提出新数学框架，让图像配准能处理断裂滑动。
A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration
- 用李群胚扩展传统方法，支持图像内断裂滑动
- 推导出控制非连续变形的欧拉-阿诺德方程
- 适合医学图像中器官分界处的精准配准
本文提出一种基于微分同胚群胚与代数胚的新数学框架，用于分段微分同胚图像配准，可建模不连续滑动。传统大变形微分同胚度量映射（LDDMM）基于李群，假设速度场连续光滑，难以处理断裂滑动。为此，我们将微分同胚李群扩展为不连续微分同胚李群胚，在同质区域保持微分同胚性的同时，允许沿滑动边界存在间断。我们对相关数学结构（包括李代数胚及其对偶）进行了严格分析，并推导出控制最优流的特定欧拉-阿诺德方程。数值实验验证了该方法的有效性。
原文摘要 · Abstract (English)
In this paper, we propose a novel mathematical framework for piecewise diffeomorphic image registration that involves discontinuous sliding motion using a diffeomorphism groupoid and algebroid approach. The traditional Large Deformation Diffeomorphic Metric Mapping (LDDMM) registration method builds on Lie groups, which assume continuity and smoothness in velocity fields, limiting its applicability in handling discontinuous sliding motion. To overcome this limitation, we extend the diffeomorphism Lie groups to a framework of discontinuous diffeomorphism Lie groupoids, allowing for discontinuities along sliding boundaries while maintaining diffeomorphism within homogeneous regions. We provide a rigorous analysis of the associated mathematical structures, including Lie algebroids and their duals, and derive specific Euler-Arnold equations to govern optimal flows for discontinuous deformations. Numerical tests are performed to validate the efficiency of the proposed approach.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。