Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decomposition

Hyperspectral images (HSIs) are often corrupted by a mixture of several types of noise during the acquisition process, e.g., Gaussian noise, impulse noise, dead lines, stripes, and many others. Such complex noise could degrade the quality of the acquired HSIs, limiting the precision of the subsequent processing. In this paper, we present a novel tensor-based HSI restoration approach by fully identifying the intrinsic structures of the clean HSI part and the mixed noise part respectively. Specifically, for the clean HSI part, we use tensor Tucker decomposition to describe the global correlation among all bands, and an anisotropic spatial-spectral total variation (SSTV) regularization to characterize the piecewise smooth structure in both spatial and spectral domains. For the mixed noise part, we adopt the $\ell_1$ norm regularization to detect the sparse noise, including stripes, impulse noise, and dead pixels. Despite that TV regulariztion has the ability of removing Gaussian noise, the Frobenius norm term is further used to model heavy Gaussian noise for some real-world scenarios. Then, we develop an efficient algorithm for solving the resulting optimization problem by using the augmented Lagrange multiplier (ALM) method. Finally, extensive experiments on simulated and real-world noise HSIs are carried out to demonstrate the superiority of the proposed method over the existing state-of-the-art ones.Comment: 15 pages, 20 figure

Complex-valued Adaptive System Identification via Low-Rank Tensor Decomposition

Machine learning (ML) and tensor-based methods have been of significant interest for the scientific community for the last few decades. In a previous work we presented a novel tensor-based system identification framework to ease the computational burden of tensor-only architectures while still being able to achieve exceptionally good performance. However, the derived approach only allows to process real-valued problems and is therefore not directly applicable on a wide range of signal processing and communications problems, which often deal with complex-valued systems. In this work we therefore derive two new architectures to allow the processing of complex-valued signals, and show that these extensions are able to surpass the trivial, complex-valued extension of the original architecture in terms of performance, while only requiring a slight overhead in computational resources to allow for complex-valued operations

Efficient ab initio auxiliary-field quantum Monte Carlo calculations in Gaussian bases via low-rank tensor decomposition

We describe an algorithm to reduce the cost of auxiliary-field quantum Monte Carlo (AFQMC) calculations for the electronic structure problem. The technique uses a nested low-rank factorization of the electron repulsion integral (ERI). While the cost of conventional AFQMC calculations in Gaussian bases scales as $\mathcal{O}(N^4)$ where $N$ is the size of the basis, we show that ground-state energies can be computed through tensor decomposition with reduced memory requirements and sub-quartic scaling. The algorithm is applied to hydrogen chains and square grids, water clusters, and hexagonal BN. In all cases we observe significant memory savings and, for larger systems, reduced, sub-quartic simulation time.Comment: 14 pages, 13 figures, expanded dataset and tex

Tucker-L2E⁡\operatorname{L_2E}: Robust Low-rank Tensor Decomposition with the L2⁡\operatorname{L_2} Criterion

The growing prevalence of tensor data, or multiway arrays, in science and engineering applications motivates the need for tensor decompositions that are robust against outliers. In this paper, we present a robust Tucker decomposition estimator based on the $\operatorname{L_2}$ criterion called the Tucker-$\operatorname{L_2E}$. Our numerical experiments demonstrate that Tucker-$\operatorname{L_2E}$ has empirically stronger recovery performance in more challenging high-rank scenarios compared with existing alternatives. The appropriate Tucker-rank can be selected in a data-driven manner with cross-validation or hold-out validation in our framework. The practical effectiveness of Tucker-$\operatorname{L_2E}$ is validated on real data applications in fMRI tensor denoising, PARAFAC analysis of fluorescence data, and feature extraction for classification of corrupted images

Sparse Low-Rank Tensor Decomposition for Metal Defect Detection Using Thermographic Imaging Diagnostics

