Diffeomorphic registration frameworks such as Large Deformation Diffeomorphic Metric Mapping (LDDMM) are used in computer graphics and the medical domain for atlas building, statistical latent modeling, and pairwise and groupwise registration. In recent years, researchers have developed neural network-based approaches regarding diffeomorphic registration to improve the accuracy and computational efficiency of traditional methods. In this work, we focus on a limitation of neural network-based atlas building and statistical latent modeling methods, namely that they either are (i) resolution dependent or (ii) disregard any data/problem-specific geometry needed for proper mean-variance analysis. In particular, we overcome this limitation by designing a novel encoder based on resolution-independent implicit neural representations. The encoder achieves resolution invariance for LDDMM-based statistical latent modeling. Additionally, the encoder adds LDDMM Riemannian geometry to resolution-independent deep learning models for statistical latent modeling. We showcase that the Riemannian geometry aspect improves latent modeling and is required for a proper mean-variance analysis. Furthermore, to showcase the benefit of resolution independence for LDDMM-based data variability modeling, we show that our approach outperforms another neural network-based LDDMM latent code model. Our work paves a way to more research into how Riemannian geometry, shape/image analysis, and deep learning can be combined.

翻译：微分同胚配准框架（如大变形微分同胚度量映射LDDMM）在计算机图形学与医学领域被广泛应用于图谱构建、统计潜变量建模及成对/组群配准。近年来，研究者开发了基于神经网络的微分同胚配准方法，以提升传统方法的精度与计算效率。本文聚焦于基于神经网络的图谱构建与统计潜变量建模方法的一个局限性——它们要么（i）依赖分辨率，要么（ii）忽略数据/问题特有的几何结构，导致无法进行恰当的均值-方差分析。具体而言，我们通过设计基于分辨率无关隐式神经表示的新型编码器克服了这一局限。该编码器实现了LDDMM统计潜变量建模的分辨率不变性，同时将LDDMM黎曼几何引入分辨率无关的深度学习模型以支持统计潜变量建模。我们证明了黎曼几何特性能够改进潜变量建模，且是进行恰当均值-方差分析的必要条件。此外，为展示分辨率无关性对LDDMM数据变异性建模的益处，我们证明所提方法优于另一种基于神经网络的LDDMM潜码模型。本研究为黎曼几何、形状/图像分析与深度学习的融合研究开辟了新路径。