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.
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在本文中,我们提出了一种用于HSI去噪的强大主成分分析的新型非耦合方法,其侧重于分别同时为低级和稀疏组分的等级和列方向稀疏性产生更准确的近似。特别是,新方法采用日志确定级别近似和新颖的$ \ ell_ {2,\ log} $常规,以便分别限制组件矩阵的本地低级或列明智地稀疏属性。对于$ \ ell_ {2,\ log} $ - 正常化的收缩问题,我们开发了一个高效的封闭式解决方案,该解决方案名为$ \ ell_ {2,\ log} $ - 收缩运算符。新的正则化和相应的操作员通常可以用于需要列明显稀疏性的其他问题。此外,我们在基于日志的非凸rpca模型中强加了空间光谱总变化正则化,这增强了从恢复的HSI中的空间和光谱视图中的全局转换平滑度和光谱一致性。关于模拟和实际HSIS的广泛实验证明了所提出的方法在去噪HSIS中的有效性。
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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.
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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。
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最近,从图像中提取的不同组件的低秩属性已经考虑在MAN Hypspectral图像去噪方法中。然而,这些方法通常将3D矩阵或1D向量展开,以利用现有信息,例如非识别空间自相似性(NSS)和全局光谱相关(GSC),其破坏了高光谱图像的内在结构相关性(HSI) )因此导致恢复质量差。此外,由于在HSI的原始高维空间中的矩阵和张量的矩阵和张量的参与,其中大多数受到重大计算负担问题。我们使用子空间表示和加权低级张量正则化(SWLRTR)进入模型中以消除高光谱图像中的混合噪声。具体地,为了在光谱频带中使用GSC,将噪声HSI投影到简化计算的低维子空间中。之后,引入加权的低级张量正则化术语以表征缩减图像子空间中的前导。此外,我们设计了一种基于交替最小化的算法来解决非耦合问题。模拟和实时数据集的实验表明,SWLRTR方法比定量和视觉上的其他高光谱去噪方法更好。
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Tensor robust principal component analysis (TRPCA) is a promising way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular values equally. However, for real-world visual data, large singular values represent more signifiant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms some existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.
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低级张力完成已广泛用于计算机视觉和机器学习。本文开发了一种新型多模态核心张量分解(MCTF)方法,与张量低秩测量和该措施的更好的非凸弛豫形式(NC-MCTF)。所提出的模型编码由Tucker和T-SVD提供的一般张量的低秩见解,因此预计将在多个方向上同时模拟光谱低秩率,并准确地恢复基于几个观察到的条目的内在低秩结构的数据。此外,我们研究了MCTF和NC-MCTF正则化最小化问题,并设计了一个有效的块连续上限最小化(BSUM)算法来解决它们。该高效的求解器可以将MCTF扩展到各种任务,例如张量完成。一系列实验,包括高光谱图像(HSI),视频和MRI完成,确认了所提出的方法的卓越性能。
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低级别在高光谱图像(HSI)降级任务中很重要。根据张量的奇异值分解定义的张量核标准(TNN)是描述HSI低级别的最新方法。但是,TNN忽略了HSI在解决deno的任务时的某些身体含义,从而导致了次优的降级性能。在本文中,我们提出了用于HSI降解任务的多模式和频率加权张量核定常(MFWTNN)和非凸MFWTNN。首先,我们研究了频率切片的物理含义,并重新考虑其权重以提高TNN的低级别表示能力。其次,我们考虑两个空间维度和HSI的光谱维度之间的相关性,并将上述改进与TNN相结合以提出MFWTNN。第三,我们使用非凸功能来近似频率张量的秩函数,并提出非MFWTNN以更好地放松MFWTNN。此外,我们自适应地选择更大的权重,用于切片,主要包含噪声信息和较小的重量,用于包含配置文件信息的切片。最后,我们开发了基于乘数(ADMM)算法的有效交替方向方法来求解所提出的模型,并在模拟和真实的HSI数据集中证实了我们的模型的有效性。
