Improved robust tensor principal component analysis for accelerating dynamic MR imaging reconstruction.

Robust tensor principal component analysis extracts the low rank and sparse component of multi-dimensional data by tensor singular value decomposition (t-SVD), which can be used for many data analysis problems. However, the current t-SVD based methods cannot fully extract the low rank component in tensor data, and low rank structure still exists in the core tensor, because t-SVD does not decompose data in the third mode. To fully exploit the low rank structure, we further extract the low rank component using low rank plus sparsity for the core matrix whose entries are from the diagonal elements of the frontal slices in the core tensor. The proposed method is applied to three groups of numerical experiments on image denoising, illumination normalization for face images and motion separation for surveillance videos, respectively, and the results show that the proposed method outperforms state-of-the-art methods in terms of both accuracy and computational complexity.

As a domain-dependent prior knowledge, side information has been introduced into Robust Principal Component Analysis (RPCA) to alleviate its degenerate or suboptimal performance in some real applications. It has recently realized that the natural structural information can be better retained if the observed data is kept in the original tensor form rather than matricizing it or other order reduction means. Hence, studies on RPCA of tensor version have attracted more and more attentions. To share the merits from both direct tensor modeling and side information, we propose three models to deal with the problem of Tensor RPCA with side information based on tensor Singular Value Decomposition (t-SVD). To solve these models, we develop an efficient algorithm with convergence guarantee using the well-known alternating direction method of multiplier. Extensive experimental studies on both synthetic and real-world tensor data have been carried out to demonstrate the superiority of the proposed models over several other state-of-the-arts. Our code is released at https://github.com/zsj9509/TPCPSF.

The major challenges during the data acquisition process in an environmental wireless sensor network (EWSN) architecture are the presence of outliers and missing data. Outliers and missing data are ubiquitous in EWSN due to sensor failures, external noise, power dwindling, communication failures etc. Robust tensor principal component analysis (RTPCA) decomposes a noisy data tensor into a low-rank tensor and a sparse tensor, which can be exploited for the data recovery in EWSNs, where the low-rank component represents the intrinsic data tensor and the sparse component represents the gross outlier tensor. In this paper, a novel probabilistic outlier modelling scheme using multivariate Chebyshev's inequality hypothesis is proposed, which maps the sample population and the associated magnitudes of outliers with the spatio-temporal correlations in the acquired data. The inherent spatio-temporal and multi-attribute correlations in the EWSN data are established using singular value methods. A tensor nuclear norm (TNN) which extracts more temporal and multi-attribute correlations in the sensory data through block circulant matricization is used as the minimization function in RTPCA. In the forest surveillance scenario, usage of RTPCA results in a reconstruction accuracy of approximately 90% for a dataset with a high missing ratio of 0.95 and the outliers having largest possible magnitudes considered. For oceanographic data, in which the variance is lower, the simulations show similar performance for different outlier contaminations, although RPTCA performs better than other matrix and tensor data completion methods in terms of reconstruction accuracy.

Chapter 6 - Tensor principal component analysis

In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].

The recent study on tensor singular value decomposition (t-SVD) that performs the Fourier transform on the tubes of a third-order tensor has gained promising performance on multidimensional data recovery problems. However, such a fixed transformation, e.g., discrete Fourier transform and discrete cosine transform, lacks being self-adapted to the change of different datasets, and thus, it is not flexible enough to exploit the low-rank and sparse property of the variety of multidimensional datasets. In this article, we consider a tube as an atom of a third-order tensor and construct a data-driven learning dictionary from the observed noisy data along the tubes of the given tensor. Then, a Bayesian dictionary learning (DL) model with tensor tubal transformed factorization, aiming to identify the underlying low-tubal-rank structure of the tensor effectively via the data-adaptive dictionary, is developed to solve the tensor robust principal component analysis (TRPCA) problem. With the defined pagewise tensor operators, a variational Bayesian DL algorithm is established and updates the posterior distributions instantaneously along the third dimension to solve the TPRCA. Extensive experiments on real-world applications, such as color image and hyperspectral image denoising and background/foreground separation problems, demonstrate both effectiveness and efficiency of the proposed approach in terms of various standard metrics.

Robust implementation of foreground extraction and vessel segmentation for X-ray coronary angiography image sequence

Probability-weighted tensor robust PCA with CP decomposition for hyperspectral image restoration

Nowadays, some tensor-based hyperspectral anomaly detection (HAD) approaches are still insufficient in utilizing the spatial-spectral structure information of hyperspectral images (HSIs), resulting in the inability to isolate the background and abnormal targets well. In this letter, an enhanced tensor low-rank representation learning (ETLR) model is proposed for HAD. Specifically, the original 3-D hyperspectral image (HSI) data is firstly decomposed into a structural background component, an anomaly component and a noise component. Among them, with the help of multi-subspace learning technology, the structural background component is reformulated by the t-product of the background dictionary tensor and the corresponding coefficient tensor. Then, tensor nuclear norm (TNN) is adopted to preserve the global low-rank property of the background component in both spatial and spectral dimensions. For the abnormal component, an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1,1</sub> -norm is designed to enhance the group sparsity of abnormal pixels. For the noise component, a tensor <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> -norm constraint is imposed to suppress the confusion of noise and anomalies. Meanwhile, a robust dictionary tensor that can adequately characterize the background is constructed by using tensor robust principal component analysis (TRPCA). Furthermore, to reduce the interference of redundant information on detection accuracy, the optimal clustering framework (OCF) method is utilized for band selection. Finally, extensive experiments on one simulated and three real HSI datasets confirm that our algorithm is superior than current HAD algorithms.

