Tensor Data Research Articles

Construction and interpretation of high-order image information based on NV optical magnetic vector detection.

Tensor imaging can provide more comprehensive information about spatial physical properties, but it is a high-dimensional physical quantity that is difficult to observe directly. This paper proposes a fast-transform magnetic tensor imaging method based on the NV magnetic detection technique. The Euler deconvolution interprets the magnetic tensor data to obtain the target three-dimensional (3D) boundary information. Fast magnetic vector imaging was performed using optical detection of magnetic resonance (ODMR) to verify the method's feasibility. The complete tensor data was obtained based on the transformation of the vector magnetic imaging data, which was subsequently solved, and the contour information of the objective was restored. In addition, a fast magnetic moment judgment model and an angular transformation model of the observation space are developed in this paper to reduce the influence of the magnetic moment direction on the results and to help interpret the magnetic tensor data. Finally, the experiment realizes the localization, judgment of magnetic moment direction, and 3D boundary identification of a micron-sized tiny magnet with a spatial resolution of 10 µm, a model accuracy of 90.1%, and a magnetic moment direction error of 4.2°.

A Novel Tensor Ring Sparsity Measurement for Image Completion.

As a promising data analysis technique, sparse modeling has gained widespread traction in the field of image processing, particularly for image recovery. The matrix rank, served as a measure of data sparsity, quantifies the sparsity within the Kronecker basis representation of a given piece of data in the matrix format. Nevertheless, in practical scenarios, much of the data are intrinsically multi-dimensional, and thus, using a matrix format for data representation will inevitably yield sub-optimal outcomes. Tensor decomposition (TD), as a high-order generalization of matrix decomposition, has been widely used to analyze multi-dimensional data. In a direct generalization to the matrix rank, low-rank tensor modeling has been developed for multi-dimensional data analysis and achieved great success. Despite its efficacy, the connection between TD rank and the sparsity of the tensor data is not direct. In this work, we introduce a novel tensor ring sparsity measurement (TRSM) for measuring the sparsity of the tensor. This metric relies on the tensor ring (TR) Kronecker basis representation of the tensor, providing a unified interpretation akin to matrix sparsity measurements, wherein the Kronecker basis serves as the foundational representation component. Moreover, TRSM can be efficiently computed by the product of the ranks of the mode-2 unfolded TR-cores. To enhance the practical performance of TRSM, the folded-concave penalty of the minimax concave penalty is introduced as a nonconvex relaxation. Lastly, we extend the TRSM to the tensor completion problem and use the alternating direction method of the multipliers scheme to solve it. Experiments on image and video data completion demonstrate the effectiveness of the proposed method.

Hydrocarbon migration and structural reservoir traps in the Western Black Sea Basin: evidence from satellite-derived gravity tensor data

Hydrocarbon migration and structural reservoir traps in the Western Black Sea Basin: evidence from satellite-derived gravity tensor data

Three-dimensional magnetotelluric and gravimetric imaging of Mount Lawu geothermal prospect area, Indonesia

Mount Lawu is a stratovolcano located on the border of Central Java and East Java Provinces in Indonesia. At least eleven manifestations indicate its geothermal potential in the form of hot water and fumaroles. This study aims to describe a geothermal system based on an integrated analysis of magnetotelluric (MT) and satellite gravity data. Some additional geochemical and petrological data analyses were also used to explain the existing hydrothermal system in this area. It is inferred from the gravity derivative analysis that fault structures exist in a location with high permeability, which acts as a controlling structure for geothermal manifestations. The 3-D resistivity inversion was carried out using MT full-impedance tensor data, while the 3-D density model was reconstructed based on the residual anomaly computed from the topography-free gravity disturbance. The 3-D resistivity model, from magnetotelluric inversion, shows a clay cap distribution, corresponds to a low resistivity anomaly, centered at the southern part of the mountain peak with a thickness of about 1 kilometer, which narrows to the western region where the hot spring shows an outflow manifestation. The gravity inversion shows several low-density bodies in the same location as the low-resistivity anomaly attributed to a clay cap. Coincident resistivity and density anomalies constrain the horizontal location of the geothermal reservoir. Despite the incapability of the model to resolve the body of the geothermal heat source, it can be inferred from the inversion results that the heat possibly emanates through the thermal conduction passage below the reservoir zone from the source located deeper down than the reconstructed models.

