN) and a three-directional log-based tensor nuclear norm (3DLogTNN) as its convex and nonconvex relaxation to provide an efficient numerical solution, respectively. Second, we propose a fibered rank minimization model for HSI mixed noise removal, in which the underlying HSI is modeled as a low-fibered-rank component. Third, we develop an efficient alternating direction method of multipliers (ADMMs)-based algorithm to solve the proposed model, especially, each subproblem within ADMM is proven to have a closed-form solution, although 3DLogTNN is nonconvex. Extensive experimental results demonstrate that the proposed method has superior denoising performance, as compared with the state-of-the-art competing methods on low-rank matrix/tensor approximation and noise modeling.

In this paper, we propose a new tensor rank, named fibered rank, by generalizing the tensor singular value decomposition (t-SVD) to the mode-k t-SVD. It factorizes a three-way tensor into two mode-k orthogonal tensors and a mode-k diagonal tensor. To efficiently remove mixed noise and finely preserve the structure information, we propose a novel hyperspectral image denoising model based on the three-directional tensor nuclear norm (3DTNN), which is a convex relaxation of the fibered rank. An efficient alternating direction method of multipliers (ADMM)-based algorithm is developed to solve the proposed model. Experimental results demonstrate the superiority of the proposed method over the compared ones.