Low-rank Tensor Factorization Research Articles

Hyperspectral Image Denoising: From Model-Driven, Data-Driven, to Model-Data-Driven.

Mixed noise pollution in HSI severely disturbs subsequent interpretations and applications. In this technical review, we first give the noise analysis in different noisy HSIs and conclude crucial points for programming HSI denoising algorithms. Then, a general HSI restoration model is formulated for optimization. Later, we comprehensively review existing HSI denoising methods, from model-driven strategy (nonlocal mean, total variation, sparse representation, low-rank matrix approximation, and low-rank tensor factorization), data-driven strategy [2-D convolutional neural network (CNN), 3-D CNN, hybrid, and unsupervised networks], to model-data-driven strategy. The advantages and disadvantages of each strategy for HSI denoising are summarized and contrasted. Behind this, we present an evaluation of the HSI denoising methods for various noisy HSIs in simulated and real experiments. The classification results of denoised HSIs and execution efficiency are depicted through these HSI denoising methods. Finally, prospects of future HSI denoising methods are listed in this technical review to guide the ongoing road for HSI denoising. The HSI denoising dataset could be found at https://qzhang95.github.io.

Hyperspectral Image Mixed Noise Removal via Double Factor Total Variation Nonlocal Low-Rank Tensor Regularization

A hyperspectral image (HSI) is often corrupted by various types of noise during image acquisition, e.g., Gaussian noise, impulse noise, stripes, deadlines, and more. Thus, as a preprocessing step, HSI denoising plays a vital role in many subsequent tasks. Recently, a variety of mixed noise removal approaches have been developed for HSI, and the methods based on spatial–spectral double factor and total variation (DFTV) regularization have achieved comparable performance. Additionally, the nonlocal low-rank tensor model (NLR) is often employed to characterize spatial nonlocal self-similarity (NSS). Generally, fully exploring prior knowledge can improve the denoising performance, but it significantly increases the computational cost when the NSS prior is employed. To solve this problem, this article proposes a novel DFTV-based NLR regularization (DFTVNLR) model for HSI mixed noise removal. The proposed model employs low-rank tensor factorization (LRTF) to characterize the spectral global low-rankness (LR), introduces 2-D and 1-D TV constraints on double-factor to characterize the spatial and spectral local smoothness (LS), respectively. Meanwhile, the NLR is applied to the spatial factor to characterize the NSS. Then, we developed an algorithm based on proximal alternating minimization (PAM) to solve the proposed model effectively. Particularly, we effectively controlled the computational cost from two aspects, namely taking small-sized double factor as regularization object and putting the time-consuming NLR model before the main loop with fewer iterations to solve it independently. Finally, considerable experiments on simulated and real noisy HSI substantiate that the proposed method is superior to the related state-of-the-art methods in balancing the denoising effect and speed.

Attention-Guided Low-Rank Tensor Factorization for Image Recovery With Poisson Observation

Attention-Guided Low-Rank Tensor Factorization for Image Recovery With Poisson Observation

Study of performance of low-rank nonnegative tensor factorization methods

Abstract In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications.

Combined Deep Priors With Low-Rank Tensor Factorization for Hyperspectral Image Restoration

Mixed noise pollution severely disturbs hyperspectral image (HSI) processing and applications. Plenty of algorithms have been developed to address this issue via two strategies: model-driven or data-driven strategy. However, model-driven methods exist in the highly time-consuming weakness of iterative optimization and unstable sensitivity of setting parameters. Data-driven methods usually perform poor due to the overfitting effects. To solve these issues, we combine both the deep denoising priors with low-rank tensor factorization (DP-LRTF) for HSI restoration. The proposed method uses Tucker tensor factorization to depict the global spectral low-rank constraint. Then the spectral orthogonal basis and spatial reduced factor are optimized by two deep denoising priors, respectively. Through this integrated strategy, we can simultaneously exploit the intrinsic low-rank property of HSI, and utilize the powerful feature extraction ability by deep learning for HSI restoration. Compared with model-driven and data-driven methods, DP-LRTF outperforms on HSI mixed noise removal and execution efficiency for various simulated/real experiments.

Bayesian robust tensor completion via CP decomposition

The real-world tensor data are inevitably missing and corrupted with noise. Some models of the low-rank tensor factorization (LRTF) add an L1 norm or L2 norm to deal with the sparse or Gaussian noise. However, the real noise are usually complex. We propose a robust Bayesian tensor completion method, called MoG BTC-CP, which could impute the missing data and remove the complex noise simultaneously. The observed tensor is assumed to be the summation of a low-rank tensor and the noise. CP decomposition is proposed to extract the low-rank structure of the tensor. We assume that the noise follows a Mixture of Gaussian (MoG) distribution. A full Bayesian framework together with a Gibbs sampling algorithm is designed to estimate the model. Extensive experiments including synthetic data and real life applications show that MoG BTC-CP outperforms the recently published leading tensor completion and denoising methods.

