Constrained $H^1$-regularization schemes for diffeomorphic image registration

TL;DR

This paper introduces new constrained $H^1$-regularization schemes for diffeomorphic image registration, enabling explicit control over deformation properties like compressibility and shear, with efficient numerical algorithms demonstrated in 2D.

Contribution

It proposes novel regularization schemes incorporating divergence constraints and shear control, along with an efficient reduced space optimization algorithm for diffeomorphic registration.

Findings

Controlled the determinant of the deformation gradient without losing registration quality.

Enabled promotion or penalization of shear in the deformation map.

Demonstrated effectiveness in 2D numerical experiments.

Abstract

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 $H^1$- and $H^2$-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.