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

Duits, Remco; Janssen, Bart; Becciu, Alessandro; van Assen, Hans

doi:10.1090/S0033-569X-2012-01313-0

Authors: Remco Duits, Bart Janssen, Alessandro Becciu and Hans van Assen
Journal: Quart. Appl. Math. 71 (2013), 1-36
MSC (2010): Primary 55R10, 49M25, 47A05; Secondary 47A10, 49M27
DOI: https://doi.org/10.1090/S0033-569X-2012-01313-0
Published electronically: October 15, 2012
MathSciNet review: 3075534
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Abstract | References | Similar Articles | Additional Information

Abstract: 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 nonsingular analytic kernel operator in order to decompose the optic flow vector field in a divergence-free and a 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.

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