Schatten P-norm Minimization Research Articles

Quaternion weighted Schatten [formula omitted]-norm minimization for color image restoration with convergence guarantee

Over the past decades, the rank approximation issue have been extensively investigated, among which weighted nuclear norm minimization (WNNM) and weighted Schatten p-norm minimization (WSNM) are two prevailing methods and have shown great superiority in various image restoration (IR) problems. However, for the complicated color image restoration (CIR) problems, traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain. Despite QWNNM’s empirical success, its convergence behavior has not been rigorously studied. The main contributions of this paper are twofold. Firstly, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. Secondly, we provide a preliminary theoretical convergence analysis. By modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove the fixed-point convergence property of the iterative sequences generated by QWNNM and QWSNM. The source code of our algorithm can be available at the website: https://github.com/qiuxuanzhizi/QWSNM.

Multi-Dimensional Low-Rank with Weighted Schatten p-Norm Minimization for Hyperspectral Anomaly Detection

Hyperspectral anomaly detection is an important unsupervised binary classification problem that aims to effectively distinguish between background and anomalies in hyperspectral images (HSIs). In recent years, methods based on low-rank tensor representations have been proposed to decompose HSIs into low-rank background and sparse anomaly tensors. However, current methods neglect the low-rank information in the spatial dimension and rely heavily on the background information contained in the dictionary. Furthermore, these algorithms show limited robustness when the dictionary information is missing or corrupted by high level noise. To address these problems, we propose a novel method called multi-dimensional low-rank (MDLR) for HSI anomaly detection. It first reconstructs three background tensors separately from three directional slices of the background tensor. Then, weighted schatten p-norm minimization is employed to enforce the low-rank constraint on the background tensor, and LF,1-norm regularization is used to describe the sparsity in the anomaly tensor. Finally, a well-designed alternating direction method of multipliers (ADMM) is employed to effectively solve the optimization problem. Extensive experiments on four real-world datasets show that our approach outperforms existing anomaly detection methods in terms of accuracy.

Total variation regularized multi-matrices weighted Schatten p-norm minimization for image denoising

Motivated by the superior performance of nonconvex nonsmooth Lp (0<p<1) norm, this paper introduces a novel method that combines the weighted Schatten p-norm, Lp-norm, and total variation regularization based on the multiple matrices denoising framework. The weighted Schatten p-norm encodes the global low-rank to multiple matrices data, while the Lp-norm provides noise robustness. Total variation regularization is incorporated to promote structural smoothness and edge preservation in the image. To solve the nonconvex and nonsmooth model, an efficient alternating direction method of multipliers (ADMM) is designed. In addition, we discuss how the value of p in the three items affects the denoising performance in simulated images. Extensive experiments on face datasets, videos, and real-world noisy images demonstrate that the proposed method significantly improves denoising performance, particularly for removing large sparse noise.

Hyperspectral fruit image restoration using non-convex optimization

Hyperspectral (HS) imaging captured at low proximity ranges has become an effective method for assessing the quality and safety of food products in recent years. The majority of Hyperspectral cameras are push-broom type, which often contributes random noise and stripes to the spectral images. As a result, the quality of the collected HS images and the precision of the subsequent processing will both suffer. Despite the fact that there are several denoising techniques that have been studied and recorded in the literature, they frequently fail to fully restore the original HS images when there are strong stripe noises and other intricate mixed noises. In this study, we introduce a new method based on tensor logarithmic Schatten-p norm minimization with anisotropic spatial spectral total variation (ASSTV) regularization for simultaneous denoising and destriping of line scan-based hyperspectral images. Tensor logarithmic Schatten-p norm improves denoising performance by deleting the smaller singular values while protecting the larger singular values. Through the application of ASSTV, stripes and Gaussian noise are eliminated while maintaining the spatial–spectral smoothness of HS images and the directional characteristics of the stripes. The augmented Lagrange multiplier (ALM) approach is used to address the ensuing optimization problem. The quantitative and visual assessments reveal that the suggested technique has improved performance than its counterparts.

