A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration
This work addresses a specific problem in medical image analysis for researchers and practitioners dealing with sliding organ motion, representing an incremental advancement over existing LDDMM methods.
The paper tackles the limitation of traditional image registration methods in handling discontinuous sliding motion by proposing a novel mathematical framework using diffeomorphism groupoids and algebroids, which allows for discontinuities along boundaries while maintaining diffeomorphism within homogeneous regions.
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.