密度和形状的切线函数典型相关分析及其在多模态成像数据中的应用
Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data
摘要：
由成像数据产生的函数数据通常假定在无限维流形中的值。揭示跨医学成像模态从相同对象提取的两个或更多个这样的非线性功能数据之间的关联可以帮助开发个性化治疗策略。我们提出了一种典型的相关分析方法，成对的概率密度或形状的封闭的平面曲线，常规用于生物医学研究，它结合了一个方便的线性化和降维的数据，使用切空间坐标。利用相应的流形是单位希尔伯特球面的子流形的事实，我们描述了如何可以很容易地计算功能数据对象的有限维表示，然后方便使用标准的多元典型相关分析方法。我们进一步构造和可视化规范变量方向直接在空间的密度或形状。该方法的效用是通过数值模拟和性能的多形性胶质母细胞瘤脑肿瘤的磁共振成像数据集证明。
英文摘要：
It is quite common for functional data arising from imaging data to assume values in infinite-dimensional manifolds. Uncovering associations between two or more such nonlinear functional data extracted from the same object across medical imaging modalities can assist development of personalized treatment strategies. We propose a method for canonical correlation analysis between paired probability densities or shapes of closed planar curves, routinely used in biomedical studies, which combines a convenient linearization and dimension reduction of the data using tangent space coordinates. Leveraging the fact that the corresponding manifolds are submanifolds of unit Hilbert spheres, we describe how finite-dimensional representations of the functional data objects can be easily computed, which then facilitates use of standard multivariate canonical correlation analysis methods. We further construct and visualize canonical variate directions directly on the space of densities or shapes. Utility of the method is demonstrated through numerical simulations and performance on a magnetic resonance imaging dataset of Glioblastoma Multiforme brain tumors.
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