Tensor Robust Principal Component Analysis With Side Information: Models and Applications

Dynamic magnetic resonance imaging (dMRI) strikes a balance between reconstruction speed and image accuracy in medical imaging field. In this paper, an improved robust tensor principal component analysis (RTPCA) method is proposed to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data. The MR reconstruction problem is formulated as a high-order low-rank tenor plus sparse tensor recovery problem, which is solved by robust tensor principal component analysis (RTPCA) with a new tensor nuclear norm (TNN). To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into TNN for enforcing the low-rank structure in MRI datasets. The experimental results show that the proposed method outperforms state-of-the-art methods in terms of both MR image reconstruction accuracy and computational efficiency on 3D and 4D experiment datasets, especially for 4D MR image reconstruction. Graphical abstract The flowchart of the proposed method to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data in the kth iteration. To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into tensor nuclear norm (TNN) for enforcing the low-rank structure in MRI datasets. In each iteration, the first step is to get low-rank tensor ℓk - 1 by using soft thresholding on the singular values of ℓk - 1 = χk - 1 - ξk - 1, and an improved tensor nuclear norm method is proposed to process the low-rank tensor ℓk - 1 firstly. Then, the shrinkage operator is applied to ξk - 1 = χk - 1 - ℓk - 1 for sparse part ξk - 1. The final reconstructed d-MRI χk is obtained by enforcing data consistency that the residual in K-space is subtracted by the sum of the reconstructed low-rank tensor and sparse tensor.

Chapter 6 - Tensor principal component analysis

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.

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 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.

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.

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.

Low-rank prior-based Fast-RPCA for clutter filtering and noise suppression in non-contrast ultrasound microvascular imaging

Many kinds of real-world multi-way signal, like color images, videos, etc., are represented in tensor form and may often be corrupted by outliers. To recover an unknown signal tensor corrupted by outliers, tensor robust principal component analysis (TRPCA) serves as a robust tensorial modification of the fundamental PCA. Recently, a successful TRPCA model based on the tubal nuclear norm (TNN) (Lu et al. in IEEE Trans Pattern Anal Mach Intell 42:925–938, 2019) has attracted much attention thanks to its superiority in many applications. However, TNN is computationally expensive due to the requirement of full singular value decompositions, seriously limiting its scalability to large tensors. To address this issue, we propose a new TRPCA model which adopts a factorization strategy. Algorithmically, an algorithm based on the non-convex augmented Lagrangian method is developed with convergence guarantee. Theoretically, we rigorously establish the sub-optimality of the proposed algorithm. We also extend the proposed model to the robust tensor completion problem. Both the effectiveness and efficiency of the proposed algorithm is demonstrated through extensive experiments on both synthetic and real data sets.

Principal component analysis (PCA) is widely used to extract features and reduce dimensionality in various computer vision and image/video processing tasks. Conventional approaches either lack robustness to outliers and corrupted data or are designed for one-dimensional signals. To address this problem, we propose a robust PCA model for two-dimensional images incorporating structured sparse priors, referred to as structured sparse 2D-PCA. This robust model considers the prior of structured and grouped pixel values in two dimensions. As the proposed formulation is jointly nonconvex and nonsmooth, which is difficult to tackle by joint optimization, we develop a two-stage alternating minimization approach to solve the problem. This approach iteratively learns the projection matrices by bidirectional decomposition and utilizes the proximal method to obtain the structured sparse outliers. By considering the structured sparsity prior, the proposed model becomes less sensitive to noisy data and outliers in two dimensions. Moreover, the computational cost indicates that the robust two-dimensional model is capable of processing quarter common intermediate format video in real time, as well as handling large-size images and videos, which is often intractable with other robust PCA approaches that involve image-to-vector conversion. Experimental results on robust face reconstruction, video background subtraction data set, and real-world videos show the effectiveness of the proposed model compared with conventional 2D-PCA and other robust PCA algorithms.

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.

