Biomedical Engineering Reference
In-Depth Information
4.3. Coupling the Left- and Right-Ventricular Models
In order to couple the two ventricular models (as described in step 3 of Table 1,
we determine the parametric values of the left-ventricular model where the right-
ventricular model attaches corresponding to the anatomical ventricular junctions
for each frame t> 0. We sample the two interfaces between the left and right
ventricles. These surfaces are defined by the surface ( u RV ,w RV ) at v RV =
0 and ( u RV ,w RV ) at v RV =1. For each sample point along those surfaces,
we determine, via conjugate gradient descent [27], the parametric values of the
left-ventricular model corresponding to that point. This information is used in
conjunction with the tag and contour information to temporally fit the right ventricle
at time t> 0 to time t =0.
4.4. Temporal Nonrigid Registration of Models
This section comprises Steps 2-6 of the algorithm presented in Table 1. To
nonrigidly register the initial model to subsequent frames of data, we define a
mapping
between the deformed frame at time t> 0, denoted by
R t , and the
χ
reference frame, denoted by
R , such that
χ : R →R t .
Within the NURBS
volumetric model, a material point
p R and its image point
p t ∈R t are related
by the mapping
χ ( p ( u, v, w )) = p t ( u, v, w ), where u , v , and w are the parameter
values of the NURBS model. This mapping allows us to use the theory developed
in the previous section on NURBS fitting to reconstruct 3D deformation fields and
corresponding strain maps.
4.4.1. Warping
R t R (Steps 2 and 3 in Table 1)
The data that comprise the spatial displacement information include:
the normal displacement of the tag planes,
the intersections of the each triplet of tag planes, and
the intersections of the contours and tag lines in both the short- and long-
axis images,
which are illustrated in Figures 7 and 8. Using these data we can warp the model at
each time frame t = i , denoted by S E ( u, v, w, i ), back to the reference frame. The
resulting model we denote as S i E ( u, v, w, 0). This is the Eulerian fitting portion
of the algorithm.
Each data type has an associated confidence value that is incorporated into our
registration procedure using weighted least squares. The contour/tag line intersec-
tion displacements describe two components of motion versus the one component
given by the normal displacements of the tag planes. Therefore, the relative confi-
dence value ratio is 2:1 (actual values = 1.0 and 0.5). The displacement information
 
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