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张量稀疏建模是一种有希望的方法,在整个科学和工程学中,取得了巨大的成功。众所周知,实际应用中的各种数据通常由多种因素产生,因此使用张量表示包含多个因素内部结构的数据。但是,与矩阵情况不同,构建合理的稀疏度量张量是一项相对困难且非常重要的任务。因此,在本文中,我们提出了一种称为张量全功能度量(FFM)的新张量稀疏度度量。它可以同时描述张量的每个维度的特征信息以及两个维度之间的相关特征,并将塔克等级与张量管等级连接。这种测量方法可以更全面地描述张量的稀疏特征。在此基础上,我们建立了其非凸放松,并将FFM应用于低级张量完成(LRTC)和张量鲁棒的主成分分析(TRPCA)。提出了基于FFM的LRTC和TRPCA模型,并开发了两种有效的交替方向乘数法(ADMM)算法来求解所提出的模型。各种实际数值实验证实了超出最先进的方法的优势。
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非凸松弛方法已被广泛用于张量恢复问题,并且与凸松弛方法相比,可以实现更好的恢复结果。在本文中,提出了一种新的非凸函数,最小值对数凹点(MLCP)函数,并分析了其某些固有属性,其中有趣的是发现对数函数是MLCP的上限功能。所提出的功能概括为张量病例,得出张量MLCP和加权张量$ l \ gamma $ -norm。考虑到将其直接应用于张量恢复问题时无法获得其明确解决方案。因此,给出了解决此类问题的相应等效定理,即张量等效的MLCP定理和等效加权张量$ l \ gamma $ -norm定理。此外,我们提出了两个基于EMLCP的经典张量恢复问题的模型,即低秩量张量完成(LRTC)和张量稳健的主组件分析(TRPCA)以及设计近端替代线性化最小化(棕榈)算法以单独解决它们。此外,基于Kurdyka - {\ l} ojasiwicz属性,证明所提出算法的溶液序列具有有限的长度并在全球范围内收敛到临界点。最后,广泛的实验表明,提出的算法取得了良好的结果,并证实MLCP函数确实比最小化问题中的对数函数更好,这与理论特性的分析一致。
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Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can only represent data on finite meshgrid due to their intrinsical discrete nature, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR), which can continuously represent data beyond meshgrid with infinite resolution. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in the LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.
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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。
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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.
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高光谱图像(HSIS)通常遭受不同类型的污染。这严重降低了HSI的质量,并限制了后续处理任务的准确性。 HSI DeNoising可以建模为低级张量降解问题。张量奇异值分解引起的张量核定标(TNN)在此问题中起重要作用。在这封信中,我们首先重新考虑了TNN中的三个不起眼但至关重要的现象。在HSI的傅立叶变换域中,不同的频率切片(FS)包含不同的信息。每个FS的不同单数值(SV)也代表不同的信息。这两个物理现象不仅处于光谱模式,而且位于空间模式下。然后,基于它们,我们提出了一个多模式和双重加权的TNN。它可以根据所有HSIS的身体含义适应FS和SVS。在乘数的交替方向方法的框架中,我们设计了一种有效的交替迭代策略来优化我们提出的模型。对合成和真实HSI数据集进行了剥落的实验证明了它们在相关方法上的优势。
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基于深度学习的高光谱图像(HSI)恢复方法因其出色的性能而广受欢迎,但每当任务更改的细节时,通常都需要昂贵的网络再培训。在本文中,我们建议使用有效的插入方法以统一的方法恢复HSI,该方法可以共同保留基于优化方法的灵活性,并利用深神经网络的强大表示能力。具体而言,我们首先开发了一个新的深HSI DeNoiser,利用了门控复发单元,短期和长期的跳过连接以及增强的噪声水平图,以更好地利用HSIS内丰富的空间光谱信息。因此,这导致在高斯和复杂的噪声设置下,在HSI DeNosing上的最新性能。然后,在处理各种HSI恢复任务之前,将提议的DeNoiser插入即插即用的框架中。通过对HSI超分辨率,压缩感测和内部进行的广泛实验,我们证明了我们的方法经常实现卓越的性能,这与每个任务上的最先进的竞争性或甚至更好任何特定任务的培训。