Infrared Small Target Detection Using Nonoverlapping Patch Spatial–Temporal Tensor Factorization With Capped Nuclear Norm Regularization

Infrared target detection technology is indispensable in precision guidance and infrared warning systems. It faces grant challenges under complex scenes, such as poor robustness against a variety of scenes, inaccurate separation of the target from similar goals, etc. To solve these problems, we present a novel small target detection method based on spatial–temporal information of infrared imageries. A nonoverlapping patch spatial–temporal tensor (NPSTT) model is established by sliding a window to obtain nonoverlapping patches in adjacent images. Moreover, the tensor capped nuclear norm (TCNN) is introduced, which approximates the tensor rank more accurately for local infrared images. TCNN regularization and NPSTT (TCNN-NPSTT) are adopted to detect the potential targets. Essentially, the procedure of extracting targets from the background is converted to the low rank and sparse tensor factorization. Besides, an efficient optimization scheme utilizing the alternating direction multiplier method (ADMM) is introduced to solve the proposed model. Experiments in various scenes show that NPSTT can obtain better performance against complex backgrounds than state-of-the-art baselines, using the evaluating indicators include the receiver operating characteristic curve (ROC), signal-to-clutter ratio gain (SCRG), and background suppression factor (BSF).

Reweighted Low-Rank Factorization With Deep Prior for Image Restoration

The low-rank recovery is a powerful tool to restore images from incomplete and corrupted observations. Conventional low-rank recovery techniques employ the reweighted nuclear norm minimization, which requires performing the full singular value decomposition and thus is computationally expensive. Using the scheme of bilinear factorization, we propose the Reweighted Low-rank Matrix Factorization (RLMF) method for single channel image restoration. The RLMF method can not only inherit the computational efficiency of bilinear factorization, but also incorporate the empirical distribution of the singular values in natural images. Then, considering the correlation between image channels, we generalize the reweighted nuclear norm from matrices to tensors, and develop the Reweighted Low-rank Tensor Factorization (RLTF) method for multichannel image restoration. Moreover, we enhance the RLMF and RLTF methods by introducing the deep image prior information, which is capable of capturing the implicit image structure through the neural network architecture to improve restoration accuracy. Experimental results show the computational efficiency of the proposed low-rank factorization scheme, and the superior restoration accuracy of the proposed methods compared with the state-of-the-art methods.

Spatial Low-Rank Tensor Factorization and Unmixing of Hyperspectral Images

This work presents a method for hyperspectral image unmixing based on non-negative tensor factorization. While traditional approaches may process spectral information without regard for spatial structures in the dataset, tensor factorization preserves the spectral-spatial relationship which we intend to exploit. We used a rank-(L, L, 1) decomposition, which approximates the original tensor as a sum of R components. Each component is a tensor resulting from the multiplication of a low-rank spatial representation and a spectral vector. Our approach uses spatial factors to identify high abundance areas where pure pixels (endmembers) may lie. Unmixing is done by applying Fully Constrained Least Squares such that abundance maps are produced for each inferred endmember. The results of this method are compared against other approaches based on non-negative matrix and tensor factorization. We observed a significant reduction of spectral angle distance for extracted endmembers and equal or better RMSE for abundance maps as compared with existing benchmarks.

Prema: Principled Tensor Data Recovery From Multiple Aggregated Views

Multidimensional data have become ubiquitous and are frequently encountered in situations where the information is aggregated over multiple data atoms. The aggregation can be over time or other features, such as geographical location. We often have access to multiple aggregated views of the same data, each aggregated in one or more dimensions, especially when data are collected or measured by different agencies. For instance, item sales can be aggregated temporally, and over groups of stores based on their location or affiliation. However, data mining and machine learning models benefit from detailed data for personalized analysis and prediction. Thus, data disaggregation algorithms are becoming increasingly important in various domains. The goal of this paper is to reconstruct finer-scale data from multiple coarse views, aggregated over different (subsets of) dimensions. The proposed method, called Prema , leverages low-rank tensor factorization tools to fuse the multiple views and provide recovery guarantees under certain conditions. Prema can tackle challenging scenarios, such as missing or partially observed data, double aggregation, and even blind disaggregation (without knowledge of the aggregation patterns) using a variant of Prema called B-Prema . To showcase the effectiveness of Prema , the paper includes extensive experiments using real data from different domains: retail sales, crime counts, and weather observations.

