In shape analysis, one of the fundamental problems is to align curves or surfaces before computing a (geodesic) distance between these shapes. To find the optimal reparametrization realizing this alignment is a computationally demanding task which leads to an optimization problem on the diffeomorphism group. In this paper, we construct approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms to solve the approximation problem. We propose a practical algorithm implemented in PyTorch which is applicable both to unparametrized curves and surfaces. We derive universal approximation results and obtain bounds for the Lipschitz constant of the obtained compositions of diffeomorphisms.

翻译：在形状分析中,一个根本问题是,在计算这些形状之间的(地貌)距离之前,对曲线或表面进行对齐。要找到实现这种对齐的最佳再平衡,就是一项计算上要求很高的任务,这会导致地貌畸形群体出现优化问题。在本文中,我们用基本地貌形态的构成来构建方向-保持地貌形态的近似值,以解决近似问题。我们提议在PyTorch实施一种实用的算法,既适用于未对称的曲线和表面。我们得出普遍近似结果,也为Lipschitz常数获得的地貌形态构成的界限。