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基于Tensor Train低秩张量分解的理论算法研究
Alternative TitleResearch on Theory and Algorithm Based on Low-rank Tensor Train Decompositon
张杨1,2
Department机器人学研究室
Thesis Advisor韩志
KeywordTT分解 图像去噪 张量补全 逐点加权 张量鲁棒主成分分析
Pages70页
Degree Discipline模式识别与智能系统
Degree Name硕士
2019-05-17
Degree Grantor中国科学院沈阳自动化研究所
Place of Conferral沈阳
Abstract近年来,张量火车(Tensor Train,TT)秩由于以更加本质的低秩属性、更加优越的性能表现而成为当下研究热点。本文以图像视频的低秩张量分解作为研究课题,针对当前图像去噪张量分解方法存在的各模式不均衡、秩评估不准确、只对张量各模式进行了加权、低秩先验信息发掘不足四个问题。在这篇文章中,我们以TT分解为手段,分别开展了如下四个工作:基于低TT秩的彩色图像去噪算法、基于TT分解因子折凹范数的张量补全算法、逐点加权低秩TT的图像补全和融合边信息的张量鲁棒主成分分析。旨在挖掘图像视频的本征低秩信息以达到数据恢复和重建的目的。本文的主要内容如下所示:1、基于低秩TT的彩色图像去噪算法研究。由于能够保留数据的空间结构信息,张量被广泛用于计算机视觉领域。由于具备均衡的展开矩阵,TT分解可以更加充分的利用原始张量的信息。因此,与传统的张量分解方法相比,TT分解在许多领域具有更好的性能。受到这些成功应用的启发,首次把低秩TT分解用来恢复被噪声严重污染的彩色图像。同时,我们提出了一种基于块坐标下降(Block Coordinate Descent,BCD)新的去噪算法。实验表明通过视觉评估和数值评估,该模型优于对比算法。2、基于TT分解因子折凹范数的张量补全算法研究。张量可以有效地表达隐藏在数据中的空间结构信息。因此,它被广泛用于数据补全任务。然而,传统的张量分解技术具有一些缺点,例如不平衡的分解矩阵,导致低秩先验信息的低效利用。同时,大多数现有方法使用核范数来近似张量的秩,但核范数是秩的有偏估计。为了克服上述两个缺点,提出了一种基于TT分解因子折凹范数的张量补全模型。针对该模型,设计了局部线性逼近的增广拉格朗日乘子算法。实验表明,该算法相比传统方法,可以更精确地恢复原始张量。3、逐点加权低秩TT的图像补全。TT分解得到了越来越多的应用。与传统的几种分解方式相比,TT分解更合理有效。然而,TT秩具有两个明显的缺点。其一是只根据模式平衡成度考虑模式权重,忽略了某些元素在非平衡模式下恢复得更好的情况。其二是当缺失率很大时,由于升阶而导致出现严重的块效应。为了解决这两个问题,本文提出了一种基于重叠升阶的逐点加权低张量分解模型用于张量补全问题。同时设计了解决该模型的优化算法。我们针对合成数据、彩色图像、不同光照下的人脸图像以及核磁共振图像的修补问题进行了广泛的试验性能评估。结果表明,我们的方法优于当前其他最先进的基于低秩张量的方法。4、带有边信息的张量鲁棒主成分分析研究。鲁棒主成分分析(Robust Principal Component Analysis,RPCA)旨在通过将数据分解为低秩子空间和稀疏残差子空间来恢复严重损坏的观测矩阵。最近,RPCA借助一些边信息,在很大程度上提高了RPCA的性能。然而,当观测信息和边信息作为张量存在时,传统基于矩阵的RPCA方法会破坏他们的自然结构。事实上,计算机视觉中的数据通常是以高阶张量的形式存在的。因此,目前的解决方案并不是最理想的。为了解决这个问题,我们基于t-SVD,提出融合边信息的张量RPCA模型。这是目前第一个融合张量边信息的工作。实验验证所提模型的有效性。
Other AbstractIn recent years, the Tensor Train (TT) rank has become a hot research topic due to its more essential low rank attribute and superior performance. In this paper, the low-rank tensor decomposition of image and video is taken as the research topic. The current image denoising methods by tensor decomposition have the following shortcomings. First, the tensor modes are unbalanced. Second, the rank evaluation is not accurate. Third, weigh only the tensor modes. Forth, insufficient exploration of low rank prior information. In this paper, we use TT decomposition to carry out the following four tasks: color image denoising algorithm based on low TT rank, tensor completion algorithm based on TT decomposition factor concave and concave norm, point-by-point weighting low Tensor TT image complementation and tensor robust principal component analysis of fused edge information. It aims to mine the intrinsic low rank information of image video for data recovery and reconstruction. The main content of this article is as follows: 1. Research on color image denoising algorithm based on low rank TT. Tensor has been widely used in computer vision due to its ability to maintain spatial structure information. Owning to the well-balanced unfolding matrices, the recently proposed tensor train (TT) decomposition can make full use of information from tensors. Thereby, tensor train representation has a better performance in many fields compared to traditional methods of tensor decomposition. Inspired by the success of tensor train, in this paper, we firstly apply low-rank tensor train to recovering noisy color images. Meanwhile, we propose a novel algorithm for noise-contaminated images based on the block coordinate descent (BCD) method. The numerical experiments demonstrate that our algorithm can give a better result in the real color image both visually and numerically. 