Abstract. Block term decomposition (BTD), which decomposes a third-order tensor into multiple terms with the multilinear rank [Formula: see text], has garnered increasing attention for high-dimensional image representation. The idea of decomposing the tensor into multiple terms has the potential to reveal the underlying different structures of the original tensor. However, BTD practically struggles to reveal these underlying different structures in the original domain, which locks the potential of the BTD. To address this problem, we propose a dictionary-based BTD (DBTD) for third-order tensors by revisiting the BTD from the convolutional dictionary learning perspective, which can better reveal the underlying different structures of the original tensor. The proposed DBTD decomposes the original tensor into multiple different terms, where each term is represented by the convolution of an adaptive dictionary and the corresponding low-rank coefficient. Herein, the adaptive dictionaries can represent distinct patterns and contribute to the DBTD’s ability to reveal the underlying different structures of the original tensor. Moreover, we establish the essential uniqueness guarantee for the DBTD. Empowered with DBTD, we suggest a high-dimensional image recovery model and develop an efficiently solving algorithm with a convergence guarantee. Numerical results on real-world high-dimensional images demonstrate that the proposed DBTD outperforms other competing decompositions in image recovery and benefits subsequent image applications.

Tensor decompositions have been successfully applied to multidimensional data recovery. However, classical tensor decompositions are not suitable for emerging spatio-irregular multidimensional data (i.e., spatio-irregular tensor), whose spatial domain is non-rectangular, e.g., spatial transcriptomics data from bioinformatics and semantic units from computer vision. By using preprocessing (e.g., zero-padding or element-wise 0-1 weighting), the spatio-irregular tensor can be converted to a spatio-regular tensor and then classical tensor decompositions can be applied, but this strategy inevitably introduces bias information, leading to artifacts. How to design a tensor-based method suitable for emerging spatio-irregular tensors is an imperative challenge. To address this challenge, we propose a learnable transform-assisted tensor singular value decomposition (LTA-TSVD) for spatio-irregular tensor recovery, which allows us to leverage the intrinsic structure behind the spatio-irregular tensor. Specifically, we design a learnable transform to project the original spatio-irregular tensor into its latent spatio-regular tensor, and then the latent low-rank structure is captured by classical TSVD on the resulting regular tensor. Empowered by LTA-TSVD, we develop spatio-irregular low-rank tensor completion (SIR-LRTC) and spatio-irregular tensor robust principal component analysis (SIR-TRPCA) models for the spatio-irregular tensor imputation and denoising respectively, and we design corresponding solving algorithms with theoretical convergence. Extensive experiments including the spatial transcriptomics data imputation and hyperspectral image denoising show SIR-LRTC and SIR-TRPCA are superior performance to competing approaches and benefit downstream applications.

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.

Multi-dimensional data are inevitably corrupted, which hinders subsequent applications (e.g., image segmentation and classification). Recently, due to the powerful ability to characterize the correlation between any two modes of tensors, fully-connected tensor network (FCTN) decomposition has received increasing attention in multi-dimensional data recovery. However, the expressive power of FCTN decomposition in the original pixel domain has yet to be fully leveraged, which can not provide satisfactory results in the recovery of details and textures, especially for low-sampling rates or heavy noise scenarios. In this work, we suggest a feature-based FCTN decomposition model (termed as F-FCTN) for multi-dimensional data recovery, which can faithfully capture the relationship between the spatial-temporal/spectral-feature modes. Compared with the original FCTN decomposition, F-FCTN can more effectively recover the details and textures and be more suitable for the subsequent high-level applications. However, F-FCTN leads to a larger-scale feature tensor as compared with the original tensor, which brings challenges in designing the solving algorithm. To harness the resulting large-scale optimization problem, we develop an efficient leverage score sampling-based proximal alternating minimization (S-PAM) algorithm and theoretically establish its relative error guarantee. Extensive numerical experiments on real-world data illustrate that the proposed method performs favorably against compared methods in data recovery and facilitates subsequent image classification.

