Low-rank Component Research Articles

An Underwater Polarization Imaging Technique Based on the Construction and Decomposition of the Low-Rank and Sparse Matrix in Stokes Space for Polarization State Imaging

Traditional underwater polarization imaging methods can only provide clear degree of polarization (DOP) and intensity images of the object but cannot provide images of the polarization state of the object. This paper proposes a method to extract clear object information from turbid water in all four Stokes parameter (I, Q, U, and V) channels by using the full Stokes camera, enabling clear polarization state image reconstruction. The method utilizes multiple images from different angles to construct a low-rank and sparse matrix. Then, by decomposing this matrix into sparse and low-rank components, clear Q, U, and V images (i.e., the full polarization state) can be obtained. Unlike traditional methods that assume the circularly polarized component (V component) to be zero, this method retains V channel information, allowing for circular polarization component measurement. The study successfully reconstructed clear underwater images of samples with inhomogeneous DOP distribution and obtained the clear polarization states of polarizers and fish in the turbid water. The results show that the proposed method can visualize and analyze the object’s polarization state quantitatively with high accuracy in turbid water for the first time, potentially extending the applicability of polarization underwater imaging in ocean exploration.

A novel bearing weak fault diagnosis method based on rank constrained low-rank and sparse decomposition

Abstract Addressing the challenge of diagnosing incipient bearing faults amidst significant noise, a novel diagnostic approach is introduced, leveraging a Rank Constrained Low-Rank and Sparse Decomposition (RCLRSD) model tailored for weak fault detection in bearings. Initially, we raised the Autocorrelation Function of the Square Envelope in Frequency Domain as an innovative method for the estimation of fault frequencies. Subsequently, we constructed a two-dimensional observation matrix, which is formulated independently of predefined assumptions. Then, we examine the configuration and distribution patterns of bearing fault signals, uncovering the low-rank nature of fault characteristics within a designated two-dimensional transform domain. Moreover, we found the noise signal to exhibit sparsity and an approximate Gaussian distribution. Based on this, a rank-constrained low-rank sparse decomposition model is established, and rank-constrained low-rank regular constraints for feature information and sparse regular constraints and Gaussian constraints for interference signals are constructed respectively. Ultimately, we employed the Gray Wolf Optimization algorithm to refine the model parameters, and we deduced the model’s solver via the Alternating Direction Method of Multipliers. The proposed RCLRSD model decomposes bearing fault data into three constituents: low-rank, sparse, and Gaussian components, effectively addressing the challenge of extracting weak fault signatures from bearings. The weak feature extraction capability of the RCLRSD model is verified using a multi-interference simulation model and experimental data of bearing failures under strong noise conditions; the generalization of the model is verified by the classification effect of the Support Vector Machine under two bearing failure datasets. Comparison with various algorithms confirms the superiority of the proposed algorithm.

Coastline target detection based on UAV hyperspectral remote sensing images

Timely and accurate monitoring of typical coastal targets using remote sensing technology is crucial for maintaining marine ecological stability. Hyperspectral target detection technology proves to be an effective tool in extracting various typical materials along the coastline. Traditional target detection methods using spectral domain information can effectively retain the intrinsic properties of the material. However, it is difficult to effectively recognize targets in homogeneous regions by using only spectral domain information, which may lead to insufficient utilization of spatial information. In this study, a detector based on signal-to-noise ratio fusion constrained energy minimization with low-rank sparse decomposition (SFLRSD) is proposed. This detector improves the separability of background and target by obtaining spatial domain information from hyperspectral images and fusing spectral domain information. First, total variation regularization and fractional Fourier transform are applied to process spatial and spectral domain information, respectively. The constrained energy minimization (CEM) detector is used to improve the separability between the target and background of the processed data. Then, the background and anomalies are represented as low-rank and sparse components, respectively, using low-rank sparse matrix factorization. This transforms the model solution into a covariance matrix problem, which is then solved using marginal distance difference (MDD) to isolate anomalous parts. Subsequently, the anomaly parts are fused with CEM detector results, weighted by their respective signal-to-noise ratios. This detection model leverages unified hyperspectral image features, enhancing spectral discreteness of anomalous targets and backgrounds. Finally, experiments on custom created hyperspectral dataset show that the proposed method outperforms other baseline methods in terms of visualization and quantitative performance. In this paper, we not only propose a new hyperspectral target detection method, but we also collect three typical marine litter of different materials by means of airborne hyperspectral remote sensing and construct four hyperspectral datasets in a real environment. All the simulation experiments in this paper are conducted in these four datasets.