Spectrum Situation Awareness for Space-Air-Ground Integrated Networks Based on Tensor Computing.

The spectrum situation awareness problem in space-air-ground integrated networks (SAGINs) is studied from a tensor-computing perspective. Tensor and tensor computing, including tensor decomposition, tensor completion and tensor eigenvalues, can satisfy the application requirements of SAGINs. Tensors can effectively handle multidimensional heterogeneous big data generated by SAGINs. Tensor computing is used to process the big data, with tensor decomposition being used for dimensionality reduction to reduce storage space, and tensor completion utilized for numeric supplementation to overcome the missing data problem. Notably, tensor eigenvalues are used to indicate the intrinsic correlations within the big data. A tensor data model is designed for space-air-ground integrated networks from multiple dimensions. Based on the multidimensional tensor data model, a novel tensor-computing-based spectrum situation awareness scheme is proposed. Two tensor eigenvalue calculation algorithms are studied to generate tensor eigenvalues. The distribution characteristics of tensor eigenvalues are used to design spectrum sensing schemes with hypothesis tests. The main advantage of this algorithm based on tensor eigenvalue distributions is that the statistics of spectrum situation awareness can be completely characterized by tensor eigenvalues. The feasibility of spectrum situation awareness based on tensor eigenvalues is evaluated by simulation results. The new application paradigm of tensor eigenvalue provides a novel direction for practical applications of tensor theory.

Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation.

Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD based low-rank approximation, which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors. Against these issues, in this article, two efficient low-rank tensor approximation approaches fusing random projection techniques are first devised under the order-d ( d ≥ 3 ) T-SVD framework. Theoretical results on error bounds for the proposed randomized algorithms are provided. On this basis, we then further investigate the robust high-order tensor completion problem, in which a double nonconvex model along with its corresponding fast optimization algorithms with convergence guarantees are developed. Experimental results on large-scale synthetic and real tensor data illustrate that the proposed method outperforms other state-of-the-art approaches in terms of both computational efficiency and estimated precision.

Сейсмотектонические деформации и сброшенные напряжения землетрясений Центрального Тянь-Шаня

Based on the seismic moment tensor (SMT) data of 270 earthquakes in the Central Tien Shan, which occurred from 1978 to 2021 and include 63 events from the Global Centroid Moment Tensor catalog and 207 events from the works of Aleksander D. Kostyuk and Naylya A. Sycheva, the focal parameters (seismotectonic deformations (STD), kinematic and dynamic parameters, source radius, and stress drop) were calculated. The STD calculation was based on the approaches proposed in the studies of Yuriy V. Riznichenko and Sergei L. Yunga. The area under consideration is characterized by such deformation modes as thrust, transpression, underpression, and oblique. The distribution of the Lode–Nadai coefficient was calculated and plotted. A significant part of the study area is characterized by simple compression deformation, dominated by simple compression and simple shear. To calculate the stress drop , the values of the scalar seismic moment M0, which are determined in the SMT calculation, and the source radii r, which are calculated on the basis of theoretical and experimental models of the dependence of the source radius on the moment magnitude, were used. The radii r and stress drops  were calculated for two models of earthquake sources: the Brune model and the Madariaga–Kaneko–Shearer model. A catalog of dynamic parameters was compiled. The comparison of the kinematic and dynamic parameters of earthquakes was carried out, and the relationship of the dropped stresses with the type of movement in the source, as well as with the Lode–Nadai coefficient, was established. The results obtained in the study can be useful for specialists in other fields of knowledge – geodesy, geology, and geophysics.

Noisy Tensor Completion via Low-Rank Tensor Ring.

Tensor completion is a fundamental tool for incomplete data analysis, where the goal is to predict missing entries from partial observations. However, existing methods often make the explicit or implicit assumption that the observed entries are noise-free to provide a theoretical guarantee of exact recovery of missing entries, which is quite restrictive in practice. To remedy such drawback, this article proposes a novel noisy tensor completion model, which complements the incompetence of existing works in handling the degeneration of high-order and noisy observations. Specifically, the tensor ring nuclear norm (TRNN) and least-squares estimator are adopted to regularize the underlying tensor and the observed entries, respectively. In addition, a nonasymptotic upper bound of estimation error is provided to depict the statistical performance of the proposed estimator. Two efficient algorithms are developed to solve the optimization problem with convergence guarantee, one of which is specially tailored to handle large-scale tensors by replacing the minimization of TRNN of the original tensor equivalently with that of a much smaller one in a heterogeneous tensor decomposition framework. Experimental results on both synthetic and real-world data demonstrate the effectiveness and efficiency of the proposed model in recovering noisy incomplete tensor data compared with state-of-the-art tensor completion models.