In the big data era, sequencing technology has produced a large number of biological sequencing data. Different views of the cancer genome data provide sufficient complementary information to explore genetic activity. The identification of differentially expressed genes from multiview cancer gene data is of great importance in cancer diagnosis and treatment. In this paper, we propose a novel method for identifying differentially expressed genes based on tensor robust principal component analysis (TRPCA), which extends the matrix method to the processing of multiway data. To identify differentially expressed genes, the plan is carried out as follows. First, multiview data containing cancer gene expression data from different sources are prepared. Second, the original tensor is decomposed into a sum of a low‐rank tensor and a sparse tensor using TRPCA. Third, the differentially expressed genes are considered to be sparse perturbed signals and then identified based on the sparse tensor. Fourth, the differentially expressed genes are evaluated using Gene Ontology and Gene Cards tools. The validity of the TRPCA method was tested using two sets of multiview data. The experimental results showed that our method is superior to the representative methods in efficiency and accuracy aspects.

Many kinds of real-world data, e.g., color images, videos, etc., are represented by tensors and may often be corrupted by outliers. Tensor robust principal component analysis (TRPCA) servers as a tensorial modification of the fundamental principal component analysis (PCA) which performs well in the presence of outliers. The recently proposed TRPCA model [12] based on tubal nuclear norm (TNN) has attracted much attention due to its superiority in many applications. However, TNN is computationally expensive, limiting the application of TRPCA for large tensors. To address this issue, we first propose a new TRPCA model by adopting a factorization strategy within the framework of tensor singular value decomposition (t-SVD). An algorithm based on the non-convex augmented Lagrangian method (ALM) is developed with convergence guarantee. Effectiveness and efficiency of the proposed algorithm is demonstrated through extensive experiments on both synthetic and real datasets.

The process monitoring and quality control of metal additive manufacturing has been a research hotspot in recent years. AI-driven laser powder bed fusion process monitoring is currently the most popular research idea. However, machine learning methods have high requirements for data sets and require a lot of cost. Therefore, this paper attempts to use a new technology, namely tensor robust principal component analysis, to directly analyze and process powder bed images, to realize the extraction of powder bed defects. The main steps include: 1) synthesizing a large number of powder bed images into a tensor; 2) separating the tensor into low-rank components and sparse components. By analyzing and processing the powder bed images collected during the two printing processes and the powder bed images with serious defects collected during a long period, the problems and suitable use scenarios of tensor robust principal component analysis in dealing with powder bed defects are discussed. It is found that it has the best effect in dealing with continuous powder bed images with variable defects.

In this paper, we propose a method for accurate 3D reconstruction based on Photometric Stereo. Instead of applying the global least square solution on the entire over-determined system, we randomly sample the images to form a set of overlapping groups and recover the surface normal for each group using the least square method. We then employ fourdimensional Tensor Robust Principal Component Analysis (TenRPCA) to obtain the accurate 3D reconstruction. Our method outperforms global least square in handling sparse noises such as shadows and specular highlights. Experiments demonstrate the reconstruction accuracy of our approach.

Reconstruction of highly accelerated dynamic magnetic resonance imaging (MRI) is of crucial importance for the medical diagnosis. The application of general robust principal component analysis (RPCA) to MRI can increase imaging speed and efficiency. However, conventional RPCA makes use of nuclear norm as convex surrogate of the rank function, whose drawbacks have been mentioned in plenty of literature. Recently, nonconvex surrogates of the rank function in RPCA have been widely investigated and proved to be tighter rank approximation than nuclear norm by the massive experimental results. Motivated by this, we propose a nonconvex alternating direction method based on nonconvex rank approximation to reconstruct dynamic MRI data from undersampled $k-t$ space data. We solve the associated nonconvex model by the alternating direction method and difference of convex programming. The convergence analysis provided guarantees the effectiveness of our algorithm. Experimental results on cardiac perfusion and cardiac cine MRI data demonstrate that our method outperforms the state-of-the-art MRI reconstruction methods in both image clarity and computation efficiency.

Tensor Robust Principal Component Analysis (TRPCA) aims at recovering the low-rank and sparse components from target tensor, which has extensive applications in multi-dimensional data recovery. However, most of the existing methods only exploit the global low-rank of image data, which result in missing local details in the recovered data. To restore the data more accurately, we propose a new TRPCA method which simultaneously combines the model-based method and data-driven approaches to preserve the global structure and fine local information. Specially, we pick the tensor nuclear norm to characterize the global correlation and a convolutional neural network(CNN) denoiser which reserves the local detail. Then, a flexible alternating direction method of multipliers (ADMM) is designed to deal with the proposed optimization model. Extensive experiments on various types of tensor data illustrate that our model enhances performance compared to state-of-the-art methods.