A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

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.

Figures (9)

• Figure 1: Illustration of the discontinuous image registration based on Lie groupoid and algebroid. (left) Illustration of the geodesic flow on groupoids $\mathrm{DDiff}(M) \rightrightarrows \mathrm{V}(M)$ of discontinuous diffeomorphisms with respect to the metric $\left\langle \cdot, \cdot \right\rangle_{\text{DVect}(M)}$, governed by the Euler-Arnold equation \ref{['eq: Euler-Arnold']}. (right) The action of discontinuous diffeomorphism groupoid on the moving image $I_m$.
• Figure 2: An illustration of image registration.
• Figure 3: (left) Illustration of the geodesic flow on groups $\mathrm{Diff}(\Omega)$ of diffeomorphisms with respect to the right-invariant metric $\left\langle \cdot, \cdot \right\rangle_{\text{Vect}(\Omega)}$. (right) The action of the diffeomorphism group on the moving image $I_m$.
• Figure 4: The structure of a groupoid $\mathcal{G} \rightrightarrows B$.
• Figure 5: Illustration of the composition rule of two groupoid elements $(\Gamma_1, \Gamma_2, \phi^+, \phi^-)$ and $(\Gamma_2, \Gamma_3, \psi^+, \psi^-)$ in the Lie groupoid $\mathrm{DDiff}(M)$, resulting in $(\Gamma_1, \Gamma_3, \psi^+ \phi^+, \psi^- \phi^-)$.

Theorems & Definitions (35)

• Definition 2.1: Groupoid izosimov2018vortex
• Definition 2.2: Lie groupoid izosimov2018vortex
• Definition 2.3: Lie algebroid izosimov2018vortex
• Definition 3.1
• Theorem 3.2
• proof
• Lemma 3.3: izosimov2018vortex
• Lemma 3.4
• proof
• Theorem 3.5