MG-TCCA: Tensor Canonical Correlation Analysis across Multiple Groups.

Tensor Canonical Correlation Analysis (TCCA) is a commonly employed statistical method utilized to examine linear associations between two sets of tensor datasets. However, the existing TCCA models fail to adequately address the heterogeneity present in real-world tensor data, such as brain imaging data collected from diverse groups characterized by factors like sex and race. Consequently, these models may yield biased outcomes. In order to surmount this constraint, we propose a novel approach called Multi-Group TCCA (MG-TCCA), which enables the joint analysis of multiple subgroups. By incorporating a dual sparsity structure and a block coordinate ascent algorithm, our MG-TCCA method effectively addresses heterogeneity and leverages information across different groups to identify consistent signals. This novel approach facilitates the quantification of shared and individual structures, reduces data dimensionality, and enables visual exploration. To empirically validate our approach, we conduct a study focused on investigating correlations between two brain positron emission tomography (PET) modalities (AV-45 and FDG) within an Alzheimer's disease (AD) cohort. Our results demonstrate that MG-TCCA surpasses traditional TCCA and Sparse TCCA (STCCA) in identifying sex-specific cross-modality imaging correlations. This heightened performance of MG-TCCA provides valuable insights for the characterization of multimodal imaging biomarkers in AD.

Balanced Unfolding Induced Tensor Nuclear Norms for High-Order Tensor Completion.

The recently proposed tensor tubal rank has been witnessed to obtain extraordinary success in real-world tensor data completion. However, existing works usually fix the transform orientation along the third mode and may fail to turn multidimensional low-tubal-rank structure into account. To alleviate these bottlenecks, we introduce two unfolding induced tensor nuclear norms (TNNs) for the tensor completion (TC) problem, which naturally extends tensor tubal rank to high-order data. Specifically, we show how multidimensional low-tubal-rank structure can be captured by utilizing a novel balanced unfolding strategy, upon which two TNNs, namely, overlapped TNN (OTNN) and latent TNN (LTNN), are developed. We also show the immediate relationship between the tubal rank of unfolding tensor and the existing tensor network (TN) rank, e.g., CANDECOMP/PARAFAC (CP) rank, Tucker rank, and tensor ring (TR) rank, to demonstrate its efficiency and practicality. Two efficient TC models are then proposed with theoretical guarantees by analyzing a unified nonasymptotic upper bound. To solve optimization problems, we develop two alternating direction methods of multipliers (ADMM) based algorithms. The proposed models have been demonstrated to exhibit superior performance based on experimental findings involving synthetic and real-world tensors, including facial images, light field images, and video sequences.

Non-negative Tucker decomposition with double constraints for multiway dimensionality reduction

<p>Nonnegative Tucker decomposition (NTD) is one of the renowned techniques in feature extraction and representation for nonnegative high-dimensional tensor data. The main focus behind the NTD-like model was how to factorize the data to get ahold of a high quality data representation from multidimensional directions. However, existing NTD-like models do not consider relationship and properties between the factor matrix of columns while preserving the geometric structure of the data space. In this paper, we managed to capture nonlinear local features of data space and further enhance expressiveness of the NTD clustering method by syncretizing organically approximately orthogonal constraint and graph regularized constraint. First, based on the uni-side and bi-side approximate orthogonality, we flexibly proposed two novel approximately orthogonal NTD with graph regularized models, which not only in part make the factor matrix tend to be orthogonality, but also preserve the geometrical information from high-dimensional tensor data. Second, we developed the iterative updating algorithm dependent on the multiplicative update rule to solve the proposed models, and provided its convergence and computational complexity. Finally, we used numerical experimental results to demonstrate the effectiveness, robustness, and efficiency of the proposed new methods on the real-world image datasets.</p>

Effective Streaming Low-Tubal-Rank Tensor Approximation via Frequent Directions.

