# Constrained H1-regularization schemes for diffeomorphic image registration

arXiv: https://arxiv.org/abs/1503.00757 · HTML: https://mlsanity.com/paper/1503.00757/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2015.

## Built on by (most cited first, 6 of 6)

- [textttGradICON: Approximate Diffeomorphisms via Gradient Inverse Consistency](https://mlsanity.com/paper/2206.05897/) (2022)
- [CLAIRE -- Parallelized Diffeomorphic Image Registration for Large-Scale Biomedical Imaging Applications](https://mlsanity.com/paper/2209.08189/) (2022)
- [Mono-Modalizing Extremely Heterogeneous Multi-Modal Medical Image Registration](https://mlsanity.com/paper/2506.15596/) (2025)
- [VPREG: An Optimal Control Formulation for Diffeomorphic Image Registration Based on the Variational Principle Grid Generation Method](https://mlsanity.com/paper/2510.13109/) (2025)
- [Multimodal Diffeomorphic Registration with Neural ODEs and Structural Descriptors](https://mlsanity.com/paper/2512.22689/) (2025)
- [A Diffeomorphism Groupoid and Algebroid Framework for Discontinuous Image Registration](https://mlsanity.com/paper/2603.11806/) (2026)

## Abstract (arXiv)

We propose regularization schemes for deformable registration and efficient algorithms for their numerical approximation. We treat image registration as a variational optimal control problem. The deformation map is parametrized by its velocity. Tikhonov regularization ensures well-posedness. Our scheme augments standard smoothness regularization operators based on H1- and H2-seminorms with a constraint on the divergence of the velocity field, which resembles variational formulations for Stokes incompressible flows. In our formulation, we invert for a stationary velocity field and a mass source map. This allows us to explicitly control the compressibility of the deformation map and by that the determinant of the deformation gradient. We also introduce a new regularization scheme that allows us to control shear. We use a globalized, preconditioned, matrix-free, reduced space (Gauss--)Newton--Krylov scheme for numerical optimization. We exploit variable elimination techniques to reduce the number of unknowns of our system; we only iterate on the reduced space of the velocity field. Our current implementation is limited to the two-dimensional case. The numerical experiments demonstrate that we can control the determinant of the deformation gradient without compromising registration quality. This additional control allows us to avoid oversmoothing of the deformation map. We also demonstrate that we can promote or penalize shear while controlling the determinant of the deformation gradient.