Development of a unified framework of low-rank approximation and deep neural networks for predicting the spatial variability of SSC in `Spania' watermelons using vis/NIR hyperspectral imaging

Soluble Solids Content (SSC) is an important quality attribute that represents the internal quality of fruits. Visible and near-infrared (Vis/NIR) combined with chemometric algorithms are now popular methods for non-invasive measurement and visualization of SSC. However, in fruits with a thick rind and a large flesh volume, such as watermelon, SSC is not evenly distributed across the fruit. The variability in SSC across the fruit flesh may have an adverse effect on the accuracy of quality analysis algorithms. Thus, this paper presents an accurate and efficient approach for predicting the spatial variation of SSC in watermelons using a combined framework of low-rank approximation and deep neural networks. Watermelon ‘Spania’ was selected for this study, and hyperspectral images of each watermelon were taken from various views, including the top, bottom, and two lateral views. The low-rank property is employed as a constraint in this approach to eliminate the unwanted variations in the spectral data. The low-rank component of the spectral data, free of unwanted variations, is then fed into a fully connected neural network (FNN) for the prediction of watermelon SSC values. The proposed approach obtained optimal performance in calibration and prediction with RC2=0.982, RMSEP = 0.132, and RP2=0.945, RMSEP = 0.195 respectively. Further, the prediction outcomes of the proposed method were compared with state-of-the-art such as Partial Least Square Regression (PLSR), Support Vector Regression (SVR), Multiple Linear Regression (MLR), Decision Tree (DT), and Random Forest (RF). The overall results showed that combining HSI data with a low-rank and deep neural network framework is an efficient method for accurately predicting SSC in watermelons as well as correctly predicting spatial variability of SSC in individual fruits.

A robust underwater polarization image recovery based on Angle of Polarization with low-rank and sparse decomposition

Underwater imaging method based on the Angle of Polarization (AoP) is extremely popular due to its ability to remove the backscattered light effectively. However, existing methods overlook a potentially important phenomenon: the AoP contains significant noise in the real world, which leads to inaccurate estimations of key parameters. In this study, we propose a novel robust underwater polarization image recovery method based on AoP with low-rank and sparse decomposition. Based on the analysis of polarization, the AoP of backscattering exhibits low-rank characteristics, whereas the AoP of noise and targets demonstrates sparsity. Considering different sparsity of AoP, we combine Robust Principal Component Analysis and Alternating Direction Multiplier Method (RPCA-ADMM) with an adaptive threshold shrinkage method for low-rank and sparse decomposition, the AoP of the backscattering is obtained. Moreover, according to the low-rank component of the AoP, the Degree of Linear Polarization (DoLP) of backscattering can be effectively and automatically estimated without considering whether the background region exists, relying on a frequency prior strategy. Finally, the degraded image is recovered utilizing underwater polarization imaging model. Extensive experimental results demonstrate that the recovered images from the proposed method have abundant detail, high contrast, and minimal color distortion, which is competitive with most state-of-the-art methods.

