A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration

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.

Deformable registration is crucial in medical imaging. Several existing applications include lesion tracking, probabilistic atlas generation, and treatment response evaluation. However, current methods often lack robustness and generalizability across two key factors: spatial resolution and differences in anatomical coverage. We jointly optimize a registration field and a learnable dimensionality reduction module so that compressed VFM embeddings remain registration-relevant, and fuse these global semantic cues with MIND local descriptors. GLIDE-Reg achieves average dice similarity coefficients (DSC) across 6 anatomical structures of 0.859, 0.862, and 0.901 in two public cohorts (Lung250M and NLST) and one institution cohort (UCLA5DCT), and outperforms the state-of-the-art DEEDS (0.834, 0.858, 0.900) with relative improvements of 3.0%, 0.5%, and 0.1%. For target registration errors, GLIDE-Reg achieves 1.58 mm on Lung250M landmarks (compared to 1.25 mm on corrField and 1.91 mm on DEEDS) and 1.11 mm on NLST nodule centers (compared to 1.11 mm on DEEDS). The substantiated performance on the nodule centers also demonstrates its robustness across challenging downstream tasks, such as nodule tracking, which is an essential prior step for early-stage lung cancer diagnosis.

Deformable image registration models implicitly encode deformation priors through their parametrization and optimization. In this work, we conduct a validation study on continuous registration methods to examine how these implicit priors affect performance across different registration tasks. Classic B-Spline transformations impose locality, smoothness, and scale through their control-point structure, whereas recent INR-based methods impose different priors through neural parameterization and optimization. We compare INR-Dense (IDIR), which directly models a dense displacement field using a SIREN-based INR; INR-BSCP (SINR), which predicts B-Spline control points with an INR; D-BSCP, which directly optimizes single-scale B-Spline control points; and MR-D-BSCP, which adds a multiresolution coarse-to-fine scheme. Experiments on inter-subject brain MR registration (OASIS) and intra-subject exhale-to-inhale lung CT registration (DIR-LAB 4DCT) reveal different behavior across deformation regimes. On OASIS, where deformations are moderate but locally complex, D-BSCP matches or slightly outperforms INR-BSCP, suggesting that the B-Spline parameterization accounts for much of INR-BSCP's effectiveness. On DIR-LAB 4DCT, where respiratory motion is larger and more coherent, single-scale B-Spline methods (D-BSCP and INR-BSCP) are less suitable, while INR-Dense and MR-D-BSCP are more effective. Across both tasks, MR-D-BSCP achieves the best performance among the tested continuous parameterizations. These findings highlight that registration accuracy depends strongly on matching the induced deformation prior to the target motion pattern, and support prior-deformation matching as a practical design principle for medical image registration. Our code will be available at https://github.com/HengjieLiu/RightPriorDIR.