A Sequentially Truncated Higher Order Singular Value Decomposition-Based Algorithm for Tensor Completion

Multichannel Image Completion With Mixture Noise: Adaptive Sparse Low-Rank Tensor Subspace Meets Nonlocal Self-Similarity

Multichannel image completion with mixture noise is a common but complex problem in the fields of machine learning, image processing, and computer vision. Most existing algorithms devote to explore global low-rank information and fail to optimize local and joint-mode structures, which may lead to oversmooth restoration results or lower quality restoration details. In this study, we propose a novel model to deal with multichannel image completion with mixture noise based on adaptive sparse low-rank tensor subspace and nonlocal self-similarity (ASLTS-NS). In the proposed model, a nonlocal similar patch matching framework cooperating with Tucker decomposition is used to explore information of global and joint modes and optimize the local structure for improving restoration quality. In order to enhance the robustness of low-rank decomposition to data missing and mixture noise, we present an adaptive sparse low-rank regularization to construct robust tensor subspace for self-weighing importance of different modes and capturing a stable inherent structure. In addition, joint tensor Frobenius and l1 regularizations are exploited to control two different types of noise. Based on alternating directions method of multipliers (ADMM), a convergent learning algorithm is designed to solve this model. Experimental results on three different types of multichannel image sets demonstrate the advantages of ASLTS-NS under five complex scenarios.

Visible light fingerprint database recovery algorithm based on CP decomposition

Visible light communication(VLC) is a new method of indoor communication. It can provide an effective solution for indoor positioning. Fingerprint-based visible light positioning(VLP) has been widely studied for its feasibility and high accuracy. The acquisition of 'fingerprint database' is crucial for accurate VLP. However, sparse sensors such as photodiode(PD) can only be arranged because of the space-limited scenario and high costs. Correspondingly, it results in the loss of the fingerprint database. Therefore, it is indispensable to solve the problem of how to effectively and accurately recover the fingerprint database from measurements of sparsely arranged sensors. In this paper, we propose a spatio-temporal constraint tensor completion (SCTC) algorithm based on CANDECOMP/PARAFAC (CP) decomposition to recover the fingerprint database from measurements of sparsely arranged sensors. Specifically, we model the measurements from the spatial and temporal dimensions as a tensor, and formulate the optimization problem based on the low-rank feature of the tensor. To improve the recovery accuracy, spatial and temporal constraint matrices are introduced to effectively constrain the optimization direction when completing the tensor. Spatial constraint matrices are constructed by utilizing the mode-n expansion matrix of the tensor based on the undirected graph theory. Accordingly, the Toeplitz matrix is used as the temporal constraint matrix to excavate the temporal correlation of the tensor. Since the optimization problem is non-convex and difficult to solve, we introduce CP decomposition to decompose the tensor into several factor matrices. By solving the factor matrices, the original tensor is reconstructed. The performance of the proposed SCTC algorithm is confirmed via experimental measured data.

Robust to Rank Selection: Low-Rank Sparse Tensor-Ring Completion

Tensor-ring (TR) decomposition was recently studied and applied for low-rank tensor completion due to its powerful representation ability of high-order tensors. However, most of the existing TR-based methods tend to suffer from deterioration when the selected rank is larger than the true one. To address this issue, this article proposes a new low-rank sparse TR completion method by imposing the Frobenius norm regularization on its latent space. Specifically, we theoretically establish that the proposed method is capable of exploiting the low rankness and Kronecker-basis-representation (KBR)-based sparsity of the target tensor using the Frobenius norm of latent TR-cores. We optimize the proposed TR completion by block coordinate descent (BCD) algorithm and design a modified TR decomposition for the initialization of this algorithm. Extensive experimental results on synthetic data and visual data have demonstrated that the proposed method is able to achieve better results compared to the conventional TR-based completion methods and other state-of-the-art methods and, meanwhile, is quite robust even if the selected TR-rank increases.

Robust Matrix Completion With Column Outliers

Matrix completion, in essence, involves recovering a low-rank matrix from a subset of its entries. Most existing methods for matrix completion neglect two significant issues. First, in several practical applications, such as collaborative filtering, some samples may be corrupted completely. However, most of the robust algorithms consider only the condition that a few components of each column have been arbitrarily contaminated. Second, many real data are not static in nature. Nevertheless, the conventional batch-based matrix completion methods cannot efficiently deal with the out-of-sample, that is, the vector completion problem. In this article, we first provide a novel robust matrix completion model and then develop an efficient optimization method that only requires conducting one time singular value decomposition for a thin matrix per iteration. Furthermore, by exploiting the essence of online matrix completion algorithms, we develop a vector completion model which can help users predict the missing values of out of sample. Numerical comparisons with traditional batch-based and online matrix completion algorithms demonstrate the benefits of the proposed method on streaming data corrupted by column outliers. Moreover, we show that our model can be used to detect outliers from incomplete information. This advantage is validated via numerous experimental results on synthetic and real-world data.

