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- or 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 investigate how the Riemannian geometry improves latent modeling and is required for a proper mean-variance analysis. To highlight the benefit of resolution independence for LDDMM-based data variability modeling, we show that our approach outperforms current neural network-based LDDMM latent code models. Our work paves the way for more research into how Riemannian geometry, shape respectively 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）在计算机图形学和医学领域被用于图谱构建、统计隐变量建模以及成对与群组配准。近年来，研究者们提出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隐变量建模方法的一个局限：它们要么（i）依赖于分辨率，要么（ii）忽略了进行恰当均值-方差分析所需的数据或问题特定几何结构。为此，我们通过设计一种基于分辨率无关的隐式神经表示的新型编码器来克服这一局限。该编码器为基于LDDMM的统计隐变量建模实现了分辨率不变性。此外，编码器将LDDMM黎曼几何引入分辨率无关的深度学习模型，以用于统计隐变量建模。我们探究了黎曼几何如何改善隐变量建模，以及为何其是进行恰当均值-方差分析所必需的。为突显分辨率无关性在基于LDDMM的数据变异性建模中的优势，我们展示了本方法优于当前基于神经网络的LDDMM隐编码模型。本研究为黎曼几何、形状/图像分析与深度学习的进一步结合探索了新的路径。