The tensor train (TT) rank has received increasing attention in tensor completion due to its ability to capture the global correlation of high-order tensors ($\textrm{order} >3$). For third order visual data, direct TT rank minimization has not exploited the potential of TT rank for high-order tensors. The TT rank minimization accompany with \emph{ket augmentation}, which transforms a lower-order tensor (e.g., visual data) into a higher-order tensor, suffers from serious block-artifacts. To tackle this issue, we suggest the TT rank minimization with nonlocal self-similarity for tensor completion by simultaneously exploring the spatial, temporal/spectral, and nonlocal redundancy in visual data. More precisely, the TT rank minimization is performed on a formed higher-order tensor called group by stacking similar cubes, which naturally and fully takes advantage of the ability of TT rank for high-order tensors. Moreover, the perturbation analysis for the TT low-rankness of each group is established. We develop the alternating direction method of multipliers tailored for the specific structure to solve the proposed model. Extensive experiments demonstrate that the proposed method is superior to several existing state-of-the-art methods in terms of both qualitative and quantitative measures.
Tensor ring (TR) decomposition has been successfully used to obtain the state-of-the-art performance in the visual data completion problem. However, the existing TR-based completion methods are severely non-convex and computationally demanding. In addition, the determination of the optimal TR rank is a tough work in practice. To overcome these drawbacks, we first introduce a class of new tensor nuclear norms by using tensor circular unfolding. Then we theoretically establish connection between the rank of the circularly-unfolded matrices and the TR ranks. We also develop an efficient tensor co…
This paper presents a patch-wise low-rank based image denoising method with constrained variational model involving local and nonlocal regularization. On one hand, recent patch-wise methods can be represented as a low-rank matrix approximation problem whose convex relaxation usually depends on nuclear norm minimization (NNM). Here, we extend the NNM to the nonconvex schatten p-norm minimization with additional weights assigned to different singular values, which is referred to as the Weighted Schatten p-Norm Minimization (WSNM). An efficient algorithm is also proposed to solve the WSNM problem…
University of Electronic Science and Technology of China（电子科技大学）；Xi’an Jiaotong University（西安交通大学）
In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on the convex relaxation by penalizing the weighted sum of nuclear norm of TR unfolding matrices. However, this method treats each singular value equally and neglects their physical meanings, which usually leads to suboptimal solutions in practice. In this paper, we propose to use the logdet-based function as a nonconvex smooth relaxation of the TR rank for tenso…
University of Electronic Science and Technology of China（电子科技大学）
The wavelet frame systems have been widely investigated and applied for image restoration and many other image processing problems over the past decades, attributing to their good capability of sparsely approximating piece-wise smooth functions such as images. Most wavelet frame based models exploit the $l_1$ norm of frame coefficients for a sparsity constraint in the past. The authors in \cite{ZhangY2013, Dong2013} proposed an $l_0$ minimization model, where the $l_0$ norm of wavelet frame coefficients is penalized instead, and have demonstrated that significant improvements can be achieved c…
Low-rank matrix completion is a problem of immense practical importance. Recent works on the subject often use nuclear norm as a convex surrogate of the rank function. Despite its solid theoretical foundation, the convex version of the problem often fails to work satisfactorily in real-life applications. Real data often suffer from very few observations, with support not meeting the random requirements, ubiquitous presence of noise and potentially gross corruptions, sometimes with these simultaneously occurring. This paper proposes a Proximal Alternating Robust Subspace Minimization (PARSuMi) …
Nonlocal image representation or group sparsity has attracted considerable interest in various low-level vision tasks and has led to several state-of-the-art image denoising techniques, such as BM3D, LSSC. In the past, convex optimization with sparsity-promoting convex regularization was usually regarded as a standard scheme for estimating sparse signals in noise. However, using convex regularization can not still obtain the correct sparsity solution under some practical problems including image inverse problems. In this paper we propose a non-convex weighted $\ell_p$ minimization based group …
Tensor completion is a technique of filling missing elements of the incomplete data tensors. It being actively studied based on the convex optimization scheme such as nuclear-norm minimization. When given data tensors include some noises, the nuclear-norm minimization problem is usually converted to the nuclear-norm `regularization' problem which simultaneously minimize penalty and error terms with some trade-off parameter. However, the good value of trade-off is not easily determined because of the difference of two units and the data dependence. In the sense of trade-off tuning, the noisy te…
This paper focus on recovering multi-dimensional data called tensor from randomly corrupted incomplete observation. Inspired by reweighted $l_1$ norm minimization for sparsity enhancement, this paper proposes a reweighted singular value enhancement scheme to improve tensor low tubular rank in the tensor completion process. An efficient iterative decomposition scheme based on t-SVD is proposed which improves low-rank signal recovery significantly. The effectiveness of the proposed method is established by applying to video completion problem, and experimental results reveal that the algorithm o…
In this paper, we propose a novel model to recover a low-rank tensor by simultaneously performing double nuclear norm regularized low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An block successive upper-bound minimization algorithm is applied to solve the model. Subsequence convergence of our algorithm can be established, and our algorithm converges to the coordinate-wise minimizers in some mild conditions. Several experiments on three types of public data sets show that our algorithm can recover a variety of low-rank tensors from significantly fewer sa…
Saitama Institute of Technology（埼玉工业大学）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心）；Imperial College London（帝国理工学院）；Hangzhou Dianzi University（杭州电子科技大学）；Guangdong University of Technology（广东工业大学）
