Low-rank and sparse decomposition based methods find their use in many applications involving background modeling such as clutter suppression and object tracking. While Robust Principal Component Analysis (RPCA) has achieved great success in performing this task, it can take hundreds of iterations to converge and its performance decreases in the presence of different phenomena such as occlusion, jitter and fast motion. The recently proposed deep unfolded networks, on the other hand, have demonstrated better accuracy and improved convergence over both their iterative equivalents as well as over other neural network architectures. In this work, we propose a novel deep unfolded spatiotemporal RPCA (DUST-RPCA) network, which explicitly takes advantage of the spatial and temporal continuity in the low-rank component. Our experimental results on the moving MNIST dataset indicate that DUST-RPCA gives better accuracy when compared with the existing state of the art deep unfolded RPCA networks.

从X射线冠状动脉造影（XCA）图像序列中提取对比度的血管对于直觉诊断和治疗具有重要的临床意义。在这项研究中，XCA图像序列O被认为是三维张量输入，血管层H是稀疏张量，而背景层B是低级别张量。使用张量核标准（TNN）最小化，提出了一种基于张量的强稳定主成分分析（TRPCA）的新型血管层提取方法。此外，考虑了血管的不规则运动和周围无关组织的动态干扰，引入了总变化（TV）正规化时空约束，以分离动态背景E。 - 阶段区域生长（TSRG）方法用于血管增强和分割。全局阈值分割用作获得主分支的预处理，并使用ra样特征（RLF）滤波器来增强和连接破碎的小段，最终的容器掩模是通过结合两个中间结果来构建的。我们评估了TV-TRPCA算法的前景提取的可见性以及TSRG算法在真实临床XCA图像序列和第三方数据库上的血管分割的准确性。定性和定量结果都验证了所提出的方法比现有的最新方法的优越性。

我们通过基于压缩感测和多输出（MIMO）无线雷达来解决材料缺陷的检测，这些材料缺陷在层状材料结构内部。这里，由于层状结构的表面的反射导致的强杂波通常经常使缺陷挑战的缺陷。因此，需要改进的缺陷检测所需的复杂信号分离方法。在许多情况下，我们感兴趣的缺陷的数量是有限的，并且分层结构的信令响应可以被建模为低秩结构。因此，我们提出了对缺陷检测的关节等级和稀疏最小化。特别是，我们提出了一种基于迭代重量的核和$ \ ell_1- $规范（一种双重重量方法）的非凸法方法，与传统的核规范和$ \ ell_1- $常态最小化相比获得更高的准确性。为此，迭代算法旨在估计低级别和稀疏贡献。此外，我们建议深入学习来学习算法（即，算法展开）的参数，以提高算法的准确性和汇聚速度。我们的数值结果表明，该方法在恢复的低级别和稀疏组分的均方误差和收敛速度方面优于常规方法。

红外小目标检测是红外系统中的重要基本任务。因此，已经提出了许多红外小目标检测方法，其中低级模型已被用作强大的工具。然而，基于低级别的方法为不同的奇异值分配相同的权重，这将导致背景估计不准确。考虑到不同的奇异值具有不同的重要性，并且应判别处理，本文提出了一种用于红外小目标检测的非凸张力低秩近似（NTLA）方法。在我们的方法中，NTLA正则化将不同的权重自适应分配给不同的奇异值以进行准确背景估计。基于所提出的NTLA，我们提出了不对称的空间 - 时间总变化（ASTTV）正则化，以实现复杂场景中的更准确的背景估计。与传统的总变化方法相比，ASTTV利用不同的平滑度强度进行空间和时间正则化。我们设计了一种有效的算法来查找我们方法的最佳解决方案。与一些最先进的方法相比，所提出的方法达到各种评估指标的改进。各种复杂场景的广泛实验结果表明，我们的方法具有强大的鲁棒性和低误报率。代码可在https://github.com/liuting20a/asttv-ntla获得。

我们提出了一个健壮的主成分分析（RPCA）框架，以从时间观察中恢复低级别和稀疏矩阵。我们开发了批处理时间算法的在线版本，以处理较大的数据集或流数据。我们从经验上将提出的方法与不同的RPCA框架进行比较，并在实际情况下显示出其有效性。

