In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time,...In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time, the compatibility conditions, the sufficient and necessary conditions and the specific solution methods for the matrix solution are given. Secondly, we further consider the solvability of the second semi-tensor product equation of the matrix. For each part, several examples are given to illustrate the validity of the results.展开更多
Diffusion tensor MRI （DT-MRI or DTI） is emerging as an important non-invasive technology for elucidating intemal brain structures. It has recently been utilized to diagnose a series of diseases that affect the integ...Diffusion tensor MRI （DT-MRI or DTI） is emerging as an important non-invasive technology for elucidating intemal brain structures. It has recently been utilized to diagnose a series of diseases that affect the integrity of neural systems to provide a basis for neuroregenerative studies. Results from the present study suggested that neural tissue is reconstructed with multiple diffusion-weighted gradient directions DTI, which varies from traditional imaging methods that utilize 6 gradient directions. Simultaneously, the diffusion tensor matrix is obtained by multiple linear regressions from an equation of echo signal intensity. The condition number value and standard deviation of fractional anisotropy for each scheme can be used to evaluate image quality. Results demonstrated that increasing gradient direction to some extent resulted in improved effects. Therefore, the traditional 6 and 15 directions should not be considered optimal scan protocols for clinical DTI application. In a scheme with 20 directions, the condition number and standard deviation of fractional anisotropy of the encoding gradients matrix were significantly reduced, and resulted in more clearly and accurately displayed neural tissue. Results demonstrated that the scheme with 20 diffusion gradient directions provided better accuracy of structural renderings and could be an optimal scan protocol for clinical DTI application.展开更多
多视图聚类方法随着数据获取途径日益多样化成为研究热点,但大多数聚类方法低估了噪声和数据多结构互补性信息对聚类结果的影响,并且忽略了聚类结果对低秩张量优化过程的反向引导作用。为解决这些问题,提出了基于结构化张量学习的多视...多视图聚类方法随着数据获取途径日益多样化成为研究热点,但大多数聚类方法低估了噪声和数据多结构互补性信息对聚类结果的影响,并且忽略了聚类结果对低秩张量优化过程的反向引导作用。为解决这些问题,提出了基于结构化张量学习的多视图聚类(multi-view clustering based on structured tensor learning,MCSTL)。首先,对初始表示张量进行再次去噪使其更具准确性和鲁棒性;同时,互补地学习局部结构、全局结构和各视图间的高阶相关性,提高表示张量与原始数据本质簇结构的一致性;然后,从跨视图信息融合的亲和矩阵中学习到统一的特征矩阵,利用其隐含的聚类结构信息反向引导表示张量的优化过程;最后,对特征矩阵施加了正交约束,使其提供数据的软标签信息,并对模型进行直接聚类解释。实验表明,MCSTL在6种聚类评价指标上均表现优异,30个指标数据中有27个达到最优,从而充分验证了MCSTL的有效性和优越性。展开更多
Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of Cartesian coordinates, which turns out to be a func...Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of Cartesian coordinates, which turns out to be a function of Lame coefficient and unit vector. By using transform matrix between Cartesian coordinates and orthogonal eurvilinear coordinates, we have deduced a mathematical expression for correcting displacement vector differential in orthogonal curvilinear coordinates, and given a general expression of strain tensor in orthogonal curvilinear coordinates.展开更多
The modified matrix method of construction of wavefield on the free surface of an anisotropic medium is proposed. The earthquake source represented by a randomly oriented force or a seismic moment tensor is placed on ...The modified matrix method of construction of wavefield on the free surface of an anisotropic medium is proposed. The earthquake source represented by a randomly oriented force or a seismic moment tensor is placed on an arbitrary boundary of a layered anisotropic medium. The theory of the matrix propagator in a homogeneous anisotropic medium by introducing a ＂wave propagator＂ is presented. It is shown that the matrix propagator can be represented by a ＂wave propagator＂ in each layer for anisotropic layered medium. The matrix propagator P（z, z0=0） acts on the free surface of the layered medium and generates stress-displacement vector at depth z. The displacement field on the free surface of an anisotropic medium is obtained from the received system of equations considering the radiation condition and that the free surface is stressless. The new method determining source time function in anisotropic medium for three different types of seismic source is validated.展开更多
We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.
近年来,基于张量补全的频谱制图得到了广泛研究.目前用于频谱制图的张量补全算法大多隐含地假设张量具有平衡特性,而对于非平衡张量,难以利用其低秩性估计完整的张量信息,导致补全算法性能受损.本文提出基于重叠Ket增强(Overlapping Ket...近年来,基于张量补全的频谱制图得到了广泛研究.目前用于频谱制图的张量补全算法大多隐含地假设张量具有平衡特性,而对于非平衡张量,难以利用其低秩性估计完整的张量信息,导致补全算法性能受损.本文提出基于重叠Ket增强(Overlapping Ket Augmentation,OKA)和张量列车(Tensor Train,TT)的非平衡频谱制图算法,以解决非平衡张量在应用传统张量补全算法时性能下降的问题.首先使用OKA将低阶高维张量表示为高阶低维张量,在无信息损耗的情况下解决非平衡张量无法利用其低秩性进行张量补全的问题;然后使用TT矩阵化得到较平衡的矩阵,在维度较平衡条件下提高补全算法的精确度;最后利用高阶低维张量的低秩性,使用并行矩阵分解或基于F范数的无奇异值分解(Singular Value Decomposition Free,SVDFree)算法完成张量补全.仿真结果表明,针对非平衡张量,所提方案与现有的张量补全算法相比,可以获得更精确的无线电地图,同时所提SVDFree算法具有更低的计算复杂度.展开更多
文摘In this paper, the solution of the matrix second semi-tensor product equation A∘lX∘lB=Cis studied. Firstly, the solvability of the matrix-vector second semi-tensor product equation is investigated. At the same time, the compatibility conditions, the sufficient and necessary conditions and the specific solution methods for the matrix solution are given. Secondly, we further consider the solvability of the second semi-tensor product equation of the matrix. For each part, several examples are given to illustrate the validity of the results.
