  In this paper, we propose a method for tracking structures (e.g., ventricles
and myocardium) in cardiac images (e.g., magnetic resonance) by propagating
forward in time a previous estimate of the structures via a new deformation
estimation scheme that is motivated by physical constraints of fluid motion.
The method employs within structure motion estimation (so that differing
motions among different structures are not mixed) while simultaneously
satisfying the physical constraint in fluid motion that at the interface
between a fluid and a medium, the normal component of the fluid's motion must
match the normal component of the motion of the medium. We show how to estimate
the motion according to the previous considerations in a variational framework,
and in particular, show that these conditions lead to PDEs with boundary
conditions at the interface that resemble Robin boundary conditions and induce
coupling between structures. We illustrate the use of this motion estimation
scheme in propagating a segmentation across frames and show that it leads to
more accurate segmentation than traditional motion estimation that does not
make use of physical constraints. Further, the method is naturally suited to
interactive segmentation methods, which are prominently used in practice in
commercial applications for cardiac analysis, where typically a segmentation
from the previous frame is used to predict a segmentation in the next frame. We
show that our propagation scheme reduces the amount of user interaction by
predicting more accurate segmentations than commonly used and recent
interactive commercial techniques.