With the increasing use of induction thermography (IT) for non-destructive testing (NDT) in the mechanical and rail industry, it becomes necessary for the manufactures to rapidly and accurately monitor the health of specimens. The most general problem for IT detection is due to strong noise interference. In order to counter it, general post-processing is carried out. However, due to the more complex nature of noise and irregular shape specimens, this task becomes difficult and challenging. In this paper, a low-rank tensor with a sparse mixture of Gaussian (MoG) (LRTSMoG) decomposition algorithm for natural crack detection is proposed. The proposed algorithm models jointly the low rank tensor and sparse pattern by using a tensor decomposition framework. In particular, the weak natural crack information can be extracted from strong noise. Low-rank tensor based iterative sparse MoG noise modeling is carried out to enhance the weak natural crack information as well as reducing the computational cost. In order to show the robustness and efficacy of the model, experiments are conducted for natural crack detection on a variety of specimens. A comparative analysis is presented with general tensor decomposition algorithms. The algorithms are evaluated quantitatively based on signal-to-noise-ratio (SNR) along with the visual comparative analysis

Estimating Joint Probability Distribution With Low-Rank Tensor Decomposition, Radon Transforms and Dictionaries

In this paper, we describe a method for estimating the joint probability density from data samples by assuming that the underlying distribution can be decomposed as a mixture of product densities with few mixture components. Prior works have used such a decomposition to estimate the joint density from lower-dimensional marginals, which can be estimated more reliably with the same number of samples. We combine two key ideas: dictionaries to represent 1-D densities, and random projections to estimate the joint distribution from 1-D marginals, explored separately in prior work. Our algorithm benefits from improved sample complexity over the previous dictionary-based approach by using 1-D marginals for reconstruction. We evaluate the performance of our method on estimating synthetic probability densities and compare it with the previous dictionary-based approach and Gaussian Mixture Models (GMMs). Our algorithm outperforms these other approaches in all the experimental settings

Tensor Decomposition Based Beamspace ESPRIT for Millimeter Wave MIMO Channel Estimation

We propose a search-free beamspace tensor-ESPRIT algorithm for millimeter wave MIMO channel estimation. It is a multidimensional generalization of beamspace-ESPRIT method by exploiting the multiple invariance structure of the measurements. Geometry-based channel model is considered to contain the channel sparsity feature. In our framework, an alternating least squares problem is solved for low rank tensor decomposition and the multidimensional parameters are automatically associated. The performance of the proposed algorithm is evaluated by considering different transformation schemes

Hyperspectral Image Denoising With Group Sparse and Low-Rank Tensor Decomposition

Hyperspectral image (HSI) is usually corrupted by various types of noise, including Gaussian noise, impulse noise, stripes, deadlines, and so on. Recently, sparse and low-rank matrix decomposition (SLRMD) has demonstrated to be an effective tool in HSI denoising. However, the matrix-based SLRMD technique cannot fully take the advantage of spatial and spectral information in a 3-D HSI data. In this paper, a novel group sparse and low-rank tensor decomposition (GSLRTD) method is proposed to remove different kinds of noise in HSI, while still well preserving spectral and spatial characteristics. Since a clean 3-D HSI data can be regarded as a 3-D tensor, the proposed GSLRTD method formulates a HSI recovery problem into a sparse and low-rank tensor decomposition framework. Specifically, the HSI is first divided into a set of overlapping 3-D tensor cubes, which are then clustered into groups by K-means algorithm. Then, each group contains similar tensor cubes, which can be constructed as a new tensor by unfolding these similar tensors into a set of matrices and stacking them. Finally, the SLRTD model is introduced to generate noisefree estimation for each group tensor. By aggregating all reconstructed group tensors, we can reconstruct a denoised HSI. Experiments on both simulated and real HSI data sets demonstrate the effectiveness of the proposed method.This paper was supported in part by the National Natural Science Foundation of China under Grant 61301255, Grant 61771192, and Grant 61471167, in part by the National Natural Science Fund of China for Distinguished Young Scholars under Grant 61325007, in part by the National Natural Science Fund of China for International Cooperation and Exchanges under Grant 61520106001, and in part by the Science and Technology Plan Project Fund of Hunan Province under Grant 2015WK3001 and Grant 2017RS3024.Peer Reviewe