A Novel Reconstruction of the Sparse-View CBCT Algorithm for Correcting Artifacts and Reducing Noise

X-ray tomography is often affected by noise and artifacts during the reconstruction process, such as detector offset, calibration errors, metal artifacts, etc. Conventional algorithms, including FDK and SART, are unable to satisfy the sampling theorem requirements for 3D reconstruction under sparse-view constraints, exacerbating the impact of noise and artifacts. This paper proposes a novel 3D reconstruction algorithm tailored to sparse-view cone-beam computed tomography (CBCT). Drawing upon compressed sensing theory, we incorporate the weighted Schatten p-norm minimization (WSNM) algorithm for 2D image denoising and the adaptive steepest descent projection onto convex sets (ASD-POCS) algorithm, which employs a total variation (TV) regularization term. These inclusions serve to reduce noise and ameliorate artifacts. Our proposed algorithm extends the WSNM approach into three-dimensional space and integrates the ASD-POCS algorithm, enabling 3D reconstruction with digital brain phantoms, clinical medical data, and real projections from our portable CBCT system. The performance of our algorithm surpasses traditional methods when evaluated using root mean square error (RMSE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM) metrics. Furthermore, our approach demonstrates marked enhancements in artifact reduction and noise suppression.

The Rank Residual Constraint Model with Weighted Schatten p-Norm Minimization for Image Denoising

The Rank Residual Constraint Model with Weighted Schatten p-Norm Minimization for Image Denoising

Joint angular-frequency distribution estimation via spatial-temporal sparse sampling and low-rank matrix recovery

This paper is committed to the joint angular-frequency distribution (JAFD) estimation for incoherently distributed wideband (IDW) sources by using spatial-temporal sparse sampling (STSS). The STSS increases dramatically the degrees of freedom (DOFs) of the measure system, and can solve the ambiguity problem between angular and frequency parameters. Then, it is observed that the discrete representation of the JAFD gives rise to a helpful low-rank matrix. To estimate the JAFD for IDW sources, a rank minimization problem is further formulated as the modified Schatten-p norm minimization (MSpNM) problem, which can be resolved efficiently by the accelerated iterative singular value thresholding algorithm (AISVTA). Compared with traditional methods based on spatial and temporal Nyquist sampling (STNS), the proposed method based on STSS reduces the sampling rate requirement on hardware and provides better JAFD estimation precision. Experiment simulations indicate its effectiveness and confirm the superior performance of the proposed method based on STSS.

Bi-Nuclear Tensor Schatten-p Norm Minimization for Multi-View Subspace Clustering.

Multi-view subspace clustering aims to integrate the complementary information contained in different views to facilitate data representation. Currently, low-rank representation (LRR) serves as a benchmark method. However, we observe that these LRR-based methods would suffer from two issues: limited clustering performance and high computational cost since (1) they usually adopt the nuclear norm with biased estimation to explore the low-rank structures; (2) the singular value decomposition of large-scale matrices is inevitably involved. Moreover, LRR may not achieve low-rank properties in both intra-views and inter-views simultaneously. To address the above issues, this paper proposes the Bi-nuclear tensor Schatten- p norm minimization for multi-view subspace clustering (BTMSC). Specifically, BTMSC constructs a third-order tensor from the view dimension to explore the high-order correlation and the subspace structures of multi-view features. The Bi-Nuclear Quasi-Norm (BiN) factorization form of the Schatten- p norm is utilized to factorize the third-order tensor as the product of two small-scale third-order tensors, which not only captures the low-rank property of the third-order tensor but also improves the computational efficiency. Finally, an efficient alternating optimization algorithm is designed to solve the BTMSC model. Extensive experiments with ten datasets of texts and images illustrate the performance superiority of the proposed BTMSC method over state-of-the-art methods.

