In this paper, we introduce a deformation based representation space for curved shapes in $\mathbb {R}^{n}$. Given an ordered set of points sampled from a curved shape, the proposed method represents the set as an element of a finite dimensional matrix Lie group. Variation due to scale and location are filtered in a preprocessing stage, while shapes that vary only in rotation are identified by an equivalence relationship. The use of a finite dimensional matrix Lie group leads to a similarity metric with an explicit geodesic solution. Subsequently, we discuss some of the properties of the metric and its relationship with a deformation by least action. Furthermore, invariance to reparametrization or estimation of point correspondence between shapes is formulated as an estimation of sampling function. Thereafter, two possible approaches are presented to solve the point correspondence estimation problem. Finally, we propose an adaptation of k-means clustering for shape analysis in the proposed representation space. Experimental results show that the proposed representation is robust to uninformative cues, e.g., local shape perturbation and displacement. In comparison to state of the art methods, it achieves a high precision on the Swedish and the Flavia leaf datasets and a comparable result on MPEG-7, Kimia99 and Kimia216 datasets.

中文翻译：

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基于变形的弯曲形状表示

在本文中，我们介绍了基于变形的曲面形状表示空间 $ \ mathbb {R} ^ {n} $。给定从曲线形状采样的点的有序集合，所提出的方法将该集合表示为有限维矩阵李群的元素。由比例和位置引起的变化在预处理阶段中进行过滤，而仅在旋转中变化的形状由等价关系标识。有限维矩阵李群的使用导致了具有显式测地线解的相似性度量。随后，我们讨论了指标的一些属性及其与最小动作变形。此外，将重新参数化的不变性或形状之间的点对应关系的估计表示为采样函数的估计。此后，提出了两种可能的方法来解决点对应估计问题。最后，我们提出了一种k-均值聚类的自适应方法，用于在提出的表示空间中进行形状分析。实验结果表明，所提出的表示对于非信息性提示（例如局部形状扰动和位移）具有鲁棒性。与最先进的方法相比，它在瑞典和Flavia叶数据集上具有很高的精度，在MPEG-7，Kimia99和Kimia216数据集上具有可比的结果。