Spatiotemporal traffic data imputation is of great significance in intelligent transportation systems and data-driven decision-making processes. To make an accurate reconstruction on partially observed traffic data, we assert the importance of characterizing both global and local trends in traffic time series. In the literature, substantial prior works have demonstrated the effectiveness of utilizing low-rankness property of traffic data by matrix/tensor completion models. In this study, we first introduce a Laplacian kernel to temporal regularization for characterizing local trends in traffic time series, which can be formulated in the form of circular convolution. Then, we develop a low-rank Laplacian convolutional representation (LCR) model by putting the nuclear norm of a circulant matrix and the Laplacian temporal regularization together, which is proved to meet a unified framework that takes a fast Fourier transform solution in a relatively low time complexity. Through extensive experiments on some traffic datasets, we demonstrate the superiority of LCR for imputing traffic time series of various time series behaviors (e.g., data noises and strong/weak periodicity). The proposed LCR model is an efficient and effective solution to large-scale traffic data imputation over the existing baseline models. The adapted datasets and Python implementation are publicly available at https://github.com/xinychen/transdim.

Low-rankness plays an important role in traditional machine learning, but is not so popular in deep learning. Most previous low-rank network compression methods compress the networks by approximating pre-trained models and re-training. However, the optimal solution in the Euclidean space may be quite different from the one in the low-rank manifold. A well-pre-trained model is not a good initialization for the model with low-rank constraints. Thus, the performance of a low-rank compressed network degrades significantly. Compared to other network compression methods such as pruning, low-rank methods attracts less attention in recent years. In this paper, we devise a new training method, low-rank projection with energy transfer (LRPET), that trains low-rank compressed networks from scratch and achieves competitive performance. First, we propose to alternately perform stochastic gradient descent training and projection onto the low-rank manifold. Compared to re-training on the compact model, this enables full utilization of model capacity since solution space is relaxed back to Euclidean space after projection. Second, the matrix energy (the sum of squares of singular values) reduction caused by projection is compensated by energy transfer. We uniformly transfer the energy of the pruned singular values to the remaining ones. We theoretically show that energy transfer eases the trend of gradient vanishing caused by projection. Third, we propose batch normalization (BN) rectification to cut off its effect on the optimal low-rank approximation of the weight matrix, which further improves the performance. Comprehensive experiments on CIFAR-10 and ImageNet have justified that our method is superior to other low-rank compression methods and also outperforms recent state-of-the-art pruning methods. Our code is available at https://github.com/BZQLin/LRPET.

近年来，健壮的主成分分析（PCA）受到了广泛关注。它的目的是从其总和中恢复一个低级别矩阵和稀疏矩阵。本文提出了一种新型的非凸强壮的PCA算法，即Riemannian Cur（Riecur），它利用了Riemannian优化和强大的CUR分解观念。该算法与迭代的鲁棒cur具有相同的计算复杂性，后者目前是最新的，但对离群值更强。Riecur还能够忍受大量的异常值，并且与加速的交替预测相媲美，该预测具有很高的离群公差，但计算复杂性比提议的方法差。因此，所提出的算法在计算复杂性和异常耐受性方面都可以在鲁棒PCA上实现最新性能。

Deep unfolding networks (DUNs) have proven to be a viable approach to compressive sensing (CS). In this work, we propose a DUN called low-rank CS network (LR-CSNet) for natural image CS. Real-world image patches are often well-represented by low-rank approximations. LR-CSNet exploits this property by adding a low-rank prior to the CS optimization task. We derive a corresponding iterative optimization procedure using variable splitting, which is then translated to a new DUN architecture. The architecture uses low-rank generation modules (LRGMs), which learn low-rank matrix factorizations, as well as gradient descent and proximal mappings (GDPMs), which are proposed to extract high-frequency features to refine image details. In addition, the deep features generated at each reconstruction stage in the DUN are transferred between stages to boost the performance. Our extensive experiments on three widely considered datasets demonstrate the promising performance of LR-CSNet compared to state-of-the-art methods in natural image CS.

Deep neural networks provide unprecedented performance gains in many real world problems in signal and image processing. Despite these gains, future development and practical deployment of deep networks is hindered by their blackbox nature, i.e., lack of interpretability, and by the need for very large training sets. An emerging technique called algorithm unrolling or unfolding offers promise in eliminating these issues by providing a concrete and systematic connection between iterative algorithms that are used widely in signal processing and deep neural networks. Unrolling methods were first proposed to develop fast neural network approximations for sparse coding. More recently, this direction has attracted enormous attention and is rapidly growing both in theoretic investigations and practical applications. The growing popularity of unrolled deep networks is due in part to their potential in developing efficient, high-performance and yet interpretable network architectures from reasonable size training sets. In this article, we review algorithm unrolling for signal and image processing. We extensively cover popular techniques for algorithm unrolling in various domains of signal and image processing including imaging, vision and recognition, and speech processing. By reviewing previous works, we reveal the connections between iterative algorithms and neural networks and present recent theoretical results. Finally, we provide a discussion on current limitations of unrolling and suggest possible future research directions.