TIHN: Tensor Improved Huber Norm for low-rank tensor recovery

Tensor Robust Principal Component Analysis (TRPCA) has received much attention in many real-world applications which aims to recover a low-rank tensor corrupted by sparse noise. Most existing TRPCA methods usually regularize the low-rank component by minimizing its Tensor Nuclear Norm (TNN). However, the original TNN shrinks all of the singular values with the same penalty which generally leads to suboptimal results. The reason is that the large singular values should be less penalized since they usually correspond to salient information of the real-world tensor. In this work, we develop a Tensor Improved Huber Norm (TIHN) which considers the difference between the singular values of tensor. The TIHN can achieve better performance by recovering the large and significant singular values exactly. We also establish the Huber TRPCA (HTRPCA) method by utilizing the proposed TIHN. In addition, an efficient optimization algorithm based on the half-quadratic theory and Alternating Direction Method of Multipliers (ADMM) framework is designed to implement the HTRPCA. Finally, experiments on color image recovery and video recovery validate the effectiveness of the proposed method.

Block-diagonal idiosyncratic covariance estimation in high-dimensional factor models for financial time series

Estimation of high-dimensional covariance matrices in latent factor models is an important topic in many fields and especially in finance. Since the number of financial assets grows while the estimation window length remains of limited size, the often used sample estimator yields noisy estimates which are not even positive definite. Under the assumption of latent factor models, the covariance matrix is decomposed into a common low-rank component and a full-rank idiosyncratic component. In this paper we focus on the estimation of the idiosyncratic component, under the assumption of a grouped structure of the time series, which may arise due to specific factors such as industries, asset classes or countries. We propose a generalized methodology for estimation of the block-diagonal idiosyncratic component by clustering the residual series and applying shrinkage to the obtained blocks in order to ensure positive definiteness. We derive two different estimators based on different clustering methods and test their performance using simulation and historical data. The proposed methods are shown to provide reliable estimates and outperform other state-of-the-art estimators based on thresholding methods.

Robust sparse representation based on fitting error decomposition

Sparse representation (SR) of a signal aims at finding the minimum number of atoms for its representation. In several practical scenarios, the signal is vulnerable to outliers and thus robustness is required for SR based algorithms. However, most existing robust schemes are designed on the assumption that the anomalies are independently distributed, which may not perform well encountering complex interferences, especially the correlated ones. To deal with this drawback, a robust SR model is proposed, where both independent and correlated outliers are considered. Specifically, the fitting error is decomposed as the combination of a low-rank component and a sparse part, corresponding to the correlated gross error and independent outlier, respectively. Then, the group sparsity of the representation coefficient is utilized. Moreover, ℓ0-norm and ℓ2,0-norm are adopted as the sparsity regularization for the sparse outlier and representation coefficients, respectively. The solutions to ℓ0/ℓ2,0-norm minimization are generated by the hard-thresholding strategy, where the decision threshold is adaptively determined using median absolute deviation operator. As for the low-rank regularization, due to the NP-hardness of rank minimization, we employ matrix logarithmic norm as the rank surrogate to lessen the approximation gap. Finally, we apply the proposed model to face recognition task, and the excellent performance demonstrates its effectiveness.

Tensor robust principal component analysis via dual lp quasi-norm sparse constraints

As a pioneering method, tensor robust principal component analysis (TRPCA) can separate an underlying low-rank component and a sparse component from the original data by minimizing a convex objective function composed of tensor nuclear norm and l1-norm. However, it has two limitations. One is that tensor nuclear norm, as a constraint on the low-rank component, treats all singular values uniformly, ignoring the differences among singular values. In essence, this constraint is a sparse constraint that is imposed on singular values of the low-rank component by l1-norm. The other is that l1-norm is used as a constraint for the sparse component. However, l1-norm is a loose constraint, leading to the solutions of TRPCA deviating from the authentic ones. To alleviate these issues, we propose a TPRCA model called p-TRPCA that utilizes the lp-norm to impose sparse constraints on both the singular values and the sparse component simultaneously. The lp quasi-norm (0<p<1) is a tighter constraint than l1-norm, which enhances the low-rankness and sparsity of the proposed model. To solve p-TRPCA, we present an effective algorithm based on the alternating direction method of multipliers and also analyze its convergence. Extensive experiments are performed on data simulation, image recovery, and background modeling. Experimental results show that our p-TRPCA outperforms TRPCA and its variants.

