A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

R. Duits, B.J Janssen, A. Becciu, H.C. van Assen

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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 single algorithm and show the practical benefit of this approach by an experiment on real cardiac images.
Original languageEnglish
Pages (from-to)1-36
Number of pages36
JournalQuarterly of Applied Mathematics
Issue number1
Publication statusPublished - 2013


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