LogDemons revisited: consistent regularisation and incompressibility constraint for soft tissue tracking in medical...

Non-linear image registration is a standard approach to track soft tissues in medical images. By estimating spatial transformations between images, visible structures can be followed over time. For clinical applications the model of transformation must be consistent with the properties of the biological tissue, such as incompressibility. LogDemons is a fast non-linear registration algorithm that provides diffusion-like diffeomorphic transformations parameterised by stationary velocity fields. Yet, its use for tissue tracking has been limited because of the ad-hoc Gaussian regularisation that prevents implementing other transformation models. In this paper, we propose a mathematical formulation of demons regularisation that fits into LogDemons framework. This formulation enables to ensure volume-preserving deformations by minimising the energy functional directly under the linear divergence-free constraint, yielding little computational overhead. Tests on synthetic incompressible fields showed that our approach outperforms the original logDemons in terms of incompressible deformation recovery. The algorithm showed promising results on one patient for the automatic recovery of myocardium strain from cardiac anatomical and 3D tagged MRI.