Recently, the classical tensor-tensor product (T-product) has attracted considerable attention for capturing the interactions between tensor factors. However, the mode-3 consistency in the T-product restricts its flexibility and expressive ability. To break the restriction, we suggest a tensor-tensor product for arbitrary mode-3 dimension (termed as Art-product) which enables us to flexibly and expressively capture the interactions between tensor factors. Concretely, by leveraging the exclusive hierarchical nonlinear transforms along the third mode, two tensor factors with inconsistency dimensions are first transformed into the corresponding latent factors with consistency dimensions. The face-wise product is then performed between these latent factors with consistency dimensions. Empowered with this Art-product, we can readily deconstruct and reconstruct new tensor network decomposition from an interaction perspective. As a representative example, we redesign the tensor train decomposition which can benefit from the advantage of the Art-product. Extensive experiments on multi-spectral images, color videos, and light field data sustain the superiority of tensor train decomposition equipped with Art-product over classic tensor decomposition.

Thick cloud removal is a long-standing and critical challenge in remote sensing (RS) image processing, with the increasing availability of multi-temporal RS images offering new opportunities to address this problem. The main limitation of existing cloud removal methods is that the classical fidelity only considers original pixel domain or handcrafted/pretrained filtered domains, overlooking the individuality filters and the corresponding feature behind each RS image, which leads to evident detail discrepancies. To address this issue, we suggest a data-aware fidelity based on the untrained neural network, which encourages deep data-aware feature matching between the contaminated image and the guidance image. Complementary to the data-aware fidelity, we design the deep self-representation to implicitly impose regularization benefiting from the same untrained neural network. Equipped with the elaborately designed fidelity and regularization, we propose a unified data-aware fidelity and regularization learning (called DAFRL) paradigm for thick cloud removal that flexibly adapts to diverse multi-temporal RS images. Under this paradigm, the fidelity and regularization are empowered by the same untrained neural network, serving distinct functions while collaborating organically. Experimental results on both simulated and real datasets show that the proposed DAFRL effectively preserves fine details and outperforms the compared methods.

Recently, the tensor nuclear norm (TNN)-based tensor robust principle component analysis (TRPCA) has achieved impressive performance in multidimensional data processing. The underlying assumption in TNN is the low-rankness of frontal slices of the tensor in the transformed domain (e.g., Fourier domain). However, the low-rankness assumption is usually violative for real-world multidimensional data (e.g., video and image) due to their intrinsically nonlinear structure. How to effectively and efficiently exploit the intrinsic structure of multidimensional data remains a challenge. In this article, we first suggest a kernelized TNN (KTNN) by leveraging the nonlinear kernel mapping in the transform domain, which faithfully captures the intrinsic structure (i.e., implicit low-rankness) of multidimensional data and is computed at a lower cost by introducing kernel trick. Armed with KTNN, we propose a tensor robust kernel PCA (TRKPCA) model for handling multidimensional data, which decomposes the observed tensor into an implicit low-rank component and a sparse component. To tackle the nonlinear and nonconvex model, we develop an efficient alternating direction method of multipliers (ADMM)-based algorithm. Extensive experiments on real-world applications collectively verify that TRKPCA achieves superiority over the state-of-the-art RPCA methods.

Tensor decomposition-based models have received increasing attention in hyperspectral image (HSI) denoising. However, tensor decompositions (e.g., Tucker decomposition and tensor singular value decomposition) in these HSI denoising models ignore exploiting the multiple components of the HSI, resulting in unsatisfactory denoising performance. To fully exploit the multiple components of the HSI, we develop a sparsity regularized rank-(L,M,N) block term decomposition (SR-BTD). In SR-BTD, the clean HSI is decomposed as the sum of multiple components, where each component is a sparse core tensor multiplied by matrices along each mode. The sparse regularization on each core tensor can benefit determining the low-rankness of each component with the unknown rank-(L,M,N) in the real world, leading to more accurately exploiting each component. Equipped with SR-BTD, we establish the HSI denoising model and design a hierarchical alternating least squares-based algorithm to efficiently solve the resulting model. Extensive experiments on simulated and real HSI denoising tasks demonstrate SR-BTD is superior to the competing tensor decompositions in terms of numerical results and visual quality.

Since the information across all bands of the cloud-contaminated region is missing, thick cloud removal for remote sensing images (RSIs) is still a challenging problem. Recently, the availability of rich spatial–spectral–temporal information for multitemporal RSIs provides the possibility for addressing the thick cloud removal problem. However, existing methods explore the holistic redundancy of multitemporal RSIs and neglect the important semantic clue of multitemporal images. In this letter, we propose a superpixel-oriented thick cloud removal (STORM) model for multitemporal images, where the multitemporal superpixel as the generic unit allows us to exploit redundancy with semantic clue in a low-rank optimization problem. To harness the resultant irregular fourth-order tensor (i.e., multitemporal superpixels) in the optimization problem, we cleverly introduce the weighted tensor to transform the irregular tensor into the regular tensor, which naturally leads to a standard low-rank tensor optimization problem. To tackle the tensor optimization problem, we develop a proximal alternating minimization (PAM)-based algorithm. Extensive simulated and real experiments on multitemporal RSIs acquired by Sentinel-2 and Landsat-8 satellites demonstrate the superior performance of the proposed method over the comparison methods.