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.

Spaceborne Synthetic Aperture Radar(SAR) can be mounted on space vehicles to collect information on the entire planet with all-day and all-weather imaging capacity. However, the spaceborne SAR sensor may suffer from severe interferences resulting in image degradation, which puts forward an urgent need for interference identification and mitigation. This paper proposes a high-precision interference identification method for spaceborne SAR images, named HPII-NET. The network is trained with simulation and real measurement data, and the effectiveness of the proposed method is verified by both simulation and the Setinel-1 satellite SAR images. Compared with the traditional identification networks, experimental results show that the HPII-NET can achieve more than 97% interference identification accuracy and thus improve the interference identification performance.

Inverse Synthetic Aperture Radar (ISAR) target recognition is an important branch of ISAR image research. The traditional ISAR recognition mission is done based on monostatic radar. However, the monostatic ISAR can only generate a single view image of the target. In this paper, to improve the recognition accuracy, a method of joint target recognition for Multi-station ISAR (MS-ISAR) via Multi-station ISAR Image Recognition (MIIR) network is proposed. In this scheme, the spatial matching algorithm and SURF algorithm are exploited to achieve multi-view fusion. The MIIR is present to achieve high accuracy recognition. To validate the proposed method, we use electromagnetic simulation software to obtain multi-view echo data of six types of aircraft targets. Then the proposed method and the traditional method are used for recognition respectively. Finally, we acquire the real experiment data of a model aircraft to validate the effectiveness of the proposed method. The results demonstrate our method provides a higher recognition accuracy rate.

The wide application of automotive radar greatly increases the risk of mutual interference between vehicles. To address this problem, this paper proposes an efficient interference suppression framework based on frequency domain sparsity. Firstly, The linear time-domain signal model is transformed into an optimal solution to the problem of extracting targets. Moreover, we utilize the orthogonal property of the Fourier matrix to avoid complex inverse matrix calculations and greatly reduce the computational memory while maintaining interference suppression performance. Both simulation and measured data validate the effectiveness of our approach, showing that our method not only suppresses mutual interference between automotive radars but also extracts range information from multiple targets.

Recently, block-term decomposition with rank-(Lr,Lr,1) (termed as LL1 decomposition), which is physically inspired by linear spectral unmixing, has received increasing attention in hyperspectral images (HSIs) denoising. However, due to the intrinsic nonlinear structure of real-world HSIs, the low-rankness of HSIs is usually implicit. Moreover, the essential uniqueness guarantee is usually violated with the low-rank assumption of the abundance maps unsupported in real scenarios, which hampers the successful deployment of LL1 decomposition. Inspired by the nonlinear spectral unmixing, we propose a nonlinear learnable transform-based LL1 decomposition (NT-LL1) for characterizing the implicit low-rank structure of real-world HSIs. More concretely, the nonlinear learnable transform in NT-LL1 decomposition is a composed transform consisting of a linear semi-orthogonal transform and a component-wise nonlinear transform, which collaboratively enhances the low-rankness of the abundance maps. Empowering with the NT-LL1 decomposition, we propose an NT-LL1 decomposition-based model for HSIs denoising. To tackle the resulting model, we develop an efficient proximal alternating minimization-based algorithm with a convergence guarantee. Extensive experimental results including simulated and real data collectively verify the superiority of the proposed method as compared with the competing methods.

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.

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.

In this article, we propose a novel tensor learning and coding model for third-order data completion. The aim of our model is to learn a data-adaptive dictionary from given observations and determine the coding coefficients of third-order tensor tubes. In the completion process, we minimize the low-rankness of each tensor slice containing the coding coefficients. By comparison with the traditional predefined transform basis, the advantages of the proposed model are that: 1) the dictionary can be learned based on the given data observations so that the basis can be more adaptively and accurately constructed and 2) the low-rankness of the coding coefficients can allow the linear combination of dictionary features more effectively. Also we develop a multiblock proximal alternating minimization algorithm for solving such tensor learning and coding model and show that the sequence generated by the algorithm can globally converge to a critical point. Extensive experimental results for real datasets such as videos, hyperspectral images, and traffic data are reported to demonstrate these advantages and show that the performance of the proposed tensor learning and coding method is significantly better than the other tensor completion methods in terms of several evaluation metrics.

