Hilbert流形上均值的非参数估计及等高线均值的外生性分析
Nonparametric Estimation of Means on Hilbert Manifolds and Extrinsic Analysis of Mean Shapes of Contours
摘要：
针对高层次数字图像分析中非参数推断问题，提出了一种基于Hilbert流形的非参数数据分析方法，重点研究了在这种样本空间上的概率分布.为了对这些方法进行推断，我们从功能数据分析中引入邻域假设的概念，并推导出单样本检验。然后，我们考虑分析的形状轮廓躺在平面上。通过将这些形状的相应样本空间（Hilbert流形）嵌入到Hilbert Schmidt算子空间中，我们可以定义平面轮廓的外平均形状及其样本类似物。我们适用于这个问题的一般方法，同时考虑到利用数字成像数据时所面临的计算限制。比较了另一种轮廓形状分析方法的计算量。
英文摘要：
Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert-Schmidt operators, we can define extrinsic mean shapes of planar contours and their sample analogues. We apply the general methods to this problem while considering the computational restrictions faced when utilizing digital imaging data. Comparisons of computational cost are provided to another method for analyzing shapes of contours.
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