Thick cloud contamination in remote sensing (RS) images significantly hinders their utility in subsequent applications. Traditional cloud removal methods predominantly focus on the design of the regularization term, while neglecting the design of the fidelity term. Recently, the proposal of gradient-domain fidelity has highlighted the significance of the fidelity term design, dedicated to maintaining textures in the gradient domain. However, the handcrafted gradient-domain fidelity still has limitations in capturing invariant and delicate features behind multitemporal RS (MTRS) images, leading to unsatisfactory detail preservation. To address the problem, we suggest a deep domain fidelity to capture deep features by leveraging a pretrained deep network, which matches the intrinsic deep feature between the original images and reference images, instead of matching the shallow features in gradient domain fidelity. Empowered with the deep domain fidelity, we propose a thick cloud removal model (called DFTR) for MTRS images, organically integrating the deep domain fidelity term with a low-rank (LR) regularization term (i.e., tensor ring (TR) decomposition), offering fine detail preservation. Extensive simulated and real experiments on MTRS images demonstrate that the proposed method outperforms the compared methods, including the origin domain-based and the gradient domain-based methods, in thick cloud removal, especially for detail preservation.

Recently, due to the ability of exploiting perceptual information, superpixel-based methods have received attention for hyperspectral image (HSI) denoising. However, existing superpixel-based denoising methods unfold the irregular 3D superpixels into the matrices along the spectral mode, which inevitably destroys the intrinsic structure of the irregular 3D superpixels. To tackle the irregular 3D superpixels, we introduce the irregular tensor representation for superpixels-guided HSI denoising. More concretely, by introducing the weighted tensor, we suggest a tensor representation of each irregular 3D superpixel, which can preserve the intrinsic structure of the irregular 3D superpixel. Equipped with the irregular tensor representation, we establish a superpixel-guided tensor optimization model for HSI denoising, which simultaneously exploits the perceptual information and low-rank structure within 3D superpixels. To solve the resulting tensor optimization problem, we develop an inexact augmented Lagrange multiplier (IALM) algorithm. Experimental results show that the proposed method outperforms other state-of-the-art HSI denoising methods, particularly matrix-based methods, on both simulated and real data.

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.

The existing nonblind cloud and cloud shadow (cloud/shadow) removal methods for remote sensing (RS) images are based on the assumption that cloud/shadow masks are accurately given. Since the masks are usually manually labeled or detected by cloud detection methods, whose accuracy cannot be well guaranteed, the cloud/shadow removal effect may be affected. In this article, we suggest a robust thick cloud/shadow removal (RTCR) method that meets the problem with an inaccurate mask. To faithfully reconstruct the multitemporal information, a coupled tensor factorization is used to explore the relationship between the abundances of the multitemporal images in the same scene. Moreover, an efficient algorithm is developed to solve the proposed model based on the augmented Lagrange multiplier method. The experimental results under accurate masks and inaccurate masks demonstrate its robustness and superiority for thick cloud/shadow removal.

The hyperspectral unmixing (HU) aims at estimating the spectral signatures of endmembers (or materials) and their corresponding abundance maps of the hyperspectral image (HSI). In this work, we treat the HSI as an 3D cube and propose a new tensor-based HU method. We decompose an HSI data as the sum of several multilinear rank-(Lr, Lr, 1) terms (or LL1 model). Based on the fact that the latent factors of LL1 model are physical meaningful (i.e., abundance maps and spectral signatures), we build a nonnegative tensor decomposition optimization model with the low-rank constraint and impose an implicit regularizer to exploit the nonlocal self-similarity prior of abundance maps-whose related subproblem can be easily solved under the plug-and-play framework. We develop an alternating direction method of multipliers algorithm to solve the proposed model. Numerical experiments demonstrate the effectiveness of our algorithm.

For multi-temporal remote sensing (RS) images, the distribution of surface materials is constant concerning time and the same material shows different spectral features at different times. Decomposing the image at each time into an abundance tensor and temporal features, there is a strong similarity between abundance tensors of all time. Based on this observation, we suggest a blind thick cloud/shadow removal model, which exploits the sparsity of the cloud component and the similarity between abundance tensors, achieving both cloud detection and multi-temporal information restoration. Moreover, a mask refinement strategy is designed to pursue the optimal cloud/shadow mask. We develop an efficient algorithm to solve the proposed model based on the augmented Lagrange multiplier method. The results of simulated experiments in different scenarios verify the superiority of the proposed method for thick cloud/shadow removal.

In hyperspectral image (HSI) denoising, subspace-based denoising methods can reduce the computational complexity of the denoising algorithm. However, the existing matrix subspaces, which are generated by the unfolding matrix of the HSI tensor, cannot completely represent a tensor since the unfolding operation will destroy the tensor structure. To overcome this, we design a novel basis tensor that is directly learned from the original tensor and present a tensor subspace representation (TenSR), which is a more authentic representation for delivering the intrinsic structure of the tensor than a matrix subspace representation. Equipped with the TenSR, we then propose a TenSR-based HSI denoising (TenSRDe) model, which simultaneously considers the low-tubal rankness of the HSI tensor and the nonlocal self-similarity of the coefficient tensor. Moreover, we develop an efficient proximal alternating minimization (PAM) algorithm to solve the proposed nonconvex model and theoretically prove that the algorithm globally converges to a critical point. Experiments implemented on simulated and real data sets substantiate the denoising effect and efficiency of the proposed method.