Video deraining via nonlocal low-rank regularization

Outdoor videos captured in rainy weather may be significantly corrupted by the undesired rain streaks, which severely affect the video processing tasks in outdoor computer vision systems. In this paper, we propose a tensor-based video rain streaks removal method using the nonlocal low-rank regularization. Specifically, we first divide videos into overlapped spatial–temporal patches. Then for each patch, we group its nonlocal similar spatial–temporal patches to form a third-order tensor. To model the clean videos, we characterize the wealth redundancy by adopting the tensor nuclear norm to regularize the low-rankness of the third-order tensors formed by similar spatial–temporal patches of clean videos. We also consider the piecewise smoothness and the temporal continuity of clean videos and utilize the unidirectional total variation to enhance the smoothness and continuity. Moreover, as rain streaks are sparse and smooth along the rain direction, we model the rain streaks by employing an ℓ₁ norm and the unidirectional total variation penalty to boost the sparsity and directional smoothness, respectively. We develop an efficient alternating direction method of multipliers to solve the proposed model. Experimental results on both synthetic and real rainy videos show that our method outperforms the state-of-the-art methods quantitatively and qualitatively.     

P1 finite element methods for a weighted elliptic state-constrained optimal control problem

Numerical Algorithms - Recently, low-rank regularization has achieved great success in tensor completion. However, only considering the global low-rankness is not sufficient, especially for a low...     

Low-rank tensor completion via smooth matrix factorization

Low-rank modeling has achieved great success in tensor completion. However, the low-rank prior is not sufficient for the recovery of the underlying tensor, especially when the sampling rate (SR) is extremely low. Fortunately, many real world data exhibit the piecewise smoothness prior along both the spatial and the third modes (e.g., the temporal mode in video data and the spectral mode in hyperspectral data). Motivated by this observation, we propose a novel low-rank tensor completion model using smooth matrix factorization (SMF-LRTC), which exploits the piecewise smoothness prior along all modes of the underlying tensor by introducing smoothness constraints on the factor matrices. An efficient block successive upper-bound minimization (BSUM)-based algorithm is developed to solve the proposed model. The developed algorithm converges to the set of the coordinate-wise minimizers under some mild conditions. Extensive experimental results demonstrate the superiority of the proposed method over the compared ones.     

一个基于张量火车分解的张量填充方法及在图像恢复中的应用

低秩张量填充在数据恢复中有广泛应用, 基于张量火车(TT) 分解的张量填充模型在彩色图像和视频以及互联网数据恢复中应用效果良好。本文提出一个基于三阶张量TT分解的填充模型。在模型中, 引入稀疏正则项与时空正则项, 分别刻画核张量的稀疏性和数据固有的块相似性。根据问题的结构特点, 引入辅助变量将原模型等价转化成可分离形式, 并采用临近交替极小化(PAM) 与交替方向乘子法(ADMM) 相结合的方法求解模型。数值实验表明, 两正则项的引入有利于提高数据恢复的稳定性和实际效果, 所提出方法优于其他方法。在采样率较低或图像出现结构性缺失时, 其方法效果较为显著。     

一个基于张量火车分解的张量填充方法及在图像恢复中的应用

低秩张量填充在数据恢复中有广泛应用, 基于张量火车(TT) 分解的张量填充模型在彩色图像和视频以及互联网数据恢复中应用效果良好。本文提出一个基于三阶张量TT分解的填充模型。在模型中, 引入稀疏正则项与时空正则项, 分别刻画核张量的稀疏性和数据固有的块相似性。根据问题的结构特点, 引入辅助变量将原模型等价转化成可分离形式, 并采用临近交替极小化(PAM) 与交替方向乘子法(ADMM) 相结合的方法求解模型。数值实验表明, 两正则项的引入有利于提高数据恢复的稳定性和实际效果, 所提出方法优于其他方法。在采样率较低或图像出现结构性缺失时, 其方法效果较为显著。     

Low Tucker rank tensor completion using a symmetric block coordinate descent method

Low Tucker rank tensor completion has wide applications in science and engineering. Many existing approaches dealt with the Tucker rank by unfolding matrix rank. However, unfolding a tensor to a matrix would destroy the data's original multi-way structure, resulting in vital information loss and degraded performance. In this article, we establish a relationship between the Tucker ranks and the ranks of the factor matrices in Tucker decomposition. Then, we reformulate the low Tucker rank tensor completion problem as a multilinear low rank matrix completion problem. For the reformulated problem, a symmetric block coordinate descent method is customized. For each matrix rank minimization subproblem, the classical truncated nuclear norm minimization is adopted. Furthermore, temporal characteristics in image and video data are introduced to such a model, which benefits the performance of the method. Numerical simulations illustrate the efficiency of our proposed models and methods.     

