Physics-informed diffeomorphic image registration

Medical image registration is the process of finding a spatial transformation that aligns a medical scan or image (X-ray, CT, MR, US, PET etc.) to another scan or image for comparison. Example use of image registration includes mapping one patient's brain MRI onto another's, or tracking organ motion across breathing cycles in lung CT. Accurate registration is a primary requirement of a wide range of clinical workflows, including disease progression monitoring, treatment planning, and atlas-based segmentation.

Diffeomorphic registration methods constrain the transformation to be smooth, invertible, and topology-preserving. This is essential for anatomical plausibility: a registration that folds tissue or creates singularities is clinically meaningless, regardless of how well the images appear aligned. Classical diffeomorphic approaches such as LDDMM and SyN [1] achieve this by modelling deformation as the flow of a velocity field governed by a partial differential equation (PDE) rooted in fluid mechanics. Modern deep-learning methods such as VoxelMorph, TransMorph, and the recently proposed flow Map or velocity-based methods have dramatically accelerated inference, but most still treat the regularization of deformations as a purely image based computational problem such as imposing gradient smoothness penalties rather than encoding the actual biomechanics of soft tissue.

Physics-Informed Neural Networks (PINNs) embed governing physical laws, such as elasticity equilibrium equations, incompressibility constraints, or hyper-elastic strain energy etc. directly into the training process to teach the neural network underline physical dynamics. This paradigm has shown strong results in fluid dynamics, solid mechanics, and biomechanical simulation, but its application to diffeomorphic image registration remains underexplored.

The project begins with a rigorous like-for-like empirical comparison between physics-informed and non-physics-informed diffeomorphic registration, using publicly available benchmark datasets.  
Following baseline frameworks and datasets will be explored:

1. Traditional optimization-based methods.

2. Deep-learning based methods that may include VoxelMorph, Transmorph, GradICON, SigDIR

3. Physics informed methods [Amiri-Hezaveh et al. (2025), Min et al. (2023,2034)]

4. Datasets: OASIS, ACDC, LungCT etc.

Evaluation metrics will include Dice similarity coefficient, percentage of voxels with negative Jacobian determinant (|J|<0%), Target Registration Error (TRE) for lung CT, and standard deviation of log-Jacobian (SDLogJ) for smoothness. This empirical phase constitutes the undergraduate student's primary deliverable.

If the empirical comparison confirms that physics constraints improve anatomical plausibility, the second stage will integrate these constraints into a flow map-based framework — replacing multiple hand-crafted penalties with a single unified physics-informed loss. This positions the work as a novel contribution suitable for a MICCAI or Medical Image Analysis submission and provides a natural PhD research trajectory for participating students.

The project investigates which tissue-specific physics constraints best ensure anatomically plausible (smooth, invertible, topology-preserving) diffeomorphic registration, and whether such constraints can be learned in an amortized framework that generalizes to unseen image pairs. Three concrete objectives:

1. Evaluate deep-learning and traditional baselines on the chosen datasets to characterize the quality of diffeomorphism each produces.

2. Compare these against physics-informed methods under a controlled design, assessing both diffeomorphism quality and out-of-distribution generalizability.

3. Consolidate the full evaluation into a manuscript for a high-impact venue (MICCAI, IPMI, or Medical Image Analysis).