Hyperspectral Image Denoising Based on Global and Nonlocal Low-Rank Factorizations

The ever-increasing spectral resolution of hyperspectral images (HSIs) is often obtained at the cost of a decrease in the signal-to-noise ratio of the measurements, thus calling for effective denoising techniques. HSIs from the real world lie in low-dimensional subspaces and are self-similar. The low dimensionality stems from the high correlation existing among the reflectance vectors, and self-similarity is common in real-world images. In this article, we exploit the above two properties. The low dimensionality is a global property that enables the denoising to be formulated just with respect to the subspace representation coefficients, thus greatly improving the denoising performance and reducing the computational complexity during processing. The self-similarity is exploited via a low-rank tensor factorization of nonlocal similar 3-D patches. The proposed factorization hinges on the optimal shrinkage/thresholding of the singular value decomposition (SVD) singular values of low-rank tensor unfoldings. As a result, the proposed method is user friendly and insensitive to its parameters. Its effectiveness is illustrated in a comparison with state-of-the-art competitors. A MATLAB demo of this work is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/LinaZhuang</uri> for the sake of reproducibility.

Sparse and Low-Rank Constrained Tensor Factorization for Hyperspectral Image Unmixing

Third-order tensors have been widely used in hyperspectral remote sensing because of their ability to maintain the 3-D structure of hyperspectral images. In recent years, hyperspectral unmixing algorithms based on tensor factorization have emerged, but these decomposition processes may be inconsistent with physical mechanism of unmixing. To solve this problem, this article proposes a sparse and low-rank constrained tensor factorization unmixing algorithm based on a matrix-vector nonnegative tensor factorization (MV-NTF) framework. Considering the fact that each component tensor obtained by the image decomposition contains only one endmember and the corresponding abundance matrix has sparse property, a sparse constraint is imposed to ensure the accuracy of abundance maps. Since abundance maps also have low-rank attribute, in order to avoid the strict low-rank constraint in the original MV-NTF framework, a low-rank tensor regularization is introduced to flexibly express the low-rank characteristics of the abundance tensors, making the resulting abundance maps more in line with the actual scene. Then, the optimization problem is solved by using the alternating direction method of multipliers. In experiments, simulated datasets are adopted to demonstrate the effectiveness of the sparse and low-rank constraints of the proposed algorithm, and real datasets from different sensors and different scenarios are used to verify its applicability.

A High-Quality Photon-Counting CT Technique Based on Weight Adaptive Total-Variation and Image-Spectral Tensor Factorization for Small Animals Imaging

Photon-counting X-ray computed tomography (CT) has been attracting great attention in tissue characterization, material discrimination, and so on. The emitting X-ray energy spectrum cutting into several energy bins that can result in only a part of X-ray photons can be collected within each narrow bin. This can compromise the image quality. In this case, how to obtain high-quality tomography is a big challenge. In this study, to overcome these issues, we mainly focus on developing an advanced imaging software based on the latest photon-counting CT system (MARS scanner). Specifically, we first design a weight adaptive total variation (TV) using compressed sensing theory. Then, combining the weight adaptive TV and nonlocal low-rank tensor factorization to formulate a new weight adaptive total-variation and image-spectral tensor factorization (WATITF) model for high-quality imaging. Finally, the optimization model is performed to obtain its solution. The studies including the numerical and preclinical mice are performed to validate and evaluate its outperformance.

Overlap matrix concentration in optimal Bayesian inference

AbstractWe consider models of Bayesian inference of signals with vectorial components of finite dimensionality. We show that under a proper perturbation, these models are replica symmetric in the sense that the overlap matrix concentrates. The overlap matrix is the order parameter in these models and is directly related to error metrics such as minimum mean-square errors. Our proof is valid in the optimal Bayesian inference setting. This means that it relies on the assumption that the model and all its hyper-parameters are known so that the posterior distribution can be written exactly. Examples of important problems in high-dimensional inference and learning to which our results apply are low-rank tensor factorization, the committee machine neural network with a finite number of hidden neurons in the teacher–student scenario or multi-layer versions of the generalized linear model.

A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships

Low-rank tensor factorization can not only mine the implicit relationships between data but also fill in the missing data when working with complex data. Compared with the traditional collaborative filtering (CF) algorithm, the changes are essentially proposed, from traditional matrix analysis to three-dimensional spatial analysis. Based on low-rank tensor factorization, this paper proposes a recommendation model that comprehensively considers local information and global information, in other words, combining the similarity between trust users and low-rank tensor factorization. First, the similarity between trusted users is measured to capture local information between users by trusting similar preferences of users when selecting items. Then, the users’ similarity is integrated into the tensor, and the low-rank tensor factorization is used to better maintain and describe the internal structure of the data to obtain global information. Furthermore, based on the idea of the alternating least squares method, the conjugate gradient (CG) optimization algorithm for the model of this paper is designed. The local and global information is used to generate the optimal expected result in an iterative process. Finally, we conducted a large number of comparative experiments on the Ciao dataset and the FilmTrust dataset. Experimental results show that the algorithm has less precision loss under the data set with lower density. Thus, not only can a perfect compromise between accuracy and coverage be achieved, but also the computational complexity can be reduced to meet the need for real-time results.