Low-tubal-rank tensor approximation has been proposed to analyze large-scale and multidimensional data. However, finding such an accurate approximation is challenging in the streaming setting, due to the limited computational resources. To alleviate this issue, this article extends a popular matrix sketching technique, namely, frequent directions (FDs), for constructing an efficient and accurate low-tubal-rank tensor approximation from streaming data based on the tensor singular value decomposition (t-SVD). Specifically, the new algorithm allows the tensor data to be observed slice by slice but only needs to maintain and incrementally update a much smaller sketch, which could capture the principal information of the original tensor. The rigorous theoretical analysis shows that the approximation error of the new algorithm can be arbitrarily small when the sketch size grows linearly. Extensive experimental results on both synthetic and real multidimensional data further reveal the superiority of the proposed algorithm compared with other sketching algorithms for getting low-tubal-rank approximation, in terms of both efficiency and accuracy.

Semi-supervised multi-sensor information fusion tailored graph embedded low-rank tensor learning machine under extremely low labeled rate

This paper investigates a demanding and meaningful task of intelligent fault diagnosis, in which multi-sensors signals are fused for semi-supervised analysis with few labeled fault data. Exploring effective strategies to solve this task from an industrial or academic perspective remains a challenging and resource-consuming task. To this issue, this study develops a new low-rank tensor-based semi-supervised classifier, called graph embedded low-rank tensor learning machine (GE-LRTLM), which can effectively alleviate the foregoing difficulties and increase the diagnosis precision in engineering applications. First, multi-sensor and multi-channel vibration signals are converted into the pixel matrices that are stacked as the multi-sensor information fusion feature tensors, making the coupling relationship between multi-sensor signals remained and achieving a reasonable fusion of multi-source features. Additionally, an advanced tensor decomposition method, tensor nuclear norm (TNN), is introduced in GE-LRTLM model to obtain the low-rank structure information of each feature tensor, implementing the extraction of the most important feature and patterns from the tensor data while ensuring that the tensor data structure remains unchanged. Ultimately, the manifold regularization and tensor-based graph construction method are introduced to obtain potential label information from unlabeled samples, achieving better description of the geometry similarity and distribution of tensor data. Numerous semi-supervised experiments are conducted across multiple datasets, and the results demonstrate that the proposed method can achieve a classification accuracy of 97% even when the number of labeled samples is extremely limited. Simultaneously, it also verifies that the combination of the constructed labeled and unlabeled multi-sensor information fusion tensor samples can promote the improvement of model accuracy.

TriSig: Evaluating the statistical significance of triclusters

Tensor data analysis allows researchers to uncover novel patterns and relationships that cannot be obtained from tabular data alone. The information inferred from multi-way patterns can offer valuable insights into disease progression, bioproduction processes, behavioral responses, weather fluctuations, or social dynamics. However, spurious patterns often hamper this process. This work aims at proposing a statistical frame to assess the probability of patterns in tensor data to deviate from null expectations, extending well-established principles for assessing the statistical significance of patterns in tabular data. A principled discussion on binomial testing to mitigate false positive discoveries is entailed at the light of: variable dependencies, temporal associations and misalignments, and multi-hypothesis correction. Results gathered from the application of triclustering algorithms over distinct real-world case studies in biotechnological domains confer validity to the proposed statistical frame while revealing vulnerabilities of reference triclustering searches. The proposed assessment can be incorporated into existing triclustering algorithms to minimize spurious occurrences, rank patterns, and further prune the search space, reducing their computational complexity.

Low-tubal-rank tensor completion via local and nonlocal knowledge

Many previous studies have indicated that exploiting prior information about the underlying tensor data is a sound approach for completing missing or damaged entries. This prior information can be divided into three commonly used categories: low-rankness, local piecewise smoothness, and nonlocal self-similarity (NSS) priors. Although existing methods based on these priors have gained considerable attention, the majority of studies utilize only one or two of these priors, leading to inadequate extraction of structural information from tensors. To address this limitation and comprehensively depict the inherent structural information of underlying tensor data, this article develops a novel tensor completion framework that can simultaneously utilize the three abovementioned priors within a plug-and-play framework. More precisely, we adopt the tensor correlated total variation (t-CTV) norm as a robust representation for capturing the combined effects of low-rankness and local piecewise smoothness priors, eliminating the need for a trade-off parameter in the process; meanwhile, we introduce an advanced denoiser to explore the NSS prior. Furthermore, to address the presented optimization model, we design an alternating direction method of multipliers (ADMM) algorithm and innovatively provide its corresponding global convergence guarantees. Extensive numerical experiments on real tensor data, including color, medical and hyperspectral images, demonstrate that the proposed method surpasses various advanced approaches in terms of both quality metrics and visual effects.

