Low-rank Property Research Articles

Exploiting Low-Rank and Sparse Properties in Strided Convolution Matrix for Pansharpening

Fusion of low spatial resolution multispectral (LR MS) and panchromatic (PAN) images to acquire high spatial resolution multispectral (HR MS) images has attracted increasing attention in recent years. In this article, we first utilize the form of convolution matrix (CM) to formulate the image fusion problem. In order to reduce the complexity of CM, the step size is introduced and strided convolution matrix (SCM) is constructed. Then, we explore the low-rank property in SCM and impose the prior on the spatial and spectral degradation model of LR MS and PAN images. Meanwhile, sparsity in SCM is considered to further enhance the local structures in the fused image. Finally, the proposed model is optimized efficiently by the alternative direction method of multipliers. By exploiting the low-rank and sparse priors in SCM of HR MS image, the local and global structures can be better preserved. The experimental results on the reduced-resolution and full-resolution datasets also show that the proposed method behaves well in qualitative and quantitative assessments.

Review of Low-Rank Data-Driven Methods Applied to Synchrophasor Measurement

There is a growing acceptance of using synchrophasor data collected over large power systems in control centers to enhance the reliability of power system operations. The spatial and temporal nature of power system ambient and disturbance response allows the analysis of large amount of synchrophasor data by low-rank methods. This paper provides an overview of several applications of synchrophasor data utilizing the low-rank property. The tools to capitalize on the low-rank property include matrix completion methods, tensor analysis, adaptive filtering, and machine learning. The applications include missing data recovery, bad data correction, and disturbance recognition.

Logarithmic Norm Regularized Low-Rank Factorization for Matrix and Tensor Completion.

Matrix and tensor completion aim to recover the incomplete two- and higher-dimensional observations using the low-rank property. Conventional techniques usually minimize the convex surrogate of rank (such as the nuclear norm), which, however, leads to the suboptimal solution for the low-rank recovery. In this paper, we propose a new definition of matrix/tensor logarithmic norm to induce a sparsity-driven surrogate for rank. More importantly, the factor matrix/tensor norm surrogate theorems are derived, which are capable of factoring the norm of large-scale matrix/tensor into those of small-scale matrices/tensors equivalently. Based upon surrogate theorems, we propose two new algorithms called Logarithmic norm Regularized Matrix Factorization (LRMF) and Logarithmic norm Regularized Tensor Factorization (LRTF). These two algorithms incorporate the logarithmic norm regularization with the matrix/tensor factorization and hence achieve more accurate low-rank approximation and high computational efficiency. The resulting optimization problems are solved using the framework of alternating minimization with the proof of convergence. Simulation results on both synthetic and real-world data demonstrate the superior performance of the proposed LRMF and LRTF algorithms over the state-of-the-art algorithms in terms of accuracy and efficiency.

Micro-Doppler Effects Removed Sparse Aperture ISAR Imaging via Low-Rank and Double Sparsity Constrained ADMM and Linearized ADMM.

Inverse synthetic aperture radar (ISAR) imaging for the target with micro-motion parts is influenced by the micro-Doppler (m-D) effects. In this case, the radar echo is generally decomposed into the components from the main body and micro-motion parts of target, respectively, to remove the m-D effects and derive a focused ISAR image of the main body. For the sparse aperture data, however, the radar echo is intentionally or occasionally under-sampled, which defocuses the ISAR image by introducing considerable interference, and deteriorates the performance of signal decomposition for the removal of m-D effects. To address this issue, this paper proposes a novel m-D effects removed sparse aperture ISAR (SA-ISAR) imaging algorithm. Note that during a short interval of ISAR imaging, the range profiles of the main body of target from different pulses are similar, resulting in a low-rank matrix of range profile sequence of main body. For the range profiles of the micro-motion parts, they either spread in different range cells or glint in a single range cell, which results in a sparse matrix of range profile sequence. From this perspective, the low-rank and sparse properties are utilized to decompose the range profiles of the main body and micro-motion parts, respectively. Moreover, the sparsity of ISAR image is also utilized as a constraint to eliminate the interference caused by sparse aperture. Hence, SA-ISAR imaging with the removal of m-D effects is modeled as a triply constrained underdetermined optimization problem. The alternating direction method of multipliers (ADMM) and linearized ADMM (L-ADMM) are further utilized to solve the problem with high efficiency. Experimental results based on both simulated and measured data validate the effectiveness of the proposed algorithm.

Recovery of surfaces and functions in high dimensions: sampling theory and links to neural networks.

