Grassmann流形上的持久同调及其在高光谱影像分析中的应用
Persistent Homology on Grassmann Manifolds for Analysis of Hyperspectral Movies
摘要：
在复杂数据集中，特征结构或形状的存在对于数学数据分析越来越重要。这一认识激发了新工具的发展，如持久同源性，探索拓扑不变量，或功能，在大型数据集。在本文中，我们应用持续同源性的气体羽流的表征的时间依赖序列的高光谱立方体，即分析的4路阵列。我们调查高光谱电影的长波长红外数据监测实验释放到空气中的化学模拟物。我们的方法模型的高光谱数据立方体内的感兴趣的区域作为点上的真正的格拉斯曼流形$G(k, n)$（其点参数化的$k$ -维子空间的$\mathbb{R}^n$），对比我们的方法在欧几里德空间中的更标准的框架。这种方法的优点是，它允许高光谱电影中的时间片序列以这样的方式被折叠成点序列，即，切片内和切片之间的一些关键结构由格拉斯曼流形上的点编码。这促使搜索拓扑特征，与高光谱电影的帧的演变，在格拉斯曼流形上的相应点内。所提出的数学模型提供了大数据集的处理，同时保留有价值的歧视性信息。在本文中，我们将讨论如何嵌入我们的数据在格拉斯曼流形，拓扑数据分析，捕捉动态事件发生的化学羽流被释放和演变。
英文摘要：
The existence of characteristic structure, or shape, in complex data sets has been recognized as increasingly important for mathematical data analysis. This realization has motivated the development of new tools such as persistent homology for exploring topological invariants, or features, in large data sets. In this paper we apply persistent homology to the characterization of gas plumes in time dependent sequences of hyperspectral cubes, i.e. the analysis of 4-way arrays. We investigate hyperspectral movies of Long-Wavelength Infrared data monitoring an experimental release of chemical simulant into the air. Our approach models regions of interest within the hyperspectral data cubes as points on the real Grassmann manifold $G(k, n)$ (whose points parameterize the $k$-dimensional subspaces of $\mathbb{R}^n$), contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows a sequence of time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological features, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed mathematical model affords the processing of large data sets while retaining valuable discriminatory information. In this paper, we discuss how embedding our data in the Grassmann manifold, together with topological data analysis, captures dynamical events that occur as the chemical plume is released and evolves.
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