A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

2026-03-12

AI Summary

This work addresses a key limitation of traditional Lie group–based image registration methods, such as LDDMM, which assume globally smooth velocity fields and thus struggle to model deformations involving sliding discontinuities. To overcome this, the authors propose a novel framework grounded in diffeomorphic Lie groupoids and their associated Lie algebroids, extending diffeomorphic registration for the first time to piecewise-diffeomorphic settings that permit sliding discontinuities along boundaries. By employing variational principles on the Lie groupoid and its dual space, they derive Euler–Arnold equations governing the evolution of discontinuous deformation flows and provide a numerical scheme for their solution. Experiments demonstrate that the method effectively captures sliding boundaries while preserving diffeomorphic regularity within homogeneous regions, achieving a favorable balance between registration accuracy and mathematical rigor.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

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. Some numerical tests are performed to validate the efficiency of the proposed approach.

Problem

• image registration
• discontinuous deformation
• diffeomorphism
• sliding motion
• LDDMM

Innovation

• diffeomorphism groupoid
• Lie algebroid
• discontinuous image registration
• sliding motion
• Euler-Arnold equations