显式低级正则化，例如核规范的正则化已被广泛用于成像科学。但是，已经发现，在各种图像处理任务中，隐式正规化优于明确的正规化。另一个问题是，固定的显式正则化将适用性限制为广泛图像，因为不同的图像偏爱不同的显式正则化捕获的不同特征。因此，本文提出了一种新的自适应和隐式低级别正则化，从训练数据中动态捕获了较低的先验。我们新的自适应和隐式低级别正则化的核心是在基于Dirichlet Energy的正则化中参数化Laplacian矩阵，我们称之为正则化空气。从理论上讲，我们表明\ retwo {air}的自适应正则化增强了训练结束时的隐式正则化和消失。我们验证了空气对各种基准任务的有效性，表明空气对缺失条目不均匀的情况特别有利。该代码可以在https://github.com/lizhemin15/air-net上找到。

明确的低级正则化，例如核规范规则，已广泛用于成像科学。但是，已经发现隐式正则化优于各种图像处理任务中的明确正则化。另一个问题是，固定的显式正则化将适用性限制为广泛的图像，因为不同的图像有利于使用不同的显式规则化捕获的不同特征。因此，本文提出了一种新的自适应和隐式低级正则化，其从训练数据动态地捕获低秩。在我们新的自适应和隐式低级正则化的核心，正在使用神经网络参数化Laplacian矩阵，并通过神经网络调用所提出的型号\ Textit {Air-Net}。从理论上讲，我们表明，空气网的自适应正规化增强了隐含的正则化并在培训结束时消失。我们验证了对各种基准任务对各种基准任务的效果，显示空中网对缺失条目不均匀时的情况尤为好评。可以在\ href {https://github.com/lizhemin15/airair-net}} {https：//github.com/lizhemin15/airair-net}。

我们提出了一个基于一般学习的框架，用于解决非平滑和非凸图像重建问题。我们将正则函数建模为$ l_ {2,1} $ norm的组成，并将平滑但非convex功能映射参数化为深卷积神经网络。我们通过利用Nesterov的平滑技术和残留学习的概念来开发一种可证明的趋同的下降型算法来解决非平滑非概念最小化问题，并学习网络参数，以使算法的输出与培训数据中的参考匹配。我们的方法用途广泛，因为人们可以将各种现代网络结构用于正规化，而所得网络继承了算法的保证收敛性。我们还表明，所提出的网络是参数有效的，其性能与实践中各种图像重建问题中的最新方法相比有利。

It is known that the decomposition in low-rank and sparse matrices (\textbf{L+S} for short) can be achieved by several Robust PCA techniques. Besides the low rankness, the local smoothness (\textbf{LSS}) is a vitally essential prior for many real-world matrix data such as hyperspectral images and surveillance videos, which makes such matrices have low-rankness and local smoothness properties at the same time. This poses an interesting question: Can we make a matrix decomposition in terms of \textbf{L\&LSS +S } form exactly? To address this issue, we propose in this paper a new RPCA model based on three-dimensional correlated total variation regularization (3DCTV-RPCA for short) by fully exploiting and encoding the prior expression underlying such joint low-rank and local smoothness matrices. Specifically, using a modification of Golfing scheme, we prove that under some mild assumptions, the proposed 3DCTV-RPCA model can decompose both components exactly, which should be the first theoretical guarantee among all such related methods combining low rankness and local smoothness. In addition, by utilizing Fast Fourier Transform (FFT), we propose an efficient ADMM algorithm with a solid convergence guarantee for solving the resulting optimization problem. Finally, a series of experiments on both simulations and real applications are carried out to demonstrate the general validity of the proposed 3DCTV-RPCA model.