Tensorial Principal Component Analysis in Detecting Temporal Trajectories of Purchase Patterns in Loyalty Card Data: Retrospective Cohort Study.

Loyalty card data automatically collected by retailers provide an excellent source for evaluating health-related purchase behavior of customers. The data comprise information on every grocery purchase, including expenditures on product groups and the time of purchase for each customer. Such data where customers have an expenditure value for every product group for each time can be formulated as 3D tensorial data. This study aimed to use the modern tensorial principal component analysis (PCA) method to uncover the characteristics of health-related purchase patterns from loyalty card data. Another aim was to identify card holders with distinct purchase patterns. We also considered the interpretation, advantages, and challenges of tensorial PCA compared with standard PCA. Loyalty card program members from the largest retailer in Finland were invited to participate in this study. Our LoCard data consist of the purchases of 7251 card holders who consented to the use of their data from the year 2016. The purchases were reclassified into 55 product groups and aggregated across 52 weeks. The data were then analyzed using tensorial PCA, allowing us to effectively reduce the time and product group-wise dimensions simultaneously. The augmentation method was used for selecting the suitable number of principal components for the analysis. Using tensorial PCA, we were able to systematically search for typical food purchasing patterns across time and product groups as well as detect different purchasing behaviors across groups of card holders. For example, we identified customers who purchased large amounts of meat products and separated them further into groups based on time profiles, that is, customers whose purchases of meat remained stable, increased, or decreased throughout the year or varied between seasons of the year. Using tensorial PCA, we can effectively examine customers' purchasing behavior in more detail than with traditional methods because it can handle time and product group dimensions simultaneously. When interpreting the results, both time and product dimensions must be considered. In further analyses, these time and product groups can be directly associated with additional consumer characteristics such as socioeconomic and demographic predictors of dietary patterns. In addition, they can be linked to external factors that impact grocery purchases such as inflation and unexpected pandemics. This enables us to identify what types of people have specific purchasing patterns, which can help in the development of ways in which consumers can be steered toward making healthier food choices.

Fastensor: Optimise the Tensor I/O Path from SSD to GPU for Deep Learning Training

In recent years, benefiting from the increase in model size and complexity, deep learning has achieved tremendous success in computer vision (CV) and (NLP). Training deep learning models using accelerators such as GPUs often requires much iterative data to be transferred from NVMe SSD to GPU memory. Much recent work has focused on data transfer during the pre-processing phase and has introduced techniques such as multiprocessing and GPU Direct Storage (GDS) to accelerate it. However, tensor data during training (such as Checkpoints, logs, and intermediate feature maps), which is also time-consuming, is often transferred using traditional serial, long-I/O-path transfer methods. In this article, based on GDS technology, we built Fastensor, an efficient tool for tensor data transfer between the NVMe SSDs and GPUs. To achieve higher tensor data I/O throughput, we optimized the traditional data I/O process. We also proposed a data and runtime context-aware tensor I/O algorithm. Fastensor can select the most suitable data transfer tool for the current tensor from a candidate set of tools during model training. The optimal tool is derived from a dictionary generated by our adaptive exploration algorithm in the first few training iterations. We used Fastensor’s unified interface to test the read/write bandwidth and energy consumption of different transfer tools for different sizes of tensor blocks. We found that the execution efficiency of different tensor transfer tools is related to both the tensor block size and the runtime context. We then deployed Fastensor in the widely applicable Pytorch deep learning framework. We showed that Fastensor could perform superior in typical scenarios of model parameter saving and intermediate feature map transfer with the same hardware configuration. Fastensor achieves a 5.37x read performance improvement compared to torch.save () when used for model parameter saving. When used for intermediate feature map transfer, Fastensor can increase the supported training batch size by 20x, while the total read and write speed is increased by 2.96x compared to the torch I/O API.