2. Research on Tensor completion based on tensor train folded-concave penalization. Tensors can effectively express the spatial structural information hidden in the data. Therefore, it is widely used in the task of tensor completion. However, traditional tensor decomposition techniques have some drawbacks, such as unbalanced decomposed matrices, resulting the low-efficient utilization of the prior information of low rank. Meanwhile, most of current methods use nuclear norm to approximate tensor ranks, of which is a biased estimation. To overcome the above two drawbacks, this paper proposed a tensor completion via tensor train with folded-concave penalization model. A local linear approximation augmented Lagrange multiplier algorithm was also designed for the model. Experiments show that this algorithm can recover the original tensors more precisely than traditional methods. 3. Image completion by element-wise weighted low rank TT. There are more and more applications based on the tensor train (TT) as the TT decomposition is more reasonable and effective compared to traditional decompositions. However, existing methods using the TT rank have two obvious drawbacks. One is that they only consider mode weights according to mode balancing degree, even though some elements are recovered better in an unbalanced mode. The other is that serious blocking artifacts appear when the missing rate is large. To solve these two problems, this paper proposes a novel approach to tensor completion via the element-wise weighted technique. Accordingly, novel optimization formulation for tensor completion as well as a novel algorithm called tensor completion by parallel weighted matrix factorization via tensor train (TWMac-TT) for its solution is proposed. In addition, we specially take the recovery quality of edge elements from adjacent blocks into consideration. Different from conventional reshaping and ket augmentation (KA), we firstly proposes a novel tensor augmentation technique called overlapping ket augmentation (OKA), which can further avoid blocking artifacts. We conduct extensive performance evaluations on synthetic data completion, gray image completion and color image completion. Our experimental results demonstrate that the proposed method outperforms all the other methods. 4. Tensor robust principal component analysis with side information. Robust Principal Component Analysis (RPCA) is designed for recovering a severely corrupted observation matrix by decomposing data into a low-rank subspace plus a sparse residual, which is a powerful method applied in many machine learning problems. Recently, some works on RPCA with a new kind of prior, side information, have improved the performance of RPCA largely. However, the natural structural information is destroyed when the observation and side information come as tensors, which is very common in computer vision applications. Thus, the solutions of their works are often suboptimal. To remedy this, we propose two tensor models for the RPCA problem with side information based on a new tensor Singular Value Decomposition (t-SVD), as simple and elegant tensor extensions. As far as we know, we are the first to employ side information in the tensor case. By minimizing the tensor nuclear norm, excellent results are obtained by our models compared with other six state-of-the-art approaches.
Language中文
Contribution Rank1
Document Type学位论文
Identifierhttp://ir.sia.cn/handle/173321/25178
Collection机器人学研究室
Affiliation1.中国科学院沈阳自动化研究所
2.中国科学院大学
Recommended Citation
GB/T 7714
张杨. 基于Tensor Train低秩张量分解的理论算法研究[D]. 沈阳. 中国科学院沈阳自动化研究所,2019.
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