A WTC suppression algorithm for weather radar based on weighted schatten p-norm minimization

ABSTRACT The Matrix Completion (MC) method can effectively suppress the wind turbine clutter (WTC) of the weather radar, however its recovery accuracy suffers from the low signal-to-noise ratio (SNR) of the real measured weather signal. Moreover, the singular value distribution of the observation matrix constructed by the measured radar data has no sharp cut-off characteristic, which results the rank can not be well approximated and also reduces the recovery accuracy. In this paper, a non-convex MC algorithm based on weighted Schatten p-norm is proposed for the WTC suppression in Doppler frequency domain. Experiments are carried out on the measured signal data, and the results demonstrate that compared with the commonly used nuclear norm minimization (NNM) model, the weighted Schatten p-norm minimization (WSNM) model can further improve the recovery accuracy of weather signal and realizes the superior estimation performance of spectral moment.

Iteratively Reweighted Minimax-Concave Penalty Minimization for Accurate Low-rank Plus Sparse Matrix Decomposition.

Low-rank plus sparse matrix decomposition (LSD) is an important problem in computer vision and machine learning. It has been solved using convex relaxations of the matrix rank and l0-pseudo-norm, which are the nuclear norm and l1-norm, respectively. Convex approximations are known to result in biased estimates, to overcome which, nonconvex regularizers such as weighted nuclear-norm minimization and weighted Schatten p-norm minimization have been proposed. However, works employing these regularizers have used heuristic weight-selection strategies. We propose weighted minimax-concave penalty (WMCP) as the nonconvex regularizer and show that it admits an equivalent representation that enables weight adaptation. Similarly, an equivalent representation to the weighted matrix gamma norm (WMGN) enables weight adaptation for the low-rank part. The optimization algorithms are based on the alternating direction method of multipliers technique. We show that the optimization frameworks relying on the two penalties, WMCP and WMGN, coupled with a novel iterative weight update strategy, result in accurate low-rank plus sparse matrix decomposition. The algorithms are also shown to satisfy descent properties and convergence guarantees. On the applications front, we consider the problem of foreground-background separation in video sequences. Simulation experiments and validations on standard datasets, namely, I2R, CDnet 2012, and BMC 2012 show that the proposed techniques outperform the benchmark techniques.

Fast multiple graphs learning for multi-view clustering

Graph-based multi-view clustering has become an active topic due to the efficiency in characterizing both the complex structure and relationship between multimedia data. However, existing methods have the following shortcomings: (1) They are inefficient or even fail for graph learning in large scale due to the graph construction and eigen-decomposition. (2) They cannot well exploit both the complementary information and spatial structure embedded in graphs of different views. To well exploit complementary information and tackle the scalability issue plaguing graph-based multi-view clustering, we propose an efficient multiple graph learning model via a small number of anchor points and tensor Schatten p-norm minimization. Specifically, we construct a hidden and tractable large graph by anchor graph for each view and well exploit complementary information embedded in anchor graphs of different views by tensor Schatten p-norm regularizer. Finally, we develop an efficient algorithm, which scales linearly with the data size, to solve our proposed model. Extensive experimental results on several datasets indicate that our proposed method outperforms some state-of-the-art multi-view clustering algorithms.

Enhance tensor RPCA-LRX anomaly detection algorithm for hyperspectral image

In this paper, a spectral–spatial anomaly detection method based on tensor de-composition is proposed. Firstly, tensor data is used to represent hyperspectral data to retain its original spectral and spatial information. Second, this method reconstructs the hyperspectral data into low-rank tensors and sparse tensors. This method uses the weighted tensor Schatten-p norm minimization (WTSNM) to stand for rank minimization. WTSNM treats different singular values differently. Finally, the reconstructed sparse tensor is used as input data and the LRX method is used to detect abnormal targets. As this approach can effectively utilize the spectral information and spatial information of hyperspectral images, it greatly improves detection accuracy, experimental results on five real data sets demonstrate that the proposed method outperforms several state-of-the-art algorithms.