Several imaging algorithms including patch-based image denoising, image time series recovery, and convolutional neural networks can be thought of as methods that exploit the manifold structure of signals. While the empirical performance of these algorithms is impressive, the understanding of recovery of the signals and functions that live on manifold is less understood. In this paper, we focus on the recovery of signals that live on a union of surfaces. In particular, we consider signals living on a union of smooth band-limited surfaces in high dimensions. We show that an exponential mapping transforms the data to a union of low-dimensional subspaces. Using this relation, we introduce a sampling theoretical framework for the recovery of smooth surfaces from few samples and the learning of functions living on smooth surfaces. The low-rank property of the features is used to determine the number of measurements needed to recover the surface. Moreover, the low-rank property of the features also provides an efficient approach, which resembles a neural network, for the local representation of multidimensional functions on the surface. The direct representation of such a function in high dimensions often suffers from the curse of dimensionality; the large number of parameters would translate to the need for extensive training data. The low-rank property of the features can significantly reduce the number of parameters, which makes the computational structure attractive for learning and inference from limited labeled training data.

Hyperspectral-Multispectral Image Fusion via Tensor Ring and Subspace Decompositions

Fusion from a spatially low resolution hyperspectral image (LR-HSI) and a spectrally low resolution multispectral image (MSI) to produce a high spatial-spectral HSI (HR-HSI), known as hyperspectral super resolution, has risen to a preferred topic for reinforcing the spatial-spectral resolution of HSI in recent years. In this work, we propose a new model, namely, low-rank tensor ring decomposition based on tensor nuclear norm (LRTRTNN), for HSI-MSI fusion. Specifically, for each spectrally subspace cube, similar patches are grouped to exploit both the global low-rank property of LR-HSI and the nonlocal similarity of HR-MSI. Afterward, a joint optimization of all groups via the presented LRTRTNN approximation is implemented in a unified cost function. With the introduced tensor nuclear norm (TNN) constraint, all 3D tensor ring factors are no longer unfolded to suit the matrix nuclear norm used in conventional methods, and the internal tensor structure can be naturally retained. The alternating direction method of multipliers is introduced for coefficients update. Numerical and visual experiments on real data show that our LRTRTNN method outperforms most state-of-the-art algorithms in terms of fusing performance.

Fast Hyperspectral Image Recovery of Dual-Camera Compressive Hyperspectral Imaging via Non-Iterative Subspace-Based Fusion.

Coded aperture snapshot spectral imaging (CASSI) is a promising technique for capturing three-dimensional hyperspectral images (HSIs), in which algorithms are used to perform the inverse problem of HSI reconstruction from a single coded two-dimensional (2D) measurement. Due to the ill-posed nature of this problem, various regularizers have been exploited to reconstruct 3D data from 2D measurements. Unfortunately, the accuracy and computational complexity are unsatisfactory. One feasible solution is to utilize additional information such as the RGB measurement in CASSI. Considering the combined CASSI and RGB measurements, in this paper, we propose a fusion model for HSI reconstruction. Specifically, we investigate the low-dimensional spectral subspace property of HSIs composed of a spectral basis and spatial coefficients. In particular, the RGB measurement is utilized to estimate the coefficients, while the CASSI measurement is adopted to provide the spectral basis. We further propose a patch processing strategy to enhance the spectral low-rank property of HSIs. The optimization of the proposed model requires neither iteration nor the spectral sensing matrix of the RGB detector. Extensive experiments on both simulated and real HSI datasets demonstrate that our proposed method not only outperforms previous state-of-the-art (iterative algorithms) methods in quality but also speeds up the reconstruction by more than 5000 times.

Deep Amended Gradient Descent for Efficient Spectral Reconstruction From Single RGB Images

This paper investigates the problem of recovering hyperspectral (HS) images from single RGB images. To tackle such a severely ill-posed problem, we propose a physically-interpretable, compact, efficient, and end-to-end learning-based framework, namely AGD-Net. Precisely, by taking advantage of the imaging process, we first formulate the problem explicitly based on the classic gradient descent algorithm. Then, we design a lightweight neural network with a multi-stage architecture to mimic the formed amended gradient descent process, in which efficient convolution and novel spectral zero-mean normalization are proposed to effectively extract spatial-spectral features for regressing an initialization, a basic gradient, and an incremental gradient. Besides, based on the approximate low-rank property of HS images, we propose a novel rank loss to promote the similarity between the global structures of reconstructed and ground-truth HS images, which is optimized with our singular value weighting strategy during training. Moreover, AGD-Net, a single network after one-time training, is flexible to handle the reconstruction with various spectral response functions. Extensive experiments over three commonly-used benchmark datasets demonstrate that AGD-Net can improve the reconstruction quality by more than 1.0 dB on average while saving 67× parameters and 32× FLOPs, compared with state-of-the-art methods. The code will be publicly available at https://github.com/zbzhzhy/GD-Net .

