This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold spaces, which allow to formulate and tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. We study in detail the construction of kernel metrics on varifold spaces and the resulting topological properties of those metrics and then propose a mathematical model for diffeomorphic registration of varifolds under a specific group action which we formulate in the framework of optimal control theory. A second important part of the paper focuses on the discrete aspects. Specifically, we address the problem of optimal finite approximations (quantization) for those metrics and show a \(\varGamma \)-convergence property for the corresponding registration functionals. Finally, we develop numerical pipelines for quantization and registration before showing a few preliminary results for one- and two-dimensional varifolds.

本文关注的是形变的表示，近似和微形配准的曲折理论及其应用。它的目的之一是综合和扩展一些迄今为止的工作，到目前为止，主要是在子流形比较和匹配的背景下使用了此框架。在这项工作中，我们取而代之的是考虑作用于一般变量空间上的变形模型，该模型可以为更广泛的一类几何对象制定和处理微分配准问题，并导致更加通用的算法流水线。我们详细研究了在变量空间上的内核度量的构造以及这些度量的拓扑性质，然后提出了在最优控制理论框架内制定的特定群体作用下变量的微分配准的数学模型。本文的第二个重要部分集中在离散方面。具体来说，我们针对这些指标解决了最佳有限逼近（量化）问题，并展示了\（\ varGamma \）-对应注册功能的收敛属性。最后，在展示一些一维和二维变量的初步结果之前，我们开发了用于量化和配准的数字流水线。