SAR is usually subject to strong electromagnetic interference (EMI) during electronic reconnaissance missions, which will seriously weaken its ability of surveying and mapping. This paper presents a novel method for SAR image interference suppression based on the encoder-decoder network (named as ISEDnet). ISEDnet mainly consists of consecutive feature extraction net (FEN), the additional encoder-decoder network, and the image supervision mechanism. FEN is used to extract the features of interfered SAR images, and the Encoder-Decoder network (EDN) is used to suppress interference of SAR images. The image supervision mechanism is proposed to recover the target features. The network trained with simulation and real measurement data, the effectiveness of ISED-net are verified by both simulation and the Sentinel-1 satellite SAR images. Compared to the traditional notch filtering method, ISEDnet can successfully suppress different types of SAR interference and improve interference suppression performance.

The compressed sensing (CS) method is widely utilized in the field of radar sparse imaging. However, it always encounters enormous iterations and low generalizability. To solve these problems, in this paper, we propose a novel learning-based sparse imaging network architecture, i.e., Split Unfolding Sparsity-Driven Network (SSD-Net), for 3D synthetic aperture radar (SAR) imaging. By combining the model-based SAR imaging method and data-driven deep learning method, SSD-Net has favorable explainability and generalization abil-ity to produce 3D SAR images. The deep hierarchical ar-chitecture of SSD-Net is obtained by combiningthe radar nonlinear operator and the split Bregman method. The exper-iments demonstrate that the proposed SSD-Net outperforms other state-of-the-art methods in the field of SAR imaging.

The general tensor-based methods can recover missing values of multidimensional images by exploiting the low-rankness on the pixel level. However, especially when considerable pixels of an image are missing, the low-rankness is not reliable on the pixel level, resulting in some details losing in their results, which hinders the performance of subsequent image applications (e.g., image recognition and segmentation). In this article, we suggest a novel multiscale feature (MSF) tensorization by exploiting the MSFs of multidimensional images, which not only helps to recover the missing values on a higher level, that is, the feature level but also benefits subsequent image applications. By exploiting the low-rankness of the resulting MSF tensor constructed by the new tensorization, we propose the convex and nonconvex MSF tensor train rank minimization (MSF-TT) to conjointly recover the MSF tensor and the corresponding original tensor in a unified framework. We develop the alternating directional method of multipliers (ADMMs) to solve the convex MSF-TT and the proximal alternating minimization (PAM) to solve the nonconvex MSF-TT. Moreover, we establish the theoretical guarantee of convergence for the PAM algorithm. Numerical examples of real-world multidimensional images show that the proposed MSF-TT outperforms other compared approaches in image recovery and the recovered MSF tensor can benefit the subsequent image recognition.

The application of the compressed sensing (CS) method in the radar field enables the radar imaging system to satisfy both low data cost and high reconstruction quality, however, it is accompanied by enormous iterative operations and difficult adjustments of parameters. In this paper, we propose a learning-based split unfolding framework, dubbed as split iterative sparse reconstruction network (SISR-Net), for near-field 3-D millimeter-wave (mmW) radar sparse imaging. Firstly, a sparse reconstruction algorithm, i.e., SISRA, is proposed to theoretically guide the structure of the imaging framework. Subsequently, by combining the model-based CS method and data-driven deep learning method, SISR-Net is constructed by SISRA to produce 3-D mmW radar images efficiently with excellent explainability and generalization ability. Joint the radar-imaging kernel, echo-generation kernel, and the split Bregman method, the efficiency and stability of SISR-Net are guaranteed, all parameters are layer-varied and learned steadily by end-to-end training to improve the convergence and robustness of the imaging network. Simulated data and the echo from a high-resolution mmW radar dataset 3DRIED, are used to train and test the SISR-Net based on the Adam optimizer. For both simulation and extensive 3-D mmW radar measured experiments, the proposed SISR-Net outperforms other state-of-the-art imaging methods in terms of imaging accuracy and generalization ability.

