SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 445-473, January 2020.
In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat $L^2$ metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.

中文翻译：

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平面曲线空间上一般弹性度量的简化变换

SIAM 成像科学杂志，第 13 卷，第 1 期，第 445-473 页，2020 年 1 月。
在计算机视觉问题的形状分析方法中，人们将形状视为无限维黎曼流形中的点，从而促进统计计算的算法，例如形状之间的测地线距离和形状集合的平均。这些算法的性能在很大程度上取决于黎曼度量的选择。在平面曲线形状的设置中，注意力主要集中在一阶 Sobolev 度量的两个参数族上，称为弹性度量。由于存在特定参数值的简化坐标变换，例如众所周知的平方根速度变换，它们特别有用。在本文中，我们将现有文献中出现的变换扩展到一系列等距，它将任何弹性指标带到平坦的 $L^2$ 指标。我们还扩展了变换以处理分段线性曲线，并证明了在此设置中微分同胚组上存在最佳匹配。我们用多个开放曲线和封闭曲线的形状测地线示例结束了这篇论文。我们还在一个简单的分类实验中展示了我们的方法的好处。