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This paper introduces a novel mathematical framework for discontinuous image registration using diffeomorphism groupoids and algebroids, extending traditional LDDMM methods. By generalizing diffeomorphism Lie groups to discontinuous diffeomorphism Lie groupoids, the approach allows for discontinuities along sliding boundaries while preserving diffeomorphism within homogeneous regions. The authors derive Euler-Arnold equations to govern optimal flows for discontinuous deformations and validate the approach through numerical experiments.
Overcome the limitations of traditional image registration methods by enabling discontinuous sliding motion with a novel diffeomorphism groupoid and algebroid framework.
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.