Anomaly detection in PV systems using constrained low-rank and sparse decomposition

PV (photovoltaic) systems, also known as solar panel systems, play an essential role in the mitigation of greenhouse gas emissions and the promotion of renewable energy. Through the conversion of sunlight into usable energy, electricity is generated without emitting greenhouse gases and producing pollutants. Notwithstanding the evolutionary significance of PV systems, the occurrence of defects and anomalies in PV systems may result in diminished power output, consequently impeding the efficiency of the systems and potentially resulting in hazards in certain circumstances. Therefore, early detection of faults and anomalies in PV systems is imperative to guarantee the reliability, efficiency, and safety of the systems. In this article, we develop a signal decomposition for the purpose of anomaly detection in PV systems. The proposed methodology is grounded on the concept of low-rank and sparse decomposition, with consideration given to the signs of the decomposed low-rank and sparse components, as well as the smooth variations within and between periods in the mean signals. Through the implementation of Monte Carlo simulations, we showcase the efficacy of our proposed methodology in identifying anomalies of varying durations and magnitudes in PV systems. A case study is employed to validate the proposed methodology in detecting anomalies in real PV systems.

Saliency-Guided Sparse Low-Rank Tensor Approximation for Unsupervised Anomaly Detection of Hyperspectral Remote Sensing Images

Hyperspectral anomaly detection can separate sparse anomalies from the low-rank background component under an unsupervised behavior due to sufficient spectral information. Therefore, hyperspectral image anomaly detection technology has great application potential and value in public security and national defense. Currently, most existing models attempt to detect anomalous targets with a sparsity prior, without further considering the visual saliency of the targets themselves. To tackle this issue, this paper proposes a saliency-guided sparse low-rank tensor approximation model, called SSLR, to detect anomalous targets from hyperspectral remote sensing images in an unsupervised manner. Specifically, we first explore the saliency information of each pixel for regularizing the sparse anomaly matrix. We then suggest a three-directional tensor nuclear norm to obtain a low-rank background to characterize the background component. We solve the SSLR optimization problem by an efficient alternating direction method of multipliers framework. Experiments conducted on benchmark hyperspectral datasets demonstrate that the proposed SSLR outperforms some state-of-the-art anomaly detection methods.

Automated Sparse and Low-Rank Shallow Autoencoders for Recommendation

Collaborative filtering (CF) works have demonstrated the robust capabilities of Shallow Autoencoders on implicit feedback, showcasing highly competitive performance with other reasonable approaches (e.g., iALS and VAE-CF). However, despite their dual advantages of high performance and simple construction, EASE still exhibits several major shortcomings that must be addressed. To be more precise, the scalability of EASE is limited by the number of items, which determines the storage and inversion cost of a large dense matrix; the square-loss optimization objective does not consistently meet the recommendation task’s requirement for predicting personalized rankings, resulting in sub-optimal outcomes; the regularization coefficients are sensitive and require re-calibration with different datasets, leading to an exhaustive and time-consuming fine-tuning process. In order to address these obstacles, we propose a novel approach called Similarity-Structure Aware Shallow Autoencoder (AutoS 2 AE) that aims to enhance both recommendation accuracy and model efficiency. Our method introduces three similarity structures: Co-Occurrence, KNN, and NSW graphs, which replace the large dense matrix in EASE with a sparse structure, thus facilitating model compression. Additionally, we optimize the model by incorporating a low-rank training component into the matrix and applying a weighted square loss for improved ranking-oriented approximations. To automatically tune the hyperparameters, we further design two validation losses on the validation set for guidance and update the hyperparameters using the gradients of these validation losses. Both theoretical analyses regarding the introduction of similarity structures and empirical evaluations on multiple real-world datasets demonstrate the effectiveness of our proposed method, which significantly outperforms competing baselines.