Robust CP Tensor Factorization With Skew Noise

The low-rank tensor factorization (LRTF) technique has received increasing popularity in data science, especially in computer vision applications. Many robust LRTF models have been presented recently. However, none of them take the skewness of data into account. This letter proposes a novel LRTF model for skew data analysis by modeling noise as a Mixture of Asymmetric Laplacians (MoAL). The numerical experiments show that the new model MoAL-LRTF outperforms several state-of-the-art counterparts. The codes for all the experiments are available at https://xsxjtu.github.io/Projects/MoAL/main.html .

Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O~(N3/2λ) T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

Hyperspectral Restoration via $L_0$ Gradient Regularized Low-Rank Tensor Factorization

Due to the mechanism of the data acquisition process, hyperspectral imagery (HSI) are usually contaminated by various noises, e.g., Gaussian noise, impulse noise, strips, and dead lines. In this article, a spectral–spatial $L_{0}$ gradient regularized low-rank tensor factorization (LRTF $L_{0}$ ) method is proposed for hyperspectral denoising, in which the restored HSI is approximated by low-rank block term decomposition (BTD). BTD factorizes a tensor into the sum of a series of component tensors, each of which is represented by the outer product of a matrix and a vector. From subspace learning point of view, the vector and matrix can be considered as a spectral atom and its corresponding coding coefficients. In the proposed method, the correlations in both spectral and spatial domains are taken into account via the small size of atom set and low-rankness of coding matrices. In addition, HSIs also have the local structure of piecewise smoothness in both spectral and spatial domains. Motivated by the supreme virtues of $L_{0}$ gradient regularization in image structure exploitation, we develop a spectral–spatial $L_{0}$ gradient regularization and embed it into BTD to explore the spectral–spatial texture information. The proposed method can simultaneously remove various types of noises, and the experimental results on both synthetic data and real-world data show its superiority when compared with several state-of-the-art approaches.

Low-rank Bayesian tensor factorization for hyperspectral image denoising

In this paper, we present a low-rank Bayesian tensor factorization approach for hyperspectral image (HSI) denoising problem, where zero-mean white and homogeneous Gaussian additive noise is removed from a given HSI. The approach is based on two intrinsic properties underlying a HSI, i.e., the global correlation along spectrum (GCS) and nonlocal self-similarity across space (NSS). We first adaptively construct the patch-based tensor representation for the HSI to extract the NSS knowledge while preserving the property of GCS. Then, we employ the low rank property in this representation to design a hierarchical probabilistic model based on Bayesian tensor factorization to capture the inherent spatial-spectral correlation of HSI, which can be effectively solved under the variational Bayesian framework. Furthermore, through incorporating these two procedures in an iterative manner, we build an effective HSI denoising model to recover HSI from its corruption. This leads to a state-of-the-art denoising performance, consistently surpassing recently published leading HSI denoising methods in terms of both comprehensive quantitative assessments and subjective visual quality.

Non-Local Low-Rank Cube-Based Tensor Factorization for Spectral CT Reconstruction.

Spectral computed tomography (CT) reconstructs material-dependent attenuation images from the projections of multiple narrow energy windows, which is meaningful for material identification and decomposition. Unfortunately, the multi-energy projection datasets usually have lower signal-noise ratios (SNR). Very recently, a spatial-spectral cube matching frame (SSCMF) was proposed to explore the non-local spatial-spectral similarities for spectral CT. This method constructs a group by clustering up a series of non-local spatial-spectral cubes. The small size of spatial patches for such a group makes the SSCMF fail to fully encode the sparsity and low-rank properties. The hard-thresholding and collaboration filtering in the SSCMF also cause difficulty in recovering the image features and spatial edges. While all the steps are operated on 4-D group, the huge computational cost and memory load might not be affordable in practice. To avoid the above limitations and further improve the image quality, we first formulate a non-local cube-based tensor instead of group to encode the sparsity and low-rank properties. Then, as a new regularizer, the Kronecker-basis-representation tensor factorization is employed into a basic spectral CT reconstruction model to enhance the capability of image feature extraction and spatial edge preservation, generating a non-local low-rank cube-based tensor factorization (NLCTF) method. Finally, the split-Bregman method is adopted to solve the NLCTF model. Both numerical simulations and preclinical mouse studies are performed to validate and evaluate the NLCTF algorithm. The results show that the NLCTF method outperforms the other state-of-the-art competing algorithms.