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CARL: A Framework for Equivariant Image Registration
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Proposes CARL, a multi-step deep registration network with a coordinate-attention first step that makes registration equivariant to independent translations of input images (and rotations with augmentation).
Problem
Deep image registration networks that predict displacements are typically only equivariant to the same transform applied to both images, not to independent deformations of moving and fixed images, which hurts generalization to large displacements and differing fields of view.
Approach
Defines [W,U] and [U,U] equivariance for registration, proves that a two-step composition is [W,U] equivariant if the first step is [W,U] and later steps are [U,U], and builds a first step from coordinate attention (center of mass of attention masks over a translation-equivariant conv encoder).
Key results
- On Abdomen1K, CARL reaches 75.7% DICE (77.2% with instance optimization) vs. 49.6% for GradICON.
- On DirLab lung registration, CARL achieves 1.88 mTRE vs. 1.93 for GradICON, close to SOTA.
- On HCP brain, CARL attains 79.6% DICE and stays unaffected under translation while GradICON's DICE drops sharply.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the lowest-resolution displacement-predicting network with coordinate attention Ξθ (ablation) | GradICON-style multi-step registration · Abdomen1K | %DICE: 49.6 -> 75.7 | Helped |
| add instance optimization | CARL · Abdomen1K | %DICE: 75.7 -> 77.2 | Helped |
| add instance optimization | CARL · DirLab | mTRE: 1.88 -> 1.34 | Helped |
| add instance optimization | CARL · Learn2Reg Abdomen CT-CT Test set | %DICE: 50 -> 56 | Helped |
| add instance optimization | CARL · HCP | %DICE: 79.6 -> 80.4 | Helped |
| use the rotation-equivariant CARL{ROT} training procedure instead of baseline CARL (ablation) | CARL · HCP | %DICE: 79.6 -> 77.8 | Hurt |
| use the rotation-equivariant CARL{ROT} training procedure with instance optimization (ablation) | CARL(IO) · HCP | %DICE: 80.4 -> 78.9 | Hurt |
| replace coordinate attention with displacement-predicting layers (ablation) | multi-step registration · Retina Shift | DICE | Hurt |
| remove refinement layers, using single-step coordinate attention only (ablation) | coordinate-attention registration · Retina Baseline | DICE | Hurt |
| train on Baseline and evaluate on the shifted test set using coordinate-attention equivariant networks (ablation) | CARL and single-step coordinate attention · Retina Shift | DICE | Helped |
| add random rotations from SO(3) and the CARL{ROT} rotation-equivariant training changes (ablation) | CARL · HCP, rotated input images | DICE | Helped |
Surprising findings
- Common displacement-predicting networks (VoxelMorph-style, GradICON) are only [U,U] equivariant, so their accuracy collapses when fixed and moving images are shifted differently.
- Coordinate attention networks trained on translations also generalize to scale changes, despite being formally equivariant only to integer translations.
Limitations
- Formal equivariance is only to integer-voxel translations; rotation equivariance requires augmentation and approximate methods.
- Instance optimization adds substantial runtime (209 s per pair).
- Flash attention scales poorly with image size, so the coordinate attention is applied at coarse 43^3 resolution.
Experimental setup
Datasets: COPDGene DirLab Abdomen1K Learn2Reg Abdomen CT-CT HCP DRIVE
Benchmarks: DirLab COPDGene HCP Abdomen1K Learn2Reg Abdomen CT-CT
Metrics: mTRE DICE percentage of negative Jacobian determinant
Baselines: GradICON ConstrICON uniGradICON EasyReg KeyMorph ANTs VoxelMorph LapIRN SAME++ PTVReg RRN
Builds on: GradICON VoxelMorph TwoStep registration ConstrICON attention
Closest papers by meaning
- Uncertainty Estimation for Pretrained Medical Image Registration Models via Transformation Equivariance (NeurIPS 2026 · similarity 0.90)
- Metric Learning for Image Registration (CVPR 2019 · similarity 0.88)
- Deep Learning in Medical Image Registration: Magic or Mirage? (NeurIPS 2024 · similarity 0.86)
- SACB-Net: Spatial-awareness Convolutions for Medical Image Registration (CVPR 2025 · similarity 0.86)
- Learning Generalizable Shape Completion with SIM(3) Equivariance (NeurIPS 2025 · similarity 0.85)
- A Plug-and-Play Image Registration Network (ICLR 2024 · similarity 0.85)
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