Robust Principal Component Analysis with Matrix Factorization

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.

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.

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.

In this paper, we propose an efficient singing voice extraction method from a music sound consisting of a singing voice and a drum sound. The proposed method is based on robust principal component analysis (RPCA) which is a technique to separate a given matrix into a sparse matrix and a low rank matrix. Dealing with a spectrogram of the music sound as the given matrix, RPCA gives an extracted singing voice spectrogram as a sparse spectrogram. We improve the capability of RPCA method by removing a residual drum sound from the sparse spectrogram. The residual drum sound is estimated as spectral components which repeatedly arise in the sparse spectrogram. Removing the estimated residual drum sound spectrogram from the sparse spectrogram gives an improved singing voice spectrogram. Simulation results show that the proposed method can improve 10dB of GNSDR as compared with the conventional RPCA singing voice extraction method.

Separating vocal and accompaniment signals from a monaural music signal is a challenging task. Recently, robust principal component analysis (RPCA) has been proposed for use in the magnitude spectrogram domain to separate the low-rank and sparse residual matrices, which are assumed to represent the accompaniment and vocal signals, respectively. In this paper, we propose two extended methods based on the RPCA algorithm for more effective vocal separation. First, we extend the conventional RPCA and propose to use in the spectrogram decomposition framework Schatten p- and lp-norms, which are generalized versions of the nuclear norm and l1-norm used in RPCA, respectively. Second, we apply proper scale compression to the magnitude spectrogram, making it a more appropriate representation for the decomposition. Experiments using the MIR-1K dataset show that the proposed methods yield significantly better separation performance than the conventional RPCA.

RPCA (Robust Principal Component Analysis) recovers sparse and low rank components from the original observation data, RPCA commonly uses ADM (Alternate Direction Method) for solving, the efficiency of algorithm depends on the nuclear norm optimization solution, that is SVD. And the application of RPCA in computer vision, image and video and so on, large amounts of data, such as images and video, make it difficult for large-scale data SVD. In this paper, we used the random matrix algorithm to improve the SVD, respectively the Count Sketch algorithm, the Prototype Randomized k-SVD algorithm and the Faster Randomized k-SVD algorithm, The main idea is to reduce the size of the original large-scale data matrix and sample randomly. Using the random projection algorithm to obtain an approximation of the original matrix, and operate QR decomposition of the approximate matrix, get the unitary matrix corresponding to the approximate matrix, and do the corresponding SVD, finally we can get the results which is similar to the original matrix calculation. Obtaining approximation of the original data matrix. But the cost time and space have been greatly optimized. Simulation experiments based on single image and video foreground detection show that the proposed methods can greatly improve the efficiency of RPCA iterative optimization.

Recent advances in optical coherence tomography (OCT) lead to the development of OCT angiography to provide additional helpful information for diagnosis of diseases like basal cell carcinoma. In this paper, we investigate how to extract blood vessels of human skin from full-field OCT (FF-OCT) data using the robust principal component analysis (RPCA) technique. Specifically, we propose a short-time RPCA method that divides the FF-OCT data into segments and decomposes each segment into a low-rank structure representing the relatively static tissues of human skin and a sparse matrix representing the blood vessels. The method mitigates the problem associated with the slow-varying background and is free of the detection error that RPCA may have when dealing with FF-OCT data. Both short-time RPCA and RPCA methods can extract blood vessels from FF-OCT data with heavy speckle noise, but the former takes only half the computation time of the latter. We evaluate the performance of the proposed method by comparing the extracted blood vessels with the ground truth vessels labeled by a dermatologist and show that the proposed method works equally well for FF-OCT volumes of different quality. The average F-measure improvements over the correlation-mapping OCT method, the modified amplitude-decorrelation OCT angiography method, and the RPCA method, respectively, are 0.1835, 0.1032, and 0.0458.

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

Practical brain-computer interfaces need to overcome several challenges, including tolerance to signal variability within- and across sessions. We introduce Robust Principal Component Analysis (RPCA) as a potential approach to tackle intra-trial variability. Assuming that subjects undergo the same cognitive process or perform the same task in a short period (e.g., a few seconds), as a result, the signal of interest should be represented by only a few components. We verified this approach on a workload detection task, where subjects needed to pilot a simulated drone. We used RPCA as a processing step to decrease trial variability and assessed its impact on classification accuracy. Our results showed that RPCA significantly increased performance in both at group and subject level analysis. On average, class-balanced accuracy when simulating RPCA online increased from 63.9% up to 70.6% $(p~ 0 . 001)$.

Due to the spread of the Internet and improvement of the audio compression technology, we need technologies to handle a large number of audio files. Separating singing voice from music is one of them and it is used in many applications, such as lyric retrieval and singer recognition. Recently, robust principal component analysis (RPCA) has been proposed, which makes use of repetition of a phrase in an accompaniment. We assume that there are some points to be improved in RPCA. In this paper, we propose a developed method based on the RPCA algorithm. First, we propose the extended RPCA algorithm that makes use of the two features of the spectrogram. Second, we apply simple post-processing to remove noise effectively. Lastly, we experiment our algorithm using the MIR-1K dataset and confirm that the proposed algorithm shows better separation performance than the conventional RPCA method.