Ship instance segmentation of high-resolution SAR images is a valuable and challenging task due to the complex scattering and noise properties. In this article, we pioneered the construction of the low-level feature to discriminate the ships and complemented the super-resolution denoising techniques in the network modules, termed low-level feature guided network (LFG-Net), for precise ship instance segmentation in SAR images. LFG-Net consists of the low-level feature concerned pyramid (LFCP), the high-resolution interaction module (HR-FIM), and the compression recovery segmentation branch (CRSB). LFCP extends vanilla FPN with the P1layer and complements super-resolution techniques to capture the regional and texture information at the image level for small object segmentation. HR-FIM interacts the bounding box region of interest (RoI) feature and mask RoI feature at the instance level with high-resolution techniques to enhance the mask RoI feature. CRSB aims at recovering the high-resolution mask predictions to improve the ship segmentation performance. Comprehensive experiments on HRSID, PSeg-SSDD, and AirSARShip indicate that LFG-Net* achieves 11.7%, 6.3%, and 12.7% AP increments compared with the Mask R-CNN baseline, respectively. Besides, it receives 9.5%, 4.9%, and 7.3% AP increments compared with state-of-the-art method, respectively, which bridges the gap of instance segmentation precision in SAR images. In terms of the visualized instance segmentation results, LFG-Net* is capable of segmenting the complex scenes, e.g, the adjacent distributed ships and ships with strong reflection noise interference, in SAR images. Code is available at: https://github.com/Evarray/LFG-Net.

The sparse imaging method based on compressed sensing (CS) is widely used in the field of millimeter-wave (MMW) synthetic aperture radar (SAR) imaging. However, 3D sparse imaging is limited by the difficult parameter tuning, the huge computational load, and the low processing efficiency. In addition, due to the motion errors and model mismatch, it is difficult to obtain well-focused results without error correction techniques. To address these issues, we propose a deep learning framework that integrates 3D sparse imaging and autofocusing, named 3D Sparse Autofocusing Network (SAF-3DNet) for MMW SAR data processing. The network is constructed based on an auto-encoder, which can optimize parameters without effective ground truth. The backbone structure of the encoder is expanded by approximate message-passing (AMP), and the operators in the frequency domain are used to replace the traditional matrix-vector CS model, which avoids large-scale matrix multiplication and other operations, and greatly improves the operation efficiency. In addition, the 2D phase error estimation in the cross-range plane is embedded into the sparse imaging models, enabling simultaneous 3D imaging and autofocusing. The decoder is designed as a mapping from the autofocusing results to the echo data. Experimental results based on both simulated and measured data demonstrate the proposed SAF-3DNet can achieve well-focused 3D reconstruction within an ephemeral time, which expresses the potential of 3D MMW SAR real-time and high-quality imaging.

In this paper, we propose a novel low-tubal-rank tensor recovery model, which directly constrains the tubal rank prior for effectively removing the mixed Gaussian and sparse noise in hyperspectral images. The constraints of tubal-rank and sparsity can govern the solution of the denoised tensor in the recovery procedure. To solve the constrained low-tubal-rank model, we develop an iterative algorithm based on bilateral random projections to efficiently solve the proposed model. The advantage of random projections is that the approximation of the low-tubal-rank tensor can be obtained quite accurately in an inexpensive manner. Experimental examples for hyperspectral image denoising are presented to demonstrate the effectiveness and efficiency of the proposed method.