从高度不足的数据中恢复颜色图像和视频是面部识别和计算机视觉中的一项基本且具有挑战性的任务。通过颜色图像和视频的多维性质，在本文中，我们提出了一种新颖的张量完成方法，该方法能够有效探索离散余弦变换（DCT）下张量数据的稀疏性。具体而言，我们介绍了两个``稀疏 +低升级''张量完成模型，以及两种可实现的算法来找到其解决方案。第一个是基于DCT的稀疏加权核标准诱导低级最小化模型。第二个是基于DCT的稀疏加上$ P $换图映射引起的低秩优化模型。此外，我们因此提出了两种可实施的增强拉格朗日算法，以解决基础优化模型。一系列数值实验在内，包括颜色图像介入和视频数据恢复表明，我们所提出的方法的性能要比许多现有的最新张量完成方法更好，尤其是对于缺少数据比率较高的情况。

In this paper, we study the problem of a batch of linearly correlated image alignment, where the observed images are deformed by some unknown domain transformations, and corrupted by additive Gaussian noise and sparse noise simultaneously. By stacking these images as the frontal slices of a third-order tensor, we propose to utilize the tensor factorization method via transformed tensor-tensor product to explore the low-rankness of the underlying tensor, which is factorized into the product of two smaller tensors via transformed tensor-tensor product under any unitary transformation. The main advantage of transformed tensor-tensor product is that its computational complexity is lower compared with the existing literature based on transformed tensor nuclear norm. Moreover, the tensor $\ell_p$ $(0<p<1)$ norm is employed to characterize the sparsity of sparse noise and the tensor Frobenius norm is adopted to model additive Gaussian noise. A generalized Gauss-Newton algorithm is designed to solve the resulting model by linearizing the domain transformations and a proximal Gauss-Seidel algorithm is developed to solve the corresponding subproblem. Furthermore, the convergence of the proximal Gauss-Seidel algorithm is established, whose convergence rate is also analyzed based on the Kurdyka-$\L$ojasiewicz property. Extensive numerical experiments on real-world image datasets are carried out to demonstrate the superior performance of the proposed method as compared to several state-of-the-art methods in both accuracy and computational time.

As a convex relaxation of the low rank matrix factorization problem, the nuclear norm minimization has been attracting significant research interest in recent years. The standard nuclear norm minimization regularizes each singular value equally to pursue the convexity of the objective function. However, this greatly restricts its capability and flexibility in dealing with many practical problems (e.g., denoising), where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, where the singular values are assigned different weights. The solutions of the WNNM problem are analyzed under different weighting conditions. We then apply the proposed WNNM algorithm to image denoising by exploiting the image nonlocal self-similarity. Experimental results clearly show that the proposed WNNM algorithm outperforms many state-of-the-art denoising algorithms such as BM3D in terms of both quantitative measure and visual perception quality.

We propose the framework of dual convexified convolutional neural networks (DCCNNs). In this framework, we first introduce a primal learning problem motivated by convexified convolutional neural networks (CCNNs), and then construct the dual convex training program through careful analysis of the Karush-Kuhn-Tucker (KKT) conditions and Fenchel conjugates. Our approach reduces the computational overhead of constructing a large kernel matrix and more importantly, eliminates the ambiguity of factorizing the matrix. Due to the low-rank structure in CCNNs and the related subdifferential of nuclear norms, there is no closed-form expression to recover the primal solution from the dual solution. To overcome this, we propose a highly novel weight recovery algorithm, which takes the dual solution and the kernel information as the input, and recovers the linear weight and the output of convolutional layer, instead of weight parameter. Furthermore, our recovery algorithm exploits the low-rank structure and imposes a small number of filters indirectly, which reduces the parameter size. As a result, DCCNNs inherit all the statistical benefits of CCNNs, while enjoying a more formal and efficient workflow.

This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.

在各种图像处理和计算机视觉任务中经常遇到颜色图像Denoising。一种传统的策略是将RGB图像转换为较小相关的颜色空间，并分别将新空间的每个通道定义。但是，这种策略无法完全利用渠道之间的相关信息，并且不足以获得令人满意的结果。为了解决这个问题，本文提出了一个新的多通道优化模型，用于在核定标准下减去Frobenius规范最小化框架下的颜色图像Deno。具体而言，基于块匹配，将颜色图像分解为重叠的RGB补丁。对于每个补丁，我们堆叠其相似的邻居以形成相应的补丁矩阵。提出的模型是在补丁矩阵上执行的，以恢复其无噪声版本。在恢复过程中，a）引入权重矩阵以充分利用通道之间的噪声差； b）单数值是自适应缩小的，而无需分配权重。有了他们，提议的模型可以在保持简单的同时取得有希望的结果。为了解决提出的模型，基于乘数框架的交替方向方法构建了准确有效的算法。每个更新步骤的解决方案可以在封闭式中分析表达。严格的理论分析证明了所提出的算法产生的解决方案序列会收敛到其各自的固定点。合成和真实噪声数据集的实验结果证明了所提出的模型优于最先进的模型。