ElasticMorph: Plug-and-play second-order elastic regularization for medical image registration

Deformable image registration is crucial for many clinical applications, but current learning-based methods often rely on simple smoothness losses. This can produce physically implausible deformations, such as tissue folding, and creates a trade-off between registration accuracy and anatomical correctness. The objective of this study is to develop and validate a novel, physics-informed regularizer that simultaneously improves both the accuracy and physical plausibility of learning-based registration with negligible computational overhead.

We propose ElasticMorph, a plug-and-play loss function derived from the Navier-Cauchy equation. This regularizer penalizes both curvature and divergence in the deformation field. We derive an approximation based, measure theoretic bound under small strain assumptions, which links the elastic residual to an upper bound on the number of negative Jacobian voxels. This gives a principled and differentiable surrogate for folding control. The method was evaluated on two public brain MRI benchmarks (IXI and LPBA-40) by integrating it into four state-of-the-art CNN and transformer backbones. Performance was quantified using the Dice Similarity Coefficient (DSC) and the percentage of negative Jacobians (%negJ), with statistical significance assessed via paired t-tests.

Across all tested backbones and datasets, ElasticMorph consistently improved registration accuracy, increasing the mean DSC by 0.56-4.92 %. Concurrently, it suppressed folding artifacts, reducing the %negJ by 24%-42%. This was achieved with a minimal increase in training time ≤8% and memory ≤15%, and no cost at inference. Ablation studies confirmed that the combined curvature-divergence loss outperformed either component alone. A hyperparameter sweep revealed a broad optimal range, with folding artifacts monotonically decreasing as regularization strength increased, consistent with our theoretical proof.

Enforcing second-order linear elasticity offers a robust and computationally efficient strategy to overcome the limitations of conventional regularizers. ElasticMorph provides a practical, principled, and plug-and-play solution for developing more accurate and physically plausible registration models, holding significant potential for improving the reliability of downstream biomedical image analysis tasks. The code will be made publicly available.