Tensor structure-based ground clutter suppression approaches for pulse doppler radar

ABSTRACT For pulse doppler (PD) radar, two tensor structure-based methods are proposed to improve the clutter suppression performance. One is tensor singular value decomposition (TSVD) and the other is tensor robust principal component analysis (TRPCA). Two algorithms both stack the range-Doppler matrices of multiple coherent processing intervals (CPI) into a tensor structure, which helps make the most of the temporal information across CPIs. TSVD utilizes the strong temporal correlation of ground clutter data to estimate the clutter subspace. Meanwhile, TRPCA exactly separates both low-rank clutter component and sparse target components from the time series range-Doppler tensor structure. Experimental results indicate that compared with conventional matrix-based schemes such as singular value decomposition (SVD) or singular value decomposition (RPCA), the proposed methods have significant advantages in keeping the target echo while suppressing the clutter. In addition, the computational complexity of each algorithm is analysed in detail.

Tensor recovery based on Bivariate Equivalent Minimax-Concave Penalty

In tensor recovery problem, larger singular values often correspond to the primary information of the image, such as contours, sharp edges and smooth areas. The minimum–maximum concave penalty (MCP) function has been effective in preserving larger singular values and achieving good tensor recovery results. However, in the process of solving this problem, singular values may vary during iterations, and the fixed parameters of the MCP function may not sufficiently preserve all the larger singular values, which may hinder the attainment of optimal results in tensor recovery. To overcome this challenge, we propose the Bivariate Equivalent Minimax-Concave Penalty (BEMCP) theorem, which allows the parameters to adapt to the changes in singular values and more comprehensively preserve the larger singular values. For low-rank tensor completion and tensor robust principal component analysis problems, we propose BEMCP-based models. Finally, experiments with various real-world data demonstrate that the proposed methods outperform state-of-the-art methods.

Interpretable Neural Networks for Video Separation: Deep Unfolding RPCA with Foreground Masking.

We present two deep unfolding neural networks for the simultaneous tasks of background subtraction and foreground detection in video. Unlike conventional neural networks based on deep feature extraction, we incorporate domain-knowledge models by considering a masked variation of the robust principal component analysis problem (RPCA). With this approach, we separate video clips into low-rank and sparse components, respectively corresponding to the backgrounds and foreground masks indicating the presence of moving objects. Our models, coined ROMAN-S and ROMAN-R, map the iterations of two alternating direction of multipliers methods (ADMM) to trainable convolutional layers, and the proximal operators are mapped to non-linear activation functions with trainable thresholds. This approach leads to lightweight networks with enhanced interpretability that can be trained on limited data. In ROMAN-S, the correlation in time of successive binary masks is controlled with side-information based on ℓ1-ℓ1 minimization. ROMAN-R enhances the foreground detection by learning a dictionary of atoms to represent the moving foreground in a high-dimensional feature space and by using reweighted-ℓ1-ℓ1 minimization. Experiments are conducted on both synthetic and real video datasets, for which we also include an analysis of the generalization to unseen clips. Comparisons are made with existing deep unfolding RPCA neural networks, which do not use a mask formulation for the foreground, and with a 3D U-Net baseline. Results show that our proposed models outperform other deep unfolding networks, as well as the untrained optimization algorithms. ROMAN-R, in particular, is competitive with the U-Net baseline for foreground detection, with the additional advantage of providing video backgrounds and requiring substantially fewer training parameters and smaller training sets.

SAR Jamming Suppression by Exploiting Polarized Similarity With Low-Rank and Sparse Matrix Decomposition

Optimal hyper-parameter selection for low-rank and sparse matrix decomposition (LRSMD) in synthetic aperture radar jamming suppression is usually challenging. This letter proposes an effective approach to LRSMD jamming suppression by exploiting the polarized similarity. A polarimetric ratio function is established to straightforwardly determine the rank hyper-parameter. The polarimetric ratio function is defined as the energy ratio of the decomposed low-rank jamming component to the sparse target signal, which is consistent across multiple polarized channels due to the equal jamming gain distinguished from the target polarized scattering. The algorithm optimally exploits the polarized similarity of jamming components to determine the rank hyper-parameter. It provides enhanced robustness and accuracy in SAR jamming removal, confirmed by synthetic experiments.

