AI 中文总结
研究如何用闵可夫斯基张量测试球面上随机场的统计各向同性。通过构建连通分量的MTs及相关函数，生成四组随机场进行测试，能区分尺度依赖对齐及两种剪切模型，偶极调制在取向统计中信号不显著。
AI 中文摘要
我们考虑一类形态描述符——闵可夫斯基张量（MTs）如何用于测试球面上随机场的统计各向同性。MTs的定义涉及张量积分，这在弯曲流形上是个模糊操作。文献中现有估计器应用于二维球面上的场时，要么明确破坏协方差，要么因切空间几何旋转人为使MT各向同性。为避免这些问题，我们构建单个连通分量的MTs，并用以建立其相对取向的相关函数$\xi_{\pm}(\theta,\nu)$。这些相关函数由标量构建，具有协变性，可用于搜索结构对齐与尺度（角间距）的关系。我们在$S^{2}$上生成四组随机场——各向同性、偶极调制、全局剪切和局部剪切，并展示连通分量相关函数如何区分尺度依赖对齐。对于剪切场，$\xi_{+}$的衰减尺度衡量对齐的角相干性，$\xi_{-}$在大间距处的幅度衡量其全局相干分数，可区分两种剪切模型。相比之下，偶极调制在取向统计中不产生显著信号。我们通过分析表明，其对闵可夫斯基张量无迹分量的影响在调制幅度$\lambda \ll 1$时为二阶。
英文摘要
We consider how a class of morphological descriptors, the Minkowski Tensors (MTs), can be used to test the statistical isotropy of random fields on the sphere. The definition of the MTs involves an integral of a tensor, which is an ambiguous operation on a curved manifold. We find that existing estimators in the literature, when applied to fields on the two-sphere, either explicitly break covariance or artificially isotropize the MT due to a geometric rotation of tangent spaces. To evade these issues, we construct the MTs of individual connected components and use them to build correlation functions $ξ_{\pm}(θ,ν)$ of their relative orientations. These correlation functions are built from scalars and are therefore covariant, and can be used to search for alignment of structures as a function of scale (angular separation). We generate four sets of random fields on $S^{2}$ -- isotropic, dipole modulated, globally-sheared and locally-sheared, and show how the connected component correlation functions can distinguish scale dependent alignments. For the sheared fields, the decay scale of $ξ_{+}$ measures the angular coherence of the alignment, and the amplitude of $ξ_{-}$ at large separations measures its globally coherent fraction, distinguishing the two shear models. In contrast, dipole modulation generates no significant signal in the orientation statistics. We show analytically that its effect on the traceless component of the Minkowski tensor is second order in the modulation amplitude $λ\ll 1$.
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