Video background recovery is a very important task in computer vision applications. Recent research offers robust principal component analysis (RPCA) as a promising approach for solving video background recovery. RPCA works by decomposing a data matrix into a low-rank matrix and a sparse matrix. Our previous work shows that when the desired rank of the low-rank matrix is known, fixing the rank in the algorithm called FrALM (fixed-rank ALM) yields more robust and accurate results than existing RPCA algorithms. However, application of RPCA to video background recovery requires that each frame in the video is encoded as a column in the data matrix. This is impractical in real applications because the videos can be easily larger than the amount of memory in a computer. This paper presents an algorithm called iFrALM (incremental fixed-rank ALM) that computes fixed-rank RPCA incrementally by splitting the video frames into an initial batch and an incremental batch. Comprehensive tests show that iFrALM uses less memory and time compared to FrALM. Moreover, the initial batch size and batch quality can be carefully selected to ensure that iFrALM reduces memory and time complexity without sacrificing accuracy.KeywordsBackground recoveryIncremental SVDFixed-rank robust PCA

Chemical exchange saturation transfer (CEST) MRI requires the acquisition of multiple saturation-weighted images and can last several minutes. Misalignments among these images, which are often due to the inevitable motion of the subject, will corrupt CEST contrast maps and result in large quantification errors. Therefore, the registration of the CEST series is critical. However, registration is challenging since common intensity-based registration algorithms may fail to differentiate CEST signals from motion artifacts. Herein, we studied how different patterns of motion affect CEST quantification and proposed a cascaded two-step registration scheme by utilizing features extracted from the entire Z-spectral image series instead of direct registration to a single image. The proposed approach is conducted in two stages: during the first coarse registration, the Z-spectral image series is decomposed by robust principal component analysis (RPCA) to separate CEST contrast from motion. The recomposed image series using only the low-rank component, which contains minimized motion, are averaged to generate a reference for the alignment of the image series. To further remove residual misalignments, the coarse registration is followed by a refinement stage, which uses PCA iteratively to generate motionless synthetic reference series with the first few principal components (PCs) that correspond to CEST contrast. In the end, the quality check is performed to exclude the images with unsuccessful registration. The proposed registration scheme (RPCA + PCA_R) was assessed by both phantom experiments and in vivo data of tumor-bearing mouse brain, with simulated random rigid motion in different patterns applied to the acquired static Z-spectral image series. For comparison, previous correction schemes using an explicit image [either S0 or Ssat(∆ω)] as registration reference were also performed, named as S0_R and Ssat_R respectively. To illustrate the advantage of combination of RPCA and PCA, registration was also exploited using either only the RPCA-based method (RPCA_R) or only the PCA-based one (PCA_R). Compared with the above four methods, RPCA + PCA_R allowed for more accurate correction of the corrupted Z-spectral images, exhibiting smaller MTRasym(∆ω) error maps and lower residual Z-spectra referring to the static data. Among all the five correction methods, the corrected Z-spectral image series by RPCA + PCA_R and the resulting MTRasym(∆ω) maps achieved the highest correlation coefficients (CC) with respect to the static ones. The registration scheme of RPCA + PCA_R provides robust motion correction between two specific Z-spectral images and among an entire image series, through extraction of the static component from the entire Z-spectra set and inclusion of a PCA-based refinement step. Therefore, this method can help improve CEST acquisition and quantification.

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.

This paper presents a novel algorithm - robust kernel principal component analysis (robust KPCA), on the basis of the research of kernel principal component analysis (KPCA) and robust principal component analysis (RPCA). First, this algorithm sets the radius of the images of the training samples in the feature space using kernel tricks, then determines whether the samples are outliers or not, and finally analyzes the training samples which have eliminated the outliers using KPCA algorithm. The improved KPCA algorithm not only retains the non-linearity property of KPCA algorithm but also gets better robustness. Because the effects of outliers are eliminated, robust KPCA algorithm gets higher accuracy than KPCA algorithm for data analysis. The simulation experiments show that the robust KPCA algorithm developed is better than the KPCA algorithm.