Robust Tensor SVD and Recovery With Rank Estimation

Tensor singular value decomposition (t-SVD) has recently become increasingly popular for tensor recovery under partial and/or corrupted observations. However, the existing t -SVD-based methods neither make use of a rank prior nor provide an accurate rank estimation (RE), which would limit their recovery performance. From the practical perspective, the tensor RE problem is nontrivial and difficult to solve. In this article, we, therefore, aim to determine the correct rank of an intrinsic low-rank tensor from corrupted observations based on t-SVD and further improve recovery results with the estimated rank. Specifically, we first induce the equivalence of the tensor nuclear norm (TNN) of a tensor and its f -diagonal tensor. We then simultaneously minimize the reconstruction error and TNN of the f -diagonal tensor, leading to RE. Subsequently, we relax our model by removing the TNN regularizer to improve the recovery performance. Furthermore, we consider more general cases in the presence of missing data and/or gross corruptions by proposing robust tensor principal component analysis and robust tensor completion with RE. The robust methods can achieve successful recovery by refining the models with correct estimated ranks. Experimental results show that the proposed methods outperform the state-of-the-art methods with significant improvements.

A Novel Reconstruction Method for Temperature Distribution Measurement Based on Ultrasonic Tomography

The precise temperature distribution measurement is crucial in many industrial fields, where ultrasonic tomography (UT) has broad application prospects and significance. In order to improve the resolution of reconstructed temperature distribution images and maintain high accuracy, a novel two-step reconstruction method is proposed in this article. First, the problem of solving the temperature distribution is converted to an optimization problem and then solved by an improved version of the equilibrium optimizer (IEO), in which a new nonlinear time strategy and novel population update rules are deployed. Then, based on the low-resolution and high-precision images reconstructed by IEO, Gaussian process regression (GPR) is adopted to enhance image resolution and keep the reconstruction errors low. After that, the number of divided grids and the parameters of IEO are also further studied to improve the reconstruction quality. The results of numerical simulations and experiments indicate that high-resolution images with low reconstruction errors can be reconstructed effectively by the proposed IEO-GPR method, and it also shows excellent robust performance. For a complex three-peak temperature distribution, a competitive accuracy with 3.10% and 2.37% error at root-mean-square error and average relative error is achieved, respectively. In practical experiment, the root-mean-square error of IEO-GPR is 0.72%, which is at least 0.89% lower than that of conventional algorithms.

A Nonlocal Self-Similarity-Based Weighted Tensor Low-Rank Decomposition for Multichannel Image Completion With Mixture Noise

Multichannel image completion with mixture noise is a challenging problem in the fields of machine learning, computer vision, image processing, and data mining. Traditional image completion models are not appropriate to deal with this problem directly since their reconstruction priors may mismatch corruption priors. To address this issue, we propose a novel nonlocal self-similarity-based weighted tensor low-rank decomposition (NSWTLD) model that can achieve global optimization and local enhancement. In the proposed model, based on the corruption priors and the reconstruction priors, a pixel weighting strategy is given to characterize the joint effects of missing data, the Gaussian noise, and the impulse noise. To discover and utilize the accurate nonlocal self-similarity information to enhance the restoration quality of the details, the traditional nonlocal learning framework is optimized by employing improved index determination of patch group and handling strip noise caused by patch overlapping. In addition, an efficient and convergent algorithm is presented to solve the NSWTLD model. Comprehensive experiments are conducted on four types of multichannel images under various corruption scenarios. The results demonstrate the efficiency and effectiveness of the proposed model.

HOSVD-Based Algorithm for Weighted Tensor Completion

Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations.

Nonlocal Low-Rank Tensor Completion for Visual Data

In this paper, we propose a novel nonlocal patch tensor-based visual data completion algorithm and analyze its potential problems. Our algorithm consists of two steps: the first step is initializing the image with triangulation-based linear interpolation and the second step is grouping similar nonlocal patches as a tensor then applying the proposed tensor completion technique. Specifically, with treating a group of patch matrices as a tensor, we impose the low-rank constraint on the tensor through the recently proposed tensor nuclear norm. Moreover, we observe that after the first interpolation step, the image gets blurred and, thus, the similar patches we have found may not exactly match the reference. We name the problem "Patch Mismatch," and then in order to avoid the error caused by it, we further decompose the patch tensor into a low-rank tensor and a sparse tensor, which means the accepted horizontal strips in mismatched patches. Furthermore, our theoretical analysis shows that the error caused by Patch Mismatch can be decomposed into two components, one of which can be bounded by a reasonable assumption named local patch similarity, and the other part is lower than that using matrix completion. Extensive experimental results on real-world datasets verify our method's superiority to the state-of-the-art tensor-based image inpainting methods.