Tensor Robust Kernel PCA for Multidimensional Data.

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.

Video Hashing with Tensor Robust PCA and Histogram of Optical Flow for Copy Detection

Abstract This paper proposes a novel video hashing with tensor robust Principal Component Analysis (PCA) and Histogram of Optical Flow (HOF) for copy detection. In the proposed hashing, a video is divided into some video groups. For each video group, a low-rank secondary frame is constructed from the low-rank component decomposed by applying tensor robust PCA to the video group. Since the low-rank component can well indicate spatial-temporal intrinsic structure of the video group and it is slightly disturbed by digital operations, feature extraction from the low-rank secondary frames is discriminative and stable. Next, spatial features and temporal features are extracted from low-rank secondary frames by Charlier moments and HOF, respectively. Since the Charlier moments are robust to geometric transform and they can efficiently distinguish video frames with different contents, the use of Charlier moments can make robust and discriminative spatial features. As the HOF can measure the distribution of motion information between frames, the temporal features formed by HOFs can provide good discrimination. Hash is ultimately determined by quantizing the spatial and temporal features and concatenating the quantized results. Numerous experiments on open video datasets indicate that the proposed hashing is superior to some hashing baseline schemes in terms of classification and copy detection.

Infrared Small Dim Target Detection Using Group Regularized Principle Component Pursuit

The detection of an infrared small target faces the problems of background interference and non-obvious target features, which have yet to be efficiently solved. By employing the non-local self-correlation characteristic of the infrared images, the principle component pursuit (PCP)-based methods are demonstrated to be applicable to infrared small target detection in a complex scene. However, existing PCP-based methods heavily depend on the uniform distribution of the background pixels and are prone to generating a high number of false alarms under strong clutter situations. In this paper, we propose a group low-rank regularized principle component pursuit model (GPCP) to solve this problem. First, the local image patches are clustered into several groups that correspond to different grayscale distributions. These patch groups are regularized with a group low-rank constraint, enabling an independent recovery of different background regions. Then, GPCP model integrates the group low-rank components with a global sparse component to extract small targets from the background. Different singular value thresholds can be exploited for image groups corresponding to different brightness and grayscale variance, boosting the recovery of background clutters and also enhancing the detection of small targets. Finally, a customized optimization approach based on alternating direction method of multipliers is proposed to solve this model. We set three representative detection scenes, including the ground background, sea background and sky background for experiment analysis and model comparison. The evaluation results show the proposed model has superiority in background suppression and achieves better adaptability for different scenes compared with various state-of-the-art methods.

Hyperspectral Anomaly Detection with Auto-Encoder and Independent Target

As an unsupervised data representation neural network, auto-encoder (AE) has shown great potential in denoising, dimensionality reduction, and data reconstruction. Many AE-based background (BKG) modeling methods have been developed for hyperspectral anomaly detection (HAD). However, their performance is subject to their unbiased reconstruction of BKG and target pixels. This article presents a rather different low rank and sparse matrix decomposition (LRaSMD) method based on AE, named auto-encoder and independent target (AE-IT), for hyperspectral anomaly detection. First, the encoder weight matrix, obtained by a designed AE network, is utilized to construct a projector for generating a low-rank component in the encoder subspace. By adaptively and reasonably determining the number of neurons in the latent layer, the designed AE-based method can promote the reconstruction of BKG. Second, to ensure independence and representativeness, the component in the encoder orthogonal subspace is made into a sphere and followed by finding of unsupervised targets to construct an anomaly space. In order to mitigate the influence of noise on anomaly detection, sparse cardinality (SC) constraint is enforced on the component in the anomaly space for obtaining the sparse anomaly component. Finally, anomaly detector is constructed by combining Mahalanobi distance and multi-components, which include encoder component and sparse anomaly component, to detect anomalies. The experimental results demonstrate that AE-IT performs competitively compared to the LRaSMD-based models and AE-based approaches.