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.

Implicit neural representation (INR) has emerged as a powerful representation for data (e.g., multispectral images and videos). Previously, most INR methods directly represent data in the original space. However, since different frequency components are mixed in the original space, it is difficult to capture these frequency components simultaneously and accurately. To alleviate this limitation, we suggest a new frequency-aware implicit neural representation (FA-INR) working in a physically interpretable and learnable frequency space by cleverly introducing an extra frequency dimension, which allows us to readily decouple and modulate different frequency components, leading to a more accurate characterization of different frequency components in a divide-and-conquer manner. Specifically, the FA-INR consists of two important modules, i.e., the frequency module and the integration module. In the frequency module, we propose a new low-rank tensor frequency function to compactly and continuously represent the latent frequency space. In the integration module, different frequency components are adaptively integrated back to the original space. Extensive experiments on various multi-dimensional data, including multispectral images, color videos, and light field data, demonstrate that the proposed FA-INR significantly outperforms the state-of-the-art INR methods, especially for characterizing high-frequency components (e.g., textures and edges).

Hyperspectral anomaly detection (HAD) aims to distinguish anomalies from background in hyperspectral images (HSIs). Recently, low-rank representation (LRR)-based methods have attracted widespread attention, but their pixel-level design overlooks semantic information in HSIs, resulting in unsatisfactory anomaly detection performance. To address this issue, we propose a prompted tensor decomposition (ProTD) for HAD, which cleverly leverages the power of large vision models with the designed prompt to guide the decomposition of HSIs. Specifically, guided by high-level semantic information distilled from a pretrained Segment Anything Model (SAM) with the prompt, we leverage uni-deep Tucker decomposition with Spatio-Spectral Total Variation (SSTV) regularization to represent the background, while using ℓ1norm to capture anomalies. To solve this model, we design an efficient algorithm based on alternating minimization. Extensive experiments on benchmark datasets (including Airport, Urban, and Beach) demonstrate that the proposed ProTD model outperforms state-of-the-art HAD methods, especially those LRR-based methods.

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.

Hyperspectral anomaly detection is an important task in remote sensing for identifying pixels with anomalous spectral signatures that deviate from their local background. Recently, low-rank and sparse representation-based methods have garnered significant attention in hyperspectral anomaly detection, which typically employ low-rank representation to characterize the background and sparse representation to capture anomalies. Since the background and anomalies usually exhibit complex characteristics beyond the low-rankness and sparsity, low-rank and sparse representation-based methods typically do not perform satisfactorily for complex scenarios. To address the challenge, we propose an unsupervised hyperspectral anomaly detection method from a continuous perspective, which organically integrates Continuous Background representation and deep Anomaly Representation (CBAR). Specifically, the CBAR model leverages the continuous low-rank tensor function to encapsulate both the low-rankness and smoothness of the background and the deep neural network to capture the complex geometric structure of anomalies. Moreover, to mitigate the overfitting of the background and anomalies to the observed HSI, we introduce two terms as overfit-shield by exploiting the prior knowledge of the background and anomalies. To solve the CBAR model, we develop an efficient alternating minimization algorithm. Extensive experiments on benchmark datasets (including Airport, Urban, Beach, and HYDICE) demonstrate that the proposed CBAR outperforms the state-of-the-art anomaly detection methods both qualitatively and quantitatively. For reproducibility, we will release our source code at: https://github.com/Weihao-Wu/CBAR.

Recently, subspace representation-based methods have shown notable success in hyperspectral image (HSI) denoising by exploiting the spectral correlation of HSIs. However, these methods cannot fully explore the spatial correlation within HSIs, resulting in unsatisfactory results. To address this limitation, we propose a novel generalized Kronecker decomposition-based subspace representation (GKDSR) method, which can simultaneously characterize the spatial and spectral correlation of HSIs. Specifically, we first decompose an HSI as the spectral subspace and coefficients. Then we impose the face-wise generalized Kronecker decomposition to the coefficients to fully characterize the spatial correlation of the HSI. Armed with the proposed GKDSR method, we build a GKDSR-based HSI denoising model, which can not only achieve a promising spectral fidelity but also preserve spatial fine details. To solve the proposed HSI denoising model, we develop an efficient alternating minimization-based algorithm. Experimental results demonstrate the significant superiority of the proposed method compared with competing methods in terms of spectral fidelity and preservation of spatial details.

Many real-world images (e.g., hyperspectral images (HSIs) and color videos) are usually partially observed and contaminated by Poisson noise, which hinder their subsequent applications. Recently, the tensor singular value decomposition (t-SVD)-based model was suggested for image recovery with Poisson observation. However, the classic t-SVD usually fails to capture the complex nonlinear structure of real-world images. To address this problem, we suggest an attention-guided low-rank tensor factorization (AGLRTF)-based model for image recovery with Poisson observation. More concretely, we consider a self-attention network as the transform in the t-SVD framework, which can treat the frontal slices unequally, allowing us to enhance the low rankness of the transformed frontal slices. Also, the self-attention block is learned unsupervised from the given data. Extensive experiments on HSIs demonstrate that our method achieves approximately a 2-dB higher PSNR metric compared with state-of-the-art methods.

Recently, fully-connected tensor network (FCTN) decomposition, which factorizes the target tensor into a series of interconnected factor tensors, has drawn growing focus on multi-dimensional visual data processing. However, the lack of clear physical interpretation for the factor tensors hinders us from introducing handcrafted regularizers to deeply explore the potential of FCTN decomposition. To tackle this issue, we suggest a unimode hierarchical nonlinear (UHN) decomposition for each factor tensor, which can adaptively capture the complex nonlinear structure and implicitly regularize factor tensors. With this UHN decomposition of the factor tensors, we naturally propose a nested fully-connected tensor network (N-FCTN) decomposition. Attributed to the adaptive and implicit regularization inherent in UHN decomposition of factor tensors, the proposed N-FCTN decomposition is expected to perform favorably against the original FCTN decomposition. Based on the proposed N-FCTN decomposition, we build a multi-dimensional visual data recovery model and provide a theoretical error bound between the recovered tensor by our model and the underlying tensor. To address the resulting non-convex and nonlinear optimization problem, we develop an efficient proximal alternating minimization (PAM)-based algorithm and establish its theoretical convergence guarantee. Extensive experimental results on multi-spectral images, color videos, and light field data demonstrate the superior recovery performance of the proposed method compared to the state-of-the-art 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, untrained neural network priors (UNNPs) have received increasing attention for multi-dimensional image recovery. However, previous studies are based on over-parameterized untrained neural networks, which results in unstable behavior. In this letter, we propose an untrained low-rank neural network prior (ULRNNP) for multi-dimensional image recovery, which enjoys the powerful representation ability and stable behavior. More specifically, the elaborately designed nonlinear Tucker decomposition module implicitly imposes low-rank constraints on the feature tensor and can more compactly represent the feature tensor. Attributed to the suggested nonlinear Tucker decomposition module, ULRNNP can simultaneously enjoy strong representation ability and stable behavior. The friendly stable behavior allows us to design a friendly stopping criteria without the reference ground truth image as compared with classic UNNP-based methods. Extensive experiments on different multi-dimensional image datasets validate the superior performance of the proposed ULRNNP over state-of-the-art methods.

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.