This paper introduces a new framework for surface analysis derived from the general setting of elastic Riemannian metrics on shape spaces. Traditionally, those metrics are defined over the infinite dimensional manifold of immersed surfaces and satisfy specific invariance properties enabling the comparison of surfaces modulo shape preserving transformations such as reparametrizations. The specificity of our approach is to restrict the space of allowable transformations to predefined finite dimensional bases of deformation fields. These are estimated in a data-driven way so as to emulate specific types of surface transformations. This allows us to simplify the representation of the corresponding shape space to a finite dimensional latent space. However, in sharp contrast with methods involving e.g. mesh autoencoders, the latent space is equipped with a non-Euclidean Riemannian metric inherited from the family of elastic metrics. We demonstrate how this model can be effectively implemented to perform a variety of tasks on surface meshes which, importantly, does not assume these to be pre-registered or to even have a consistent mesh structure. We specifically validate our approach on human body shape and pose data as well as human face and hand scans for problems such as shape registration, interpolation, motion transfer or random pose generation.

本文介绍了一种新的表面分析框架，该框架源自形状空间的弹性黎曼度量的一般设置。传统上，这些度量是在浸没表面的无限维流形上定义的，并满足特定的不变性，从而能够比较表面的模形状保持变换，例如重新参数化。我们方法的特殊性是将允许变换的空间限制为变形场的预定义有限维基。这些是以数据驱动的方式估计的，以便模拟特定类型的表面变换。这使我们能够将相应的形状空间的表示简化为有限维度的潜在空间。然而，与涉及网格自动编码器的方法形成鲜明对比的是，潜在空间配备了从弹性度量家族继承的非欧几里得黎曼度量。我们演示了如何有效地实现此模型以在表面网格上执行各种任务，重要的是，这些网格不会假设这些网格是预先注册的，甚至不具有一致的网格结构。我们专门验证了我们的方法，用于人体形状和姿势数据以及人脸和手部扫描，以解决形状配准、插值、运动传输或随机姿势生成等问题。