Enforcing symmetry in tensor network MIMO Volterra identification

Abstract The estimation of an exponential number of model parameters in a truncated Volterra model can be circumvented by using a low-rank tensor decomposition approach. This low-rank property of the tensor decomposition can be interpreted as the assumption that all Volterra parameters are structured. In this article, we investigate whether it is possible to explicitly enforce symmetry of the Volterra kernels to the low-rank tensor decomposition. We show that low-rank symmetric Volterra identification is an ill-conditioned problem as the low-rank property of the exact symmetric kernels cannot be upheld in the presence of measurement noise. Furthermore, an algorithm is derived to compute the symmetric Volterra kernels directly in tensor network form.

Artistic Color Virtual Reality Implementation Based on Similarity Image Restoration

In this paper, exploratory and innovative research is done on the implementation technique of artistic color virtual reality for similarity image recovery. Based on similarity images, a nonlocal natural image before the regular term is proposed to deal with the single‐image blind deblurring problem. This paper designs a new artistic color virtual reality realization technology based on similarity image restoration, which exploits the low‐rank property between nonlocal similarity blocks in images and combines a strong convex term to enhance the convexity of the artistic color virtual reality model. We analyze virtual reality interaction design from the perspective of art color design, sort out the concept and content of design, analyze the elements, design principles, and evaluation criteria included in virtual reality interaction art color design, and explore the conceptual principles of virtual reality interaction art color design. A full understanding of the characteristics of the medium of the virtual reality interaction can help us to better use this medium as a tool to create works that aim to bring higher quality and experiential feeling with a perceptual communication method that is closer to natural interaction. Combining the power of technology, artistic colourful thinking, and a design approach paves the way forward. The study shows that virtual reality technology can effectively improve the status quo and promote the cultivation of professional practice ability in art color design, which is conducive to the cultivation of applied design talents.

Hyperspectral anomaly detection using a background endmember signature

Due to lacking use of prior information, the anomaly detection results are not always satisfactory. However, with the establishment of the spectral library, it becomes possible to obtain one or more spectra of the background in the image to be detected. If we can make use of such background information that is always ignored or discarded, the detection result is very likely to be improved. Hence, we proposed a hyperspectral anomaly detection method using a background endmember signature. To better separate the anomaly from the background, we first perform spectral unmixing to estimate the abundance matrix for further study instead of the original spectral data. In this process, we introduce a non-negative matrix factorization-based unmixing method and a corresponding initialization method using a background endmember. Then the low-rank property contained in the abundance matrix is exploited. A low-rank decomposition method is used to separate the anomalies. The proposed algorithm is evaluated on both synthetic and real data sets. Experiment results show the effectiveness of the proposed method and the improvement brought by the usage of a known background endmember.

An ADMM-ResNet for data recovery in wireless sensor networks with guaranteed convergence

Data collection is a basic application of wireless sensor networks (WSNs). In practice, only a subset of sensor nodes is selected for data sensing and transmission due to the bandwidth constraint of the channel, energy constraint of the nodes, or malfunctions of the nodes. Data recovery from incomplete sensing data is vital to WSNs. Many works perform data recovery by utilizing the low-rank property of the spatio-temporal correlated data. However, these methods usually converge slowly to achieve satisfactory accuracy performance. In this paper, we propose an ADMM-ResNet framework based on residual networks (ResNets) for spatio-temporal correlated data recovery. The formulated optimization problem is solved by the alternating direction method of multipliers (ADMM) algorithm. The updates of auxiliary variable in the ADMM algorithm can be replaced by ResNets, and the ADMM algorithm is unrolled into a fixed-length neural network. The proposed ADMM-ResNet significantly reduces the number of iterations compared with traditional ADMM algorithm. We theoretically prove that the proposed ADMM-ResNet can globally converge to a fixed-point. Experimental results verify the theoretical convergence and demonstrate the effectiveness.