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低等级张量完成(LRTC)问题引起了计算机视觉和信号处理的极大关注。如何获得高质量的图像恢复效果仍然是目前要解决的紧急任务。本文提出了一种新的张量$ l_ {2,1} $最小化模型(TLNM),该模型(TLNM)集成了总和核标准(SNN)方法,与经典的张量核定常(TNN)基于张量的张量完成方法不同,与$ L_ { 2,1} $ norm和卡塔尔里亚尔分解用于解决LRTC问题。为了提高图像的局部先验信息的利用率,引入了总变化(TV)正则化项,从而导致一类新的Tensor $ L_ {2,1} $ NORM Minimization,总变量模型(TLNMTV)。两个提出的模型都是凸,因此具有全局最佳解决方案。此外,我们采用交替的方向乘数法(ADMM)来获得每个变量的封闭形式解,从而确保算法的可行性。数值实验表明,这两种提出的算法是收敛性的,比较优于方法。特别是,当高光谱图像的采样率为2.5 \%时,我们的方法显着优于对比方法。
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张张量强大的主成分分析(TRPCA)旨在恢复因稀疏噪声破坏的低排名张量,在许多真实应用中引起了很多关注。本文开发了一种新的全球加权TRPCA方法(GWTRPCA),该方法是第一种同时考虑额外域内切片和额叶间切片奇异值的重要性。利用这些全球信息,GWTRPCA惩罚了较大的单数值,并为其分配了较小的权重。因此,我们的方法可以更准确地恢复低管级组件。此外,我们提出了通过改良的考奇估计量(MCE)的有效自适应学习策略,因为重量设置在GWTRPCA的成功中起着至关重要的作用。为了实现GWTRPCA方法,我们使用乘数的交替方向方法(ADMM)方法设计了一种优化算法。对现实世界数据集的实验验证了我们提出的方法的有效性。
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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。
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Deconvolution is a widely used strategy to mitigate the blurring and noisy degradation of hyperspectral images~(HSI) generated by the acquisition devices. This issue is usually addressed by solving an ill-posed inverse problem. While investigating proper image priors can enhance the deconvolution performance, it is not trivial to handcraft a powerful regularizer and to set the regularization parameters. To address these issues, in this paper we introduce a tuning-free Plug-and-Play (PnP) algorithm for HSI deconvolution. Specifically, we use the alternating direction method of multipliers (ADMM) to decompose the optimization problem into two iterative sub-problems. A flexible blind 3D denoising network (B3DDN) is designed to learn deep priors and to solve the denoising sub-problem with different noise levels. A measure of 3D residual whiteness is then investigated to adjust the penalty parameters when solving the quadratic sub-problems, as well as a stopping criterion. Experimental results on both simulated and real-world data with ground-truth demonstrate the superiority of the proposed method.
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在本文中,我们将颜色图像插入作为纯季基矩阵完成问题。在文献中,季节矩阵完成的理论保证并不确定。我们的主要目的是提出一个新的最小化问题,并将核标准和三个通道之间的二次损失相结合。为了填补理论空缺,我们获得了在干净和损坏的政权中绑定的错误,这依赖于四元素矩阵的一些新结果。在强大的完成中考虑了一般的高斯噪音,所有观察都被损坏。由于界限的动机,我们建议通过二次损失中的跨通道重量来处理不平衡或相关的噪声,这是重新平衡噪声水平或消除噪声相关性的主要目的。提供了有关合成和颜色图像数据的广泛实验结果,以确认和证明我们的理论发现。
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