基金supported by the National Natural Science Foundation of China (Key technology of neural fiber reconstruction based on MRI),No. 60703045
文摘Diffusion tensor MRI （DT-MRI or DTI） is emerging as an important non-invasive technology for elucidating intemal brain structures. It has recently been utilized to diagnose a series of diseases that affect the integrity of neural systems to provide a basis for neuroregenerative studies. Results from the present study suggested that neural tissue is reconstructed with multiple diffusion-weighted gradient directions DTI, which varies from traditional imaging methods that utilize 6 gradient directions. Simultaneously, the diffusion tensor matrix is obtained by multiple linear regressions from an equation of echo signal intensity. The condition number value and standard deviation of fractional anisotropy for each scheme can be used to evaluate image quality. Results demonstrated that increasing gradient direction to some extent resulted in improved effects. Therefore, the traditional 6 and 15 directions should not be considered optimal scan protocols for clinical DTI application. In a scheme with 20 directions, the condition number and standard deviation of fractional anisotropy of the encoding gradients matrix were significantly reduced, and resulted in more clearly and accurately displayed neural tissue. Results demonstrated that the scheme with 20 diffusion gradient directions provided better accuracy of structural renderings and could be an optimal scan protocol for clinical DTI application.
文摘多视图聚类方法随着数据获取途径日益多样化成为研究热点,但大多数聚类方法低估了噪声和数据多结构互补性信息对聚类结果的影响,并且忽略了聚类结果对低秩张量优化过程的反向引导作用。为解决这些问题,提出了基于结构化张量学习的多视图聚类(multi-view clustering based on structured tensor learning,MCSTL)。首先,对初始表示张量进行再次去噪使其更具准确性和鲁棒性;同时,互补地学习局部结构、全局结构和各视图间的高阶相关性,提高表示张量与原始数据本质簇结构的一致性;然后,从跨视图信息融合的亲和矩阵中学习到统一的特征矩阵,利用其隐含的聚类结构信息反向引导表示张量的优化过程;最后,对特征矩阵施加了正交约束,使其提供数据的软标签信息,并对模型进行直接聚类解释。实验表明,MCSTL在6种聚类评价指标上均表现优异,30个指标数据中有27个达到最优,从而充分验证了MCSTL的有效性和优越性。
文摘Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of Cartesian coordinates, which turns out to be a function of Lame coefficient and unit vector. By using transform matrix between Cartesian coordinates and orthogonal eurvilinear coordinates, we have deduced a mathematical expression for correcting displacement vector differential in orthogonal curvilinear coordinates, and given a general expression of strain tensor in orthogonal curvilinear coordinates.
文摘The modified matrix method of construction of wavefield on the free surface of an anisotropic medium is proposed. The earthquake source represented by a randomly oriented force or a seismic moment tensor is placed on an arbitrary boundary of a layered anisotropic medium. The theory of the matrix propagator in a homogeneous anisotropic medium by introducing a ＂wave propagator＂ is presented. It is shown that the matrix propagator can be represented by a ＂wave propagator＂ in each layer for anisotropic layered medium. The matrix propagator P（z, z0=0） acts on the free surface of the layered medium and generates stress-displacement vector at depth z. The displacement field on the free surface of an anisotropic medium is obtained from the received system of equations considering the radiation condition and that the free surface is stressless. The new method determining source time function in anisotropic medium for three different types of seismic source is validated.
文摘We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.
文摘近年来,基于张量补全的频谱制图得到了广泛研究.目前用于频谱制图的张量补全算法大多隐含地假设张量具有平衡特性,而对于非平衡张量,难以利用其低秩性估计完整的张量信息,导致补全算法性能受损.本文提出基于重叠Ket增强(Overlapping Ket Augmentation,OKA)和张量列车(Tensor Train,TT)的非平衡频谱制图算法,以解决非平衡张量在应用传统张量补全算法时性能下降的问题.首先使用OKA将低阶高维张量表示为高阶低维张量,在无信息损耗的情况下解决非平衡张量无法利用其低秩性进行张量补全的问题;然后使用TT矩阵化得到较平衡的矩阵,在维度较平衡条件下提高补全算法的精确度;最后利用高阶低维张量的低秩性,使用并行矩阵分解或基于F范数的无奇异值分解(Singular Value Decomposition Free,SVDFree)算法完成张量补全.仿真结果表明,针对非平衡张量,所提方案与现有的张量补全算法相比,可以获得更精确的无线电地图,同时所提SVDFree算法具有更低的计算复杂度.