一类基于L0/1软间隔损失函数的低秩支持张量机

支持向量机作为基于向量空间的一种传统的机器学习方法,不能直接处理张量类型的数据,否则不仅破坏数据的空间结构,还会造成维度灾难及小样本问题。作为支持向量机的一种高阶推广,用于处理张量数据分类的支持张量机已经引起众多学者的关注,并应用于遥感成像、视频分析、金融、故障诊断等多个领域。与支持向量机类似,已有的支持张量机模型中采用的损失函数多为L_0/1函数的代理函数。将直接使用L_0/1这一本原函数作为损失函数,并利用张量数据的低秩性,建立针对二分类问题的低秩支持张量机模型。针对这一非凸非连续的张量优化问题,设计交替方向乘子法进行求解,并通过对模拟数据和真实数据进行数值实验,验证模型与算法的有效性。     

A non-convex tensor rank approximation for tensor completion

Low-rankness has been widely exploited for the tensor completion problem. Recent advances have suggested that the tensor nuclear norm often leads to a promising approximation for the tensor rank. It treats the singular values equally to pursue the convexity of the objective function, while the singular values for the practical images have clear physical meanings with different importance and should be treated differently. In this paper, we propose a non-convex logDet function as a smooth approximation for tensor rank instead of the convex tensor nuclear norm and introduce it into the low-rank tensor completion problem. An alternating direction method of multiplier (ADMM)-based method is developed to solve the problem. Experimental results have shown that the proposed method can significantly outperform existing state-of-the-art nuclear norm-based methods for tensor completion.     

Identifying local smoothness for spatially inhomogeneous functions

We consider a problem of estimating local smoothness of a spatially inhomogeneous function from noisy data under the framework of smoothing splines. Most existing studies related to this problem deal with estimation induced by a single smoothing parameter or partially local smoothing parameters, which may not be efficient to characterize various degrees of smoothness of the underlying function when it is spatially varying. In this paper, we propose a new nonparametric method to estimate local smoothness of the function based on a moving local risk minimization coupled with spatially adaptive smoothing splines. The proposed method provides full information of the local smoothness at every location on the entire data domain, so that it is able to understand the degrees of spatial inhomogeneity of the function. A successful estimate of the local smoothness is useful for identifying abrupt changes of smoothness of the data, performing functional clustering and improving the uniformity of coverage of the confidence intervals of smoothing splines. We further consider a nontrivial extension of the local smoothness of inhomogeneous two-dimensional functions or spatial fields. Empirical performance of the proposed method is evaluated through numerical examples, which demonstrates promising results of the proposed method.     

Simultaneous state‐time approximation of the chemical master equation using tensor product formats

We apply the novel tensor product formats (tensor train, quantized TT [QTT], and QTT‐Tucker) to the solution of d‐dimensional chemical master equations for gene regulating networks (signaling cascades, toggle switches, and phage‐ λ). For some important cases, for example, signaling cascade models, we prove analytical tensor product representations of the system operator. The quantized tensor representations (QTT, QTT‐Tucker) are employed in both state space and time, and the global state‐time (d + 1)‐dimensional system is solved in the tensor product form by the alternating minimal energy iteration, the ALS‐type algorithm. This approach leads to the logarithmic dependence of the computational complexity on the volume of the state space. We investigate the proposed technique numerically and compare it with the direct chemical master equation solution and some previously known approximate schemes, where possible. We observe that the newer tensor methods demonstrate a good potential in simulation of relevant biological systems. Copyright © 2014 John Wiley & Sons, Ltd.     

A fractional order derivative based active contour model for inhomogeneous image segmentation

Segmenting intensity inhomogeneous images is a challenging task for both local and global methods. Some hybrid methods have great advantages over the traditional methods in inhomogeneous image segmentation. In this paper, a new hybrid method is presented, which incorporates image gradient, local environment and global information into a framework, called adaptive-weighting active contour model. The energy or level set functions in the framework mainly include two parts: a global term and local term. The global term aims to enhance the image contrast, and it can also accelerate the convergence rate when minimizing the energy function. The local term integrates fractional order differentiation, fractional order gradient magnitude, and difference image information into the well-known local Chan–Vese model, which has been shown to be effective and efficient in modeling the local information. The local term can also enhance low frequency information and improve the inhomogeneous image segmentation. An adaptive weighting strategy is proposed to balance the actions of the global and local terms automatically. When minimizing the level set functions, regularization can be imposed by applying Gaussian filtering to ensure smoothness in the evolution process. In addition, a corresponding stopping criterion is proposed to ensure the evolving curve automatically stops on true boundaries of objects. Dice similarity coefficient is employed as the comparative quantitative measures for the segmented results. Experiments on synthetic images as well as real images are performed to demonstrate the segmentation accuracy and computational efficiency of the presented hybrid method.     