Road traffic sensors provide us with rich multi-variable datastreams about the current traffic conditions. Occasionally, there are unusual traffic events (such as accidents, jams, severe weather, etc) that disrupt the expected road traffic conditions. Detecting the occurrence of such events in an online and real-time manner is useful to drivers in planning their routes and in the management of the transportation infrastructure. We propose a new method for detecting traffic events that impact road traffic conditions by extending the Bayesian Robust Principal Component Analysis (RPCA) approach. Our method couples multiple traffic datastreams so that they share a certain sparse structure. This sparse structure is used to localize traffic events in space and time. The traffic datastreams are measurements of different physical quantities (e.g. traffic flow, road occupancy) by different nearby sensors. Our proposed method process datastreams in an incremental way with little computational cost, and hence it is suitable to detect events in an online and real-time manner. We experimentally analyze the detection performance of the proposed coupled Bayesian RPCA using real data from loop detectors on the Minnesota I-494. We find that our method significantly improves the detection accuracy when compared with the traditional PCA and non-coupled Bayesian RPCA.

Blind monaural singing voice separation using rank-1 constraint robust principal component analysis and vocal activity detection

Due to the severe interference of background noise, the signal-to-noise ratio of desert seismic data is extremely low. In addition, due to low-frequency characteristics of sand in the Tarim desert region, the background noise in desert seismic data is mainly distributed in low-frequency band, so that the frequency spectrum aliasing of effective signals and background noise is more serious than the general land seismic data. Thus, conventional filtering methods cannot effectively suppress background noise in desert seismic data and recover effective signals. In order to overcome the problem that low-frequency background noise in desert seismic data is hard to suppress, a new method called R-TNRD based on robust principal component analysis (RPCA) algorithm and trainable nonlinear reaction diffusion (TNRD) network is proposed in this paper. By using the good sparsity of RPCA, the input noisy desert seismic data are decomposed into a low-rank matrix and a sparse matrix, and these two matrices contain background noise and effective signals. Due to the serious spectrum aliasing of desert seismic data, conventional thresholds have been unable to extract effective signals from the two matrices obtained by RPCA effectively. Therefore, we introduce TNRD network into desert seismic data denoising. By network training with a low-frequency noise set, the optimisation of TNRD network can be achieved, so as to accurately extract the effective signals from the low-rank matrix and the sparse matrix. In the experimental part, we test the performance of R-TNRD on both synthetic and real seismic data. The results demonstrate that the proposed method can suppress background noise more effectively than conventional methods.

In this letter, we first present a rank-revealing matrix factorization algorithm by using randomization called randomized truncated pivoted QLP (RTp-QLP) to approximate an input matrix. For a dense and large $n_1\times n_2$ matrix with numerical rank $k$ , RTp-QLP needs only a few passes over the matrix (regardless of $k$ ) and $\mathcal {O}(n_1n_2d)$ floating-point operations, where $d$ is much smaller than both $n_1$ and $n_2$ . Next, we develop a robust principal component analysis (RPCA) method by utilizing RTp-QLP. In addition, we propose a rank estimation technique that efficiently solves the RPCA task. RTp-QLP is highly accurate and numerically stable. Our proposed RTp-QLP-based RPCA method yields the optimal solution, and it is faster than existing methods. Our simulation results support our claims.

Biochemically recurrent prostate cancer (BCR) is the main indication to perform prostate-specific membrane antigen PET/CT. However, localizing BCR with prostate-specific membrane antigen PET/CT remains challenging in patients with low prostate-specific antigen (PSA) values. Here, we studied the impact of advanced PET image reconstruction methods on BCR localization and interobserver agreement with 18F-DCFPyL PET/CT scans in patients with BCR and low PSA values. Methods: Twenty-four patients with BCR and a PSA level of less than 2.0 ng/mL were included. PET images were reconstructed with 4-mm voxels and 2-mm voxels, both with and without point-spread function. All scans were interpreted by 4 nuclear medicine physicians. Additionally, PET examinations of 5 patients with primary prostate cancer and confirmed absence of lymph node metastases (after lymph node dissection) were included, to assess the risk of introducing false-positive findings when using advanced reconstruction. Calculation of BCR localization rates (scan positivity) was based on consensus among our readers (≥3 readers regarding a scan positive for BCR), as well as the individual scan interpretations of the readers. Results: In the consensus analysis, BCR localization rates were not higher using advanced reconstruction (62.5%-66.7%) than using 4-mm reconstruction (62.5%). On the basis of individual readings, however, more scans were positive using 2-mm reconstruction (74.0%; 95% confidence interval [CI], 65.0%-82.9%) (P = 0.027) and 2-mm reconstruction with point-spread function (75.0%; 95% CI, 66.2%-83.8%) (P = 0.014) than 4-mm reconstruction (65.6%; 95% CI, 56.0%-75.3%). A higher number of lesions was detected on the 2-mm scans (median, 2 lesions; interquartile range, 1-3) than the 4-mm scans (median, 1; interquartile range, 0-3; P = 0.008). The advanced reconstruction methods did not increase interobserver agreement (80.6%-84.7%), compared with the 4-mm scans (75.7%, P = 0.08-0.25). In the patients with primary prostate cancer, an equal number of false-positive lesions was observed among the different reconstruction methods (overall, n = 13). Conclusion: Applying advanced image reconstruction for 18F-DCFPyL PET/CT scans did not increase BCR localization in patients with BCR and low PSA values (reader consensus). Yet, the increased number of positive individual readings may imply that further development of image reconstruction methods holds potential to improve BCR localization. No improved interobserver agreement was observed with advanced reconstruction compared with standard 4-mm reconstruction.