Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of m×m symmetric random matrices, D, observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, D̄. When D̄ has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same D̄ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with m = 3, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order t ≥ 4 with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model.

中文翻译：

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具有各向同性高斯噪声的随机矩阵的特征值和扩散张量成像实验的设计。


不同物理属性的张量值和矩阵值测量在材料科学和医学成像应用中越来越可用。这种多元数据的特征值和特征向量提供了新颖且独特的信息，但代价是需要更复杂的统计分析。在这项工作中，我们推导了使用各向同性矩阵变量高斯噪声观察到的 m×m 对称随机矩阵 D 的特殊但重要情况下的特征值和特征向量的分布。这些分布的属性很大程度上取决于平均张量/矩阵 D̄ 的对称性。当 D̄ 具有重复特征值时，D 的特征值不是渐近高斯分布的，并且在相同 D̄ 特征空间对应的特征值之间观察到排斥。我们将这些结果应用于扩散张量成像（DTI），其中 m = 3，解决了检测扩散张量对称性的重要问题，并寻求可能产生各向同性高斯分布的实验设计。在 3 维情况下，当平均张量是球对称且噪声是高斯且各向同性时，前三个特征值中心矩统计量的渐近分布很简单，可以用来检验各向同性。为了应用此类测试，我们使用单位球上具有恒定权重的 t ≥ 4 阶求积规则来设计 DTI 实验，其特性是基础真张量的各向同性意味着 Fisher 信息的各向同性。我们还解释了使用模拟 DTI 数据和莱斯噪声模型的方法的潜在影响。