A Tensor Regularized Nuclear Norm Method for Image and Video Completion

Journal of Optimization Theory and Applications - In the present paper, we propose two new methods for tensor completion of third-order tensors. The proposed methods consist in minimizing the...     

Robust tensor completion using transformed tensor singular value decomposition

In this article, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in peak signal‐to‐noise ratio than that by using Fourier transform and other robust tensor completion methods.     

Tensor train completion: Local recovery guarantees via Riemannian optimization

In this work, we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with auxiliary subspace information and obtain the corresponding local convergence guarantees.     

Robust tensor train component analysis

Robust Principal Component Analysis plays a key role in various fields such as image and video processing, data mining, and hyperspectral data analysis. In this paper, we study the problem of robust tensor train (TT) principal component analysis from partial observations, which aims to decompose a given tensor into the low TT rank and sparse components. The decomposition of the proposed model is used to find the hidden factors and help alleviate the curse of dimensionality via a set of connected low-rank tensors. A relaxation model is to minimize a weighted combination of the sum of nuclear norms of unfolding matrices of core tensors and the tensor ${?}_1$ norm. A proximal alternating direction method of multipliers is developed to solve the resulting model. Furthermore, we show that any cluster point of the convergent subsequence is a Karush-Kuhn-Tucker point of the proposed model under some conditions. Extensive numerical examples on both synthetic data and real-world datasets are presented to demonstrate the effectiveness of the proposed approach.     

基于一个新的NCP函数的光滑牛顿法求解非线性互补问题

本文研究了非线性互补的光滑化问题.利用一个新的光滑NCP函数将非线性互补问题转化为等价的光滑方程组,并在此基础上建立了求解P0-函数非线性互补问题的一个完全光滑化牛顿法,获得了算法的全局收敛性和局部二次收敛性的结果.并给出数值实验验证了理论分析的正确性.     

Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence

In this paper, an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence is proposed. In the model, we consider the infection age of infectious individuals and the biological age of pathogen in the environment. It is verified that the global dynamics of the model is completely determined by the basic reproduction number. Asymptotic smoothness is verified as a necessary argument. By analyzing corresponding characteristic equations, we discuss the local stability of each of feasible steady states. Uniform persistence is shown by using the persistence theory for infinite dimensional dynamical system. The global stability of each of feasible steady states is established by using suitable Lyapunov functionals and LaSalle’s invariance principle. Numerical simulations are carried out to illustrate the theoretical results.     

Adaptive Segmentation Model for Images with Intensity Inhomogeneity based on Local Neighborhood Contrast

Segmentation of images with intensity inhomogeneity is a significant task in the field of image processing, especially in medical image processing and analysis. Some local region-based models work well on handling intensity inhomogeneity, but they are always sensitive to contour initialization and high noise. In this paper, we present an adaptive segmentation model for images with intensity inhomogeneity in the form of partial differential equation. Firstly, a global intensity fitting term and a local intensity fitting term are constructed by employing the global and local image information, respectively. Secondly, a tradeoff function is defined to adjust adaptively the weight between two fitting terms, which is based on the neighborhood contrast of image pixel. Finally, a weighted regularization term related to local entropy is used to ensure the smoothness of evolution curve. Meanwhile, a distance regularization term is added for stable level set evolution. Experimental results show that the proposed model without initial contour can segment inhomogeneous images stably and effectively, which thereby avoiding the influence of contour initialization on segmentation results. Besides, the proposed model works better on noise images comparing with two relevant segmentation models.     

低秩张量填充的加速随机临近梯度算法

本文提出了一种求解低秩张量填充问题的加速随机临近梯度算法.张量填充模型可以松弛为平均组合形式的无约束优化问题,在迭代过程中,随机选取该组合中的某一函数进行变量更新,有效减少了张量展开、矩阵折叠及奇异值分解带来的较大的计算花费.本文证明了算法的收敛率为$O (1/k^{2})$.最后,随机生成的和真实的张量填充实验结果表明新算法在CPU时间上优于现有的三种算法.     

几种基于散乱数据拟合的局部插值方法

本文首先针对散乱数据拟合的Shepard方法,结合截断多项式、B样条基函数和指数函数来构造其权函数,使新的权函数具有更高的光滑度和更好的衰减性,并且其光滑性和衰减性可以根据实际需要自由调节,从而提高了曲面的拟合质量．同时还给出一种类似的局部插值方法．另外,本文还基于多重二次插值,结合多元样条的思想,给出了两个局部插值算法．该算法较好地继承了多重二次插值曲面的性质,从而保证了拟合曲面具有好地光顺性和拟合精度．曲面整体也具有较高的光滑性．