A Generalized Lyapunov Feature for Dynamical Systems on Riemannian Manifolds
A Generalized Lyapunov Feature for Dynamical Systems on Riemannian Manifolds
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黎曼流形动力系统的广义李雅普诺夫特征
DOI：
10.5244/c.29.diffcv.4
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
P. Turaga
中科院分区：
文献类型：
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作者：
Rushil Anirudh;Vinay Venkataraman;P. Turaga
通常通过视觉传感器观察到动态现象，例如人类活动，动态场景和移动人群，从而导致随时间采样的特征轨迹。可以通过考虑时间变化并考虑变化来准确地对这种现象进行建模。对于轨迹足够不同的问题，弹性指标可以提供速度不变的距离，但是对于更复杂的问题，例如细粒度活性分类，需要利用更高阶的动力学特性。对于欧几里得空间中的功能，诸如人群监测，动态场景识别和人类运动质量分析之类的应用程序已经取得了很大的成功。在本文中，我们提出了最大的Riemannian Lyapunov指数（L-RLE），这是最大的Lyapunov指数对Riemannian歧管的首次概括。最大的Lyapunov指数是量化欧几里得空间中信号内混乱量的经典措施，并允许我们利用各种应用程序来利用高级动力学。我们显示了L-RLE对格拉斯曼和So（3）谎言组的两个流形的有效性。通过将人类的作用建模为动态过程在里曼尼亚歧管上演变，我们表明l-rle可以准确地测量每个动作中的混乱量。我们表明，我们的度量是对最大的欧几里得莱普诺诺夫指数（L-ele）的良好概括，并且不太容易受到任意扭曲的影响。
Dynamic phenomena such as human activities, dynamic scenes, and moving crowds are commonly observed through visual sensors, resulting in feature trajectories sampled in time. Such phenomena can be accurately modeled by taking the temporal variations and changes into account. For problems where the trajectories are sufficiently different, elastic metrics can provide distances that are invariant to speed, but for more complex problems such as fine grained activity classification, one needs to exploit higher order dynamical properties. For features in the Euclidean space, applications such as crowd monitoring, dynamic scene recognition and human movement quality analysis have found a lot of success this way. In this paper we propose the largest Riemannian Lyapunov exponent (L-RLE), which is the first generalization of the largest Lyapunov exponent to Riemannian manifolds. The largest Lyapunov exponent is a classic measure to quantify the amount of chaos within signals in the Euclidean space, and allows us to exploit higher order dynamics for various applications. We show the effectiveness of the L-RLE on two manifolds the Grassmann and the SO(3) lie group. By modeling human actions as dynamic processes evolving on Riemannian manifolds, we show that L-RLE can measure the amount of chaos within each action accurately. We show that our measure is a good generalization of largest Euclidean Lyapunov exponent (L-ELE), and is less susceptible to arbitrary distortions.