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather challenging to find the optimal TR decomposition. In this paper, by exploiting the low-rank structure of the TR latent space, we propose a novel tensor completion method which is robust to model selection. In contrast to imposing the low-rank constraint on the data space, we introd…
Low-rank tensor completion has been widely used in computer vision and machine learning. This paper develops a novel multi-modal core tensor factorization (MCTF) method combined with a tensor low-rankness measure and a better nonconvex relaxation form of this measure (NC-MCTF). The proposed models encode low-rank insights for general tensors provided by Tucker and T-SVD, and thus are expected to simultaneously model spectral low-rankness in multiple orientations and accurately restore the data of intrinsic low-rank structure based on few observed entries. Furthermore, we study the MCTF and NC-…
Institute of Computer Science and Technology, Peking University（北京大学计算机科学技术研究所）；Microsoft Research Asia（微软亚洲研究院）
In this paper, we propose a novel multiplanar autoregressive (AR) model to exploit the correlation in cross-dimensional planes of a similar patch group collected in an image, which has long been neglected by previous AR models. On that basis, we then present a joint multiplanar AR and low-rank based approach (MARLow) for image completion from random sampling, which exploits the nonlocal self-similarity within natural images more effectively. Specifically, the multiplanar AR model constraints the local stationarity in different cross-sections of the patch group, while the low-rank minimization …
Tensor completion and robust principal component analysis have been widely used in machine learning while the key problem relies on the minimization of a tensor rank that is very challenging. A common way to tackle this difficulty is to approximate the tensor rank with the $\ell_1-$norm of singular values based on its Tensor Singular Value Decomposition (T-SVD). Besides, the sparsity of a tensor is also measured by its $\ell_1-$norm. However, the $\ell_1$ penalty is essentially biased and thus the result will deviate. In order to sidestep the bias, we propose a novel non-convex tensor rank sur…
University of Electronic Science and Technology of China（电子科技大学）；Henan University（河南大学）
Remote sensing image (RSI) inpainting plays an important role in real applications. Recently, fully-connected tensor network (FCTN) decomposition has been shown the remarkable ability to fully characterize the global correlation. Considering the global correlation and the nonlocal self-similarity (NSS) of RSIs, this paper introduces the FCTN decomposition to the whole RSI and its NSS groups, and proposes a novel nonlocal patch-based FCTN (NL-FCTN) decomposition for RSI inpainting. Different from other nonlocal patch-based methods, the NL-FCTN decomposition-based method, which increases tensor …
Hunan University of Science and Technology（湖南科技大学）；Nanjing First Hospital, Nanjing Medical University（南京医科大学附属南京第一医院）；University of Macau（澳门大学）；Nanjing University of Posts and Telecommunications（南京邮电大学）
Compressive sensing (CS) based computed tomography (CT) image reconstruction aims at reducing the radiation risk through sparse-view projection data. It is usually challenging to achieve satisfying image quality from incomplete projections. Recently, the nonconvex ${L_ {1/2}} $-norm has achieved promising performance in sparse recovery, while the applications on imaging are unsatisfactory due to its nonconvexity. In this paper, we develop a ${L_ {1/2}} $-regularized nonlocal self-similarity (NSS) denoiser for CT reconstruction problem, which integrates low-rank approximation with group sparse …
Southwestern University of Finance and Economics（西南财经大学）；The University of Hong Kong（香港大学）；University of Electronic Science and Technology of China（电子科技大学）
The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the origin…
Northwest A&F University（西北农林科技大学）；University of Texas of the Permian Basin（德克萨斯大学帕米亚盆地分校）
The low rank tensor completion (LRTC) problem has attracted great attention in computer vision and signal processing. How to acquire high quality image recovery effect is still an urgent task to be solved at present. This paper proposes a new tensor $L_{2,1}$ norm minimization model (TLNM) that integrates sum nuclear norm (SNN) method, differing from the classical tensor nuclear norm (TNN)-based tensor completion method, with $L_{2,1}$ norm and Qatar Riyal decomposition for solving the LRTC problem. To improve the utilization rate of the local prior information of the image, a total variation …
Jiangsu Normal University（江苏师范大学）；Inner Mongolia University（内蒙古大学）；The University of Hong Kong（香港大学）；University of Electronic Science and Technology of China（电子科技大学）
The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image and has been widely applied in many recently proposed machining learning algorithms for image processing. However, there is no theoretical analysis on its working principle in the literature. In this paper, we discover a potential causality between NSS and low-rank property of color images, which is also available to grey images. A new patch group based NSS prior scheme is proposed to learn explicit NSS models of natural color images. The numerical…
Hangzhou Dianzi University（杭州电子科技大学）；Ningbo University（宁波大学）；Huawei Technologies Co., Ltd.（华为技术有限公司）；The Hong Kong Polytechnic University（香港理工大学）
Recovering color images and videos from highly undersampled data is a fundamental and challenging task in face recognition and computer vision. By the multi-dimensional nature of color images and videos, in this paper, we propose a novel tensor completion approach, which is able to efficiently explore the sparsity of tensor data under the discrete cosine transform (DCT). Specifically, we introduce two ``sparse + low-rank'' tensor completion models as well as two implementable algorithms for finding their solutions. The first one is a DCT-based sparse plus weighted nuclear norm induced low-rank…
The objective of this study is to address the problem of background/foreground separation with missing pixels by combining the video acquisition, video recovery, background/foreground separation into a single framework. To achieve this, a smooth robust tensor completion (SRTC) model is proposed to recover the data and decompose it into the static background and smooth foreground, respectively. Specifically, the static background is modeled by the low-rank tucker decomposition and the smooth foreground (moving objects) is modeled by the spatiotemporal continuity, which is enforced by the total …
