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  Quarterly of Applied Mathematics
Quarterly of Applied Mathematics
  
Online ISSN 1552-4485; Print ISSN 0033-569X
 

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


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
Published electronically: October 15, 2012
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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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Additional Information

Remco Duits
Affiliation: Department of Mathematics/Computer Science and Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Address at time of publication: Department of Mathematics/Computer Science and Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: R.Duits@tue.nl

Bart Janssen
Affiliation: Department of Mathematics/Computer Science, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: B.J.Janssen@tue.nl

Alessandro Becciu
Affiliation: Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: A.Becciu@tue.nl

Hans van Assen
Affiliation: Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: H.C.v.Assen@tue.nl

DOI: http://dx.doi.org/10.1090/S0033-569X-2012-01313-0
PII: S 0033-569X(2012)01313-0
Keywords: Covariant derivatives, fiber bundles, Helmholtz decomposition, inverse problems, optical flow methods in image analysis, scale space.
Received by editor(s): October 20, 2010
Published electronically: October 15, 2012
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.



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