Convex Robust Recovery of Corrupted Tensors via Tensor Singular Value Decomposition and Local Low-Rank Approximation

This paper discusses the recovery of tensor data corrupted by random noise. Our approach assumes that the potential structure of data is a linear combination of several low-rank tensor subspaces. The goal is to recover exactly these local low-rank tensors and remove random noise as much as possible. Non-parametric kernel smoothing technique is employed to establish an effective mathematical notion of local models. After that, each local model can be robustly separated into a low-rank tensor and a sparse tensor. The low-rank tensor can be recovered by minimizing a weighted combination of the norm and the tensor nuclear norm (TNN) obtained as the tightest convex relaxation of tensor multi-linear rank defined in Tensor Singular Value Decomposition (TSVD). Numerical simulation experiments verify that our proposed approach can effectively denoise the tensor data such as color images with random noise and has superior performance compared to the existing methods.

Comparative 3D inversion of magnetotelluric phase tensors and impedances reveals electrically anisotropic base of Gawler Craton, South Australia

Isotropic three-dimensional (3-D) inversion has become a standard tool in the interpretation of magnetotelluric (MT) data. 3-D anisotropic inversion codes are under development, yet the number of unknowns increases by a factor of 6 rendering the problem extremely ill-posed. The presence of anisotropy is usually inferred from (i) spurious sequences of conductive and resistive bodies or (ii) comparison with two-dimensional anisotropic modelling approaches. Here, we investigate the 3-D structure of the Gawler Craton down to ∼250 km depth using 282 sites of the AusLAMP array located in the southern half of South Australia. Inversions of the MT impedance as phase tensors and real and imaginary parts result in diverging structures at depths > 70 km. We demonstrate that a unifying model that explains all data types similarly well is suggestive of an anisotropic resistivity structure at the base of the Gawler Craton lithosphere at depths of 120–210 km. Depth location and orientation of the anisotropy agree well with results from the analysis of seismic receiver functions. We suggest that electric anisotropy in the Gawler Craton is a result of lattice-preferred orientation of olivine crystals and metasomatic processes with macroscopic preferential orientation. Our results illustrate that inversion of phase tensor data is superior for the direct imaging of anisotropic resistivity contrasts in otherwise isotropic resistivity models; inversion models obtained with impedances may miss such structures. “Comparable” overall RMS misfits are often meaningless when comparing inversion results for various data types since sensitivities differ between data types. Reliable inversion results consistent with the entire data set can only be recovered if data fits are assessed systematically for all data representations. We also discuss the influence of error settings for phase tensors on the inversion. Our study also revealed that, if persistent across large areas, (i) parallel orientation of phase tensor major axes, (ii) constantly high phase tensor maximum phases or (iii) diverging directions of phase tensor major axes and induction arrows are suggestive of anisotropic structures and corresponding hypotheses should be evaluated.

Multilinear subspace learning for Person Re-Identification based fusion of high order tensor features

Video surveillance image analysis and processing is an important field in computer vision and among the challenging tasks for Person Re-Identification (PRe-ID). The latter aims at finding a target person who has already been identified and appeared on a camera network using a powerful description of their pedestrian images. The success of recent research on person PRe-ID is largely backed into the effective features extraction and representation with a powerful learning of these features to correctly discriminate pedestrian images. To this end, two powerful features, Convolutional Neural Network (CNN) and Local Maximal Occurrence (LOMO) are modeled on a multidimensional data in the proposed method, High-Dimensional Feature Fusion (HDFF). Specifically, a new tensor fusion scheme is introduced to take advantage and combine two types of features in the same tensor data even if its dimensions are not the same. To improve the accuracy, we use Tensor Cross-View Quadratic Analysis (TXQDA) to perform multilinear subspace learning followed by the Cosine similarity for matching. TXQDA efficiently ensures the learning ability and reduces the high dimensionality resulting from high-order tensor data. The effectiveness of the proposed method is verified through experiments on three challenging widely-used PRe-ID datasets namely, VIPeR, GRID, and PRID450S. Extensive experiments show that the proposed method performs very well when compared with recent state-of-the-art methods.