A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

TL;DR

This paper introduces a new mathematical framework using diffeomorphism groupoids and algebroids for image registration that handles discontinuous sliding motions, extending traditional smooth methods.

Contribution

It extends the LDDMM framework to discontinuous diffeomorphisms via groupoids and algebroids, enabling registration with sliding boundaries.

Findings

Mathematical analysis of discontinuous diffeomorphism groupoids and algebroids.

Derivation of Euler-Arnold equations for optimal flows with discontinuities.

Numerical tests validate the efficiency of the proposed method.

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. Numerical tests are performed to validate the efficiency of the proposed approach.