A Variational Approach to Cardiac Motion Estimation Based on Covariant Derivatives and Multi-scale Helmholtz Decomposition

The investigation and quantification of cardiac motion is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider a new method to track cardiac motion from magnetic resonance (MR) tagged images. Tracking is achieved by following the spatial maxima in scale-space of the MR images over time. Reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev-norm expressed in covariant derivatives. These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. Furthermore, we propose a multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in one non-singular analytic kernel operator in order to decompose the optic flow vector field in a divergence free, and rotation free part. Finally, we combine both the multi-scale Helmholtz decomposition and our vector field reconstruction (based on covariant derivatives) in a 2000 Mathematics Subject Classification. Primary 55R10,49M25,47A05 ; Secondary 47A10, 49M27.