Spatiotemporal traffic data imputation via tensorial weighted Schatten‐ p norm minimization

Spatiotemporal traffic data exhibit multi-granular low-rank structure due to their periodicity among different timelines. Traditional low rank data completion methods fail to characterize such properties and produce unsatisfactory results for data imputation. In this paper, a tensorial weighted Schatten-p norm minimization (TWSN) is proposed for spatiotemporal traffic data imputation. TWSN consists of an approximation term and a low-rank regularization term over the recovered tensor data, where the latter is a combination of the weighted Schatten-p norm in the matrix form of each mode of the tensor. For each mode, TWSN utilizes a selection scheme of the mode-wise weights to capture different properties of singular values of each mode of the tensor. Overall, TWSN not only plays a balancing role between the rank function and the nuclear norm, but also captures the anisotropic correlation of singular values of each mode of the tensor. TWSN is evaluated on four real-world datasets with different ping frequencies (2, 5, 10 min) and its performance is compared with several state-of-the-art methods. The experimental results show that TWSN outperforms other methods under various data missing scenarios.

Weighted Schatten [formula omitted]-norm minimization with logarithmic constraint for subspace clustering

Rank minimization-based subspace clustering methods have been widely developed in the past decades. Although some smooth surrogates, such as the nuclear norm and Schatten-p norm mitigate the NP-hard issue to some extend, these existing methods may yield unsatisfactory results, due to the singular values of the coefficient matrix not being further suppressed. To tackle this, in this paper, we propose a novel non-convex low-rank approximation based on weighted Schatten-p norm jointed logarithmic constraint, which can suppress the small and large singular values flexibly with a tighter relaxation. Specifically, we firstly proposed a low-rank approximation termed SLog by utilizing the logarithmic to tighten the Schatten-p norm, which can shrink the large singular values in a similar trend of the real rank minimization. Furthermore, to suppress the small singular values simultaneously, usually considered noise, we propose a weighted Schatten p-norm minimization named WSLog based on SLog by introducing the weight w, which behaves more robust to the sparse noise, especially w<1. Compared with recent-proposed methods, extensive experiments in subspace clustering on real datasets demonstrate the effective performance of our methods.

Image restoration algorithm based on multiscale weighted Schatten p-norm minimization

Image restoration is the process of restoring an image from a degraded version, which is usually blurred and noisy. We are motivated by the problem of restoring blurred and noisy images using multiscale weighted Schatten p-norm minimization, which not only gives a better approximation to the original low-rank hypothesis but also considers the importance of different rank components. Similar patches are vectorized and grouped to construct a noisy low-rank matrix. Weighted Schatten p-norm minimization values of all image patch groups are simultaneously penalized by a new regularization term, which can represent both the sparsity and self-similarity of the image structure accurately. In addition, by calculating the similarity of patches on different scales of the image, the restoration effect of the image is further improved. The experimental results show that our method is superior to some existing excellent algorithms in both numerical and visual effects.

Compressed Sensing MRI by Integrating Deep Denoiser and Weighted Schatten P-Norm Minimization

To efficiently reconstruct magnetic resonance images (MRI) from highly undersampled measurements by using compressed sensing (CS), in this letter, we propose a hybrid regularization model from deep prior and low-rank prior. The local deep prior is explored by a fast flexible denoising convolutional neural network (FFDNet). To compensate for 1) the generalization capability of FFDNet on artifact noise caused by undersampling K-space and 2) the inaccurate noise estimation for various undersampling ratios, we model the low-rank prior as a weighted Schatten p-norm to obtain the global information of MRIs. The final model, combined by the local deep and low-rank priors, is solved by the alternating directional method of multipliers under the plug-and-play framework. Compared with the popular CS-MRI approaches, the experimental results demonstrate that the proposed method can achieve better reconstruction performance in terms of quality index and visual effects.