저 차원 Hankel 구조를 이용한 PMU 데이터 복구에 관한 연구

Phasor Measurement Units (PMUs) provide synchronized phasor measurements at much higher sampling rate than that in the traditional Supervisory Control And Data Acquisition (SCADA) system. Several synchrophasor-based algorithms and techniques have been and continue to be developed for real-time operation applications such as state estimation, stability analysis, disturbance detection, dynamic security assessment etc. However, synchrophasor data quality limits the incorporation of synchrophasor-based applications into control room operations environment and processes. The goal of this project is to develop methods that can improve synchrophasor data quality by recovering missing data reliably and efficiently. Data recovery refers to methods that estimate the values of missing data in the synchrophasor streams. Recently, modeless missing data recovery methods have been developed, that exploit the low-rank property of the spatial-temporal synchrophasor data blocks. A spatial-temporal synchrophasor data block can be considered as a matrix that is constructed by the measurements sampled at consecutive time instants with each row denoting the measurement of one certain channel across time. By exploiting the low-rank property of synchrophasor data matrices, the missing data recovery can be formulated as a low-rank matrix completion problem. The low-rank matrix completion problem has been extensively studied in the past few years and several algorithms have been developed to recover a low-rank matrix from partial observations. In this study, synchrophasor data analysis has been combined with low-rank matrix completion theory to develop a missing synchrophasor data recovery technique and tool.

Depth Image Denoising Using Nuclear Norm and Learning Graph Model

Depth image denoising is increasingly becoming the hot research topic nowadays, because it reflects the three-dimensional scene and can be applied in various fields of computer vision. But the depth images obtained from depth camera usually contain stains such as noise, which greatly impairs the performance of depth-related applications. In this article, considering that group-based image restoration methods are more effective in gathering the similarity among patches, a group-based nuclear norm and learning graph (GNNLG) model was proposed. For each patch, we find and group the most similar patches within a searching window. The intrinsic low-rank property of the grouped patches is exploited in our model. In addition, we studied the manifold learning method and devised an effective optimized learning strategy to obtain the graph Laplacian matrix, which reflects the topological structure of image, to further impose the smoothing priors to the denoised depth image. To achieve fast speed and high convergence, the alternating direction method of multipliers is proposed to solve our GNNLG. The experimental results show that the proposed method is superior to other current state-of-the-art denoising methods in both subjective and objective criterion.

Front-Wall Clutter Removal in Through-the-Wall Radar Based on Weighted Nuclear Norm Minimization

The front-wall clutter removal in the case of the through-the-wall radar (TWR) system is studied in this work. To remove the wall clutter, its low-rank property is utilized, and at the same time, the sparse property of the target returns is exploited to perform target reconstruction. To account for the unparalleled setting of the antenna and the wall, a weighted nuclear norm minimization (WNNM) is employed, and the resulting problem is solved in an alternating manner. In addition, different transmitted waveform signals, including monofrequency and stepped-frequency waveforms, are used to demonstrate their effects on the clutter suppression performances. The experimental results show that the proposed WNNM with stepped-frequency waveform outperforms other approaches.

A Matrix Completion Based Method for RFI Source Localization in Microwave Interferometric Radiometry

The Soil Moisture and Ocean Salinity (SMOS) mission led by the European Space Agency (ESA) is aimed to globally monitor the Earth surface moisture and ocean salinity. As the single payload of SMOS satellite, the Microwave Interferometric Radiometer with Aperture Synthesis (MIRAS) operates in the protected L-band. Nonetheless, the artificial sources emitting close to or/and fully in this band are contaminating the collected remote sensing data and deteriorating the performance of the SMOS mission. Identifying and localizing such sources is crucial to improve the quality of SMOS scientific products. In this article, we propose a method based on matrix completion (MC) for localization of radio frequency interference (RFI) sources. This method mainly exploits the low-rank property of the augmented covariance matrix (ACM) of the sparse array and addresses the ACM incompleteness (i.e., sampling data loss) due to inherent array geometry (e.g., the SMOS Y-shaped array) or potential hardware malfunction (e.g., the correlator failure). Validation results show that, compared with existing RFI localization approaches such as the discrete Fourier transformation (DFT) inversion and covariance-based direction-of-arrival (DOA) estimation, the proposed MC method possesses competitive localization accuracy, superior spatial resolution, and reduced artifacts.

