One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

翻译：形状分析中的基本问题之一是在计算形状间测地距离之前对齐曲线或曲面。实现这种对齐所需的最优重参数化是一项计算密集型任务，通常通过在微分同胚群上求解优化问题来完成。本文提出一种通过初等微分同胚复合构造保定向微分同胚逼近的算法。该算法基于PyTorch实现，同时适用于无参数曲线和曲面。此外，我们证明了所构造架构的普适逼近性质，并得到了所得微分同胚的Lipschitz常数界限。