Dynamic and reliable subtask tracker with general schatten [formula omitted]-norm regularization

Some multi-task trackers adopt an inaccurate shrink strategy to treat different rank components equally. Thus, their flexibility is vulnerable to some tracking challenges. To resolve this problem, we propose a spatial-aware reliable multi-subtask tracker via weighted Schatten p-norm regularization (SLRT-W), which dynamically chooses the suitable and reliable subset of the whole subtasks for tracking. Its major merits not only assign the flexible weights to different subtask rank components depending on their tracking contribution, but also preserve consistent spatial layout structure and correspondence of layered multi-subtask. Specifically, multiple layered subtasks correspond to different target subregions, they are cooperative and complement. A weighted Schatten p-norm is introduced to adaptively shrink different multi-subtask rank components, and emphasize important components as reliable ones. Then, a structured hyper-graph regularized term simultaneously exploits the intrinsic geometry correspondence among multiple layers of subtasks, and spatial layout structure inside each layer. We devise an alternatively generalized iterated shrinkage method to optimize the multi-subtask Schatten p-norm minimization. Finally, a robust decision-evaluation strategy is developed to choose the reliable multi-subtask tracking combination. Encouraging results on some challenging benchmarks demonstrate the proposed tracker performs favorably in robustness and accuracy, against some state-of-the-art trackers.

Enhanced Tensor RPCA and its Application

Despite the promising results, tensor robust principal component analysis (TRPCA), which aims to recover underlying low-rank structure of clean tensor data corrupted with noise/outliers by shrinking all singular values equally, cannot well preserve the salient content of image. The major reason is that, in real applications, there is a salient difference information between all singular values of a tensor image, and the larger singular values are generally associated with some salient parts in the image. Thus, the singular values should be treated differently. Inspired by this observation, we investigate whether there is a better alternative solution when using tensor rank minimization. In this paper, we develop an enhanced TRPCA (ETRPCA) which explicitly considers the salient difference information between singular values of tensor data by the weighted tensor Schatten p-norm minimization, and then propose an efficient algorithm, which has a good convergence, to solve ETRPCA. Extensive experimental results reveal that the proposed method ETRPCA is superior to several state-of-the-art variant RPCA methods in terms of performance.

Weighted Schatten p-Norm Low Rank Error Constraint for Image Denoising.

Traditional image denoising algorithms obtain prior information from noisy images that are directly based on low rank matrix restoration, which pays little attention to the nonlocal self-similarity errors between clear images and noisy images. This paper proposes a new image denoising algorithm based on low rank matrix restoration in order to solve this problem. The proposed algorithm introduces the non-local self-similarity error between the clear image and noisy image into the weighted Schatten p-norm minimization model using the non-local self-similarity of the image. In addition, the low rank error is constrained by using Schatten p-norm to obtain a better low rank matrix in order to improve the performance of the image denoising algorithm. The results demonstrate that, on the classic data set, when comparing with block matching 3D filtering (BM3D), weighted nuclear norm minimization (WNNM), weighted Schatten p-norm minimization (WSNM), and FFDNet, the proposed algorithm achieves a higher peak signal-to-noise ratio, better denoising effect, and visual effects with improved robustness and generalization.

Weighted Schatten p-norm minimization for impulse noise removal with TV regularization and its application to medical images

Noise of impulse type was common in medical images. In this paper, we modeled the denoising problem for impulse noise by Weighted Schatten p-norm minimization (WSNM) with Robust Principal Component Analysis (RPCA). The anisotropic Total Variation (TV) regularization was incorporated to preserve edge information which was important for clinic detection and diagnosis. The alternating direction method of multipliers (ADMM) algorithm was adopted for solving the formulated nonconvex optimization problem. We tested the performance on both standard natural images and medical images with additive impulse noise in different levels. Experiment results implied its competitiveness compared to traditional denoising algorithms that validated to be state-of-the-art. The propose algorithm restored images with better structure information preservation outperformed the conventional techniques in terms of visual appearances. Quantitative metrics (PSNR, SSIM and FSIM) further objectively demonstrated the effectiveness of the proposed algorithm for impulse noise removal superior to the existing ones.