Feature-Enhanced Speckle Reduction via Low-Rank and Space-Angle Continuity for Circular SAR Target Recognition

With the development of synthetic aperture radar (SAR) system, automatic target recognition (ATR) has attracted wide attention in many decision-making tasks, in which an enhanced feature of SAR image is a powerful tool to improve the recognition accuracy. However, the presence of speckle noise and natural clutter inevitably contaminates SAR images and, thus, degrades image features. In this article, we explicitly address the speckle reduction problem for the circular SAR system, in which the motion of aircraft platform causes continuous angular variations so that different SAR images can be captured with the high interrelationship. By exploiting the underlying low-rank and continuous properties among different SAR images, a method called the $\ell _{p}$ -regularized low-rank and space-angle continuity extraction ( $\ell _{p}$ -LSCE) is proposed to suppress the noise and enhance the target feature. Taking into account the interrelationship between SAR images, we arrange the images in a 3-D tensor to investigate the space-angle continuity of the targets. Furthermore, we develop a robust $\ell _{p}$ -regularized scheme to incorporate the low-rank property of targets. Then, the joint optimization problem is solved via the framework of augmented Lagrange multiplier (ALM) with efficient computation of each ALM subproblem. The experimental results of circular SAR data sets of the moving and stationary target acquisition and recognition (MSTAR) and the VideoSAR demonstrate that the proposed method can efficiently despeckle SAR images with well-preserved target features, which is conducive to the improvement of ATR performance.

Recovery of image and video based on compressive sensing via tensor approximation and Spatio-temporal correlation

In recent years, block-based compressive sensing (BCS) has been extensively studied because it can reduce computational complexity and data storage by dividing the image into smaller patches, but the performance of the reconstruction algorithm is not satisfactory. In this paper, a new reconstruction model for image and video is proposed. The model makes full use of spatio-temporal correlation and utilizes low-rank tensor approximation to improve the quality of the reconstructed image and video. For image recovery, the proposed model obtains a low-rank approximation of a tensor formed by non-local similar patches, and improves the reconstruction quality from a spatial perspective by combining non-local similarity and low-rank property. For video recovery, the reconstruction process is divided into two phases. In the first phase, each frame of the video sequence is regarded as an independent image to be reconstructed by taking advantage of spatial property. The second phase performs tensor approximation through searching similar patches within frames near the target frame, to achieve reconstruction by putting the spatio-temporal correlation into full play. The resulting model is solved by an efficient Alternating Direction Method of Multipliers (ADMM) algorithm. A series of experiments show that the quality of the proposed model is comparable to the current state-of-the-art recovery methods.

Exploiting low-rank structure in semidefinite programming by approximate operator splitting

In contrast to many other convex optimization classes, state-of-the-art semidefinite programming solvers are still unable to efficiently solve large-scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The key characteristic of the proposed algorithm is to be able to exploit the low-rank property inherent to several semidefinite programming problems. Exploiting the low-rank structure provides a substantial speedup and allows the operator splitting method to efficiently scale to larger instances. As opposed to other low-rank based methods, the proposed algorithm has convergence guarantees for general semidefinite programming problems. Additionally, an open-source semidefinite programming solver called ProxSDP is made available and its implementation details are discussed. Case studies are presented in order to evaluate the performance of the proposed methodology.

Hyperspectral Image Restoration: Where Does the Low-Rank Property Exist

Hyperspectral image (HSI) restoration is to recover the clean image from degraded version, such as the noisy, blurred, or damaged. Recent low-rank tensor-based recovery methods have been widely explored in HSIs restoration. Most of previous methods, however, neglect an inconspicuous but important phenomenon that the physical meaning and dimension along the spatial, spectral, and nonlocal mode are markedly different. In this work, we discover the low-rank property discrepancy along spatial, spectral, and nonlocal self-similarity mode in the HSIs, and argue that the intrinsic low-rank correlations along each mode contribute different to the final restoration results. Consequently, we figure out that the combination of the spectral and nonlocal-induced low-rank is most beneficial for HSIs modeling, and propose an optimal low-rank tensor (OLRT) model for HSIs restoration. Furthermore, we not only explore the low-rank property in the image component, but also in the sparse error component (stripe noise in HSIs). Thus, we extend OLRT to the OLRT-robust principal component analysis (RPCA) with low-rank tensor priors for both the HSIs and sparse error. Besides, previous methods are usually designed for one specific HSI task, which is less robust to various tasks. We prove that the proposed optimal low-rank prior is very flexible for various HSI restoration problems including denoising, deblurring, inpainting, and destriping. The proposed methods have been extensively evaluated on several benchmarks and tasks, and greatly outperform state-of-the-art (STOA). We show the simple yet effective OLRT strategy is also beneficial to STOA.