Shape encoding and shape analysis are valuable tools for comparing shapes and for dimensionality reduction. A specific framework for shape analysis is the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, which is capable of shape matching and dimensionality reduction. Researchers have recently introduced neural networks into this framework. However, these works can not match more than two objects simultaneously or have suboptimal performance in shape variability modeling. The latter limitation occurs as the works do not use state-of-the-art shape encoding methods. Moreover, the literature does not discuss the connection between the LDDMM Riemannian distance and the Riemannian geometry for deep learning literature. Our work aims to bridge this gap by demonstrating how LDDMM can integrate Riemannian geometry into deep learning. Furthermore, we discuss how deep learning solves and generalizes shape matching and dimensionality reduction formulations of LDDMM. We achieve both goals by designing a novel implicit encoder for shapes. This model extends a neural network-based algorithm for LDDMM-based pairwise registration, results in a nonlinear manifold PCA, and adds a Riemannian geometry aspect to deep learning models for shape variability modeling. Additionally, we demonstrate that the Riemannian geometry component improves the reconstruction procedure of the implicit encoder in terms of reconstruction quality and stability to noise. We hope our discussion paves the way to more research into how Riemannian geometry, shape/image analysis, and deep learning can be combined.

翻译：形状编码与形状分析是比较形状及降维的重要工具。大形变微分同胚度量映射（LDDMM）框架是形状分析的一种特定方法，能够实现形状匹配与降维。研究者近期将神经网络引入该框架，但现有工作无法同时匹配两个以上物体，或在形状变异性建模中表现欠佳。后者源于这些工作未采用最先进的形状编码方法。此外，现有文献未探讨LDDMM黎曼距离与深度学习领域中黎曼几何之间的关联。本研究旨在弥合这一鸿沟，通过展示LDDMM如何将黎曼几何融入深度学习，并探讨深度学习如何求解与泛化LDDMM的形状匹配及降维公式。我们通过设计一种新型隐式形状编码器实现上述目标。该模型扩展了基于LDDMM成对配准的神经网络算法，构建了非线性流形主成分分析，并为形状变异性建模的深度学习模型引入黎曼几何维度。进一步地，我们证明了黎曼几何成分在重构质量与噪声稳定性方面能提升隐式编码器的重构过程。希望本文的讨论能为黎曼几何、形状/图像分析与深度学习的交叉研究开辟新路径。