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首页> 外文期刊>Geoscience and Remote Sensing Letters, IEEE >Calculation of Geodesic Distances in Nonlinear Mixing Models: Application to the Generalized Bilinear Model
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Calculation of Geodesic Distances in Nonlinear Mixing Models: Application to the Generalized Bilinear Model

机译：非线性混合模型中测地距离的计算：在广义双线性模型中的应用

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摘要

Recently, several nonlinear techniques have been proposed in hyperspectral image processing for classification and unmixing applications. A popular data-driven approach for treating nonlinear problems employs the geodesic distances on the data manifold as property of interest. These geodesic distances are approximated by the shortest path distances in a nearest neighbor graph constructed in the data cloud. Although this approach often works well in practical applications, the graph-based approximation of these geodesic distances often fails to capture correctly the true nonlinear structure of the manifold, causing deviations in the subsequent algorithms. On the other hand, several model-based nonlinear techniques have been introduced as well and have the advantage that one can, in theory, calculate the geodesic distances analytically. In this letter, we demonstrate how one can calculate the true geodesics, and their lengths, on any manifold induced by a nonlinear hyperspectral mixing model. We introduce the required techniques from differential geometry, show how the constraints on the abundances can be integrated in these techniques, and present a numerical method for finding a solution of the geodesic equations. We demonstrate this technique on the recently developed generalized bilinear model, which is a flexible model for the nonlinearities introduced by secondary reflections. As an application of the technique, we demonstrate that multidimensional scaling applied to these geodesic distances can be used as a preprocessing step to linear unmixing, yielding better unmixing results on nonlinear data when compared to principal component analysis and outperforming ISOMAP.
机译：最近，在高光谱图像处理中已经提出了几种用于分类和解混应用的非线性技术。一种用于处理非线性问题的流行数据驱动方法采用数据流形上的测地距离作为感兴趣的属性。这些测地距离由数据云中构造的最近邻图中的最短路径距离近似。尽管此方法在实际应用中通常效果很好，但是这些测地距离的基于图的逼近常常无法正确捕获流形的真实非线性结构，从而导致后续算法出现偏差。另一方面，还引入了几种基于模型的非线性技术，这些技术具有理论上可以解析地计算测地距离的优点。在这封信中，我们演示了如何在非线性高光谱混合模型引起的任何流形上计算真实的测地线及其长度。我们从微分几何中介绍了所需的技术，展示了如何将丰度约束整合到这些技术中，并提出了一种找到测地线方程解的数值方法。我们在最近开发的广义双线性模型上证明了该技术，该模型是用于由二次反射引入的非线性的灵活模型。作为该技术的一项应用，我们证明了应用于这些测地距离的多维缩放比例可以用作线性分解的预处理步骤，与主成分分析和优于ISOMAP相比，在非线性数据上产生更好的分解结果。

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