A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

Lili Bao; Bin Xiao; Shihui Ying; Stefan Sommer

math.GRMar 12, 2026

A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

Authors: Lili Bao, Bin Xiao, Shihui Ying, Stefan Sommer

Organizations: Department of Computer Science, University of Copenhagen, Universitetsparken 1, Copenhagen 2100, Denmark · Institute for Medical Imaging Technology, Ruijin Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 201800, China · Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, China · School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China

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.

Figures & tables

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 ( 3.37 ). (right) The action of the 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 $(\Gamma_{1},\Gamma_{3},\psi^{+}\phi^{+},\psi^{-}\phi^{-})$ .

Figure 6: The structure of the Lie groupoid $\mathrm{DDiff}(M)\rightrightarrows\mathrm{V}(M)$ .

Panel of Figure 7 — Figure 7: Registration results on synthetic rectangle images with sliding motion. 7 and 7 are the moving image and fixed image; 7 - 7 are the deformed moving images using TV, LDDMM, and the proposed diffeomorphism groupoid, respectively; 7 - 7 are the deformed grids obtained by TV, LDDMM, and the proposed diffeomorphism groupoid, respectively.

Panel of Figure 8 — Figure 8: Registration results on synthetic wheel images with sliding motion. 8 and 8 are the moving image and fixed image; 8 - 8 are the deformed moving images using TV, LDDMM, and the proposed diffeomorphism groupoid, respectively; 8 - 8 are the deformed grids obtained by TV, LDDMM, and the proposed diffeomorphism groupoid, respectively.

Data	Methods	$Re\_SSD(\%)$	$NCC$	$SSIM$
Rectangle	Before	100	0.9228	0.8071
	TV	7.78	0.9949	0.9219
	LDDMM	10.98	0.9934	0.9396
	Proposed	3.37	0.9986	0.9925
Wheel	Before	100	0.9957	0.9690
	TV	61.09	0.9974	0.9720

Table 1: Comparison of

Re\_SSD(\%)

NCC

and

SSIM

by different methods. Bold values are the best values.

Panel of Figure 9 — Figure 9: Registration results on real lung images with sliding motion. Rows from top to bottom display axial, coronal, and sagittal views. The first and second columns present the moving and fixed images, respectively. The third and fourth columns depict the residual difference images before and after registration obtained by the proposed method, respectively. The last column shows the deformed moving images by the proposed method with deformation fields overlaid, respectively, in which the sliding boundaries are shown in yellow.

Deformable Image Registration

Deformable image registration aligns images by estimating local spatial transformations that account for anatomical change, motion, and differences between scans.

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Sep 9, 2026J/K move · Enter open · S save

Two-Parameter Flow Map Learning for Continuous-Time Diffeomorphic Image Registration

Mohammadjavad Matinkia, Nilanjan Ray

University of Alberta, Canada

Medical Image Registration Flow Map Learning Deformable Image Registration

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Diffeomorphic image registration is central to medical image analysis, enabling anatomically consistent alignment across subjects. Most learning-based diffeomorphic methods model autonomous ODEs(ordinary differential equations) by parameterizing a stationary velocity field and recovering deformations via scaling-and-squaring. While non-autonomous ODEs with time-dependent velocities increase expressiveness, existing approaches rely on numerical integration to implicitly enforce flow structure that entangles model expressiveness with discretization accuracy. We propose a framework to directly learn the continuous-time solution of a non-autonomous ODE formulated as a two-parameterflow map. By enforcing cocycle consistency, a fundamental structural property of time-varying flows, we learn the flow maps without time discretization and velocity integration during training. The framework recovers diffeomorphic mappings at inference using a small number of compositions. Our proposed framework seamlessly incorporates standard registration backbones and improves alignment accuracy consistently across nine datasets while preserving diffeomorphic structure. Notably, the proposed method achieves an average Dice improvement of 2.1% on brain MRI benchmarks, a 12% TRE reduction on lung CT, and a 2.6% Dice gain on cardiac MRI and ultrasound datasets.

Abstract

Deformable Image Registration

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