Biomedical Engineering Reference
In-Depth Information
NURBS model at time
t
=0, the previously discussed registration strategy is
employed, given by the following system:
ψ
γ,t
,
B
γ
WΓ
t
B
γ
Ψ
t
=(
B
γ
WB
γ
)
P
B
π
WΠ
t
B
π
Ψ
t
=(
B
π
WB
π
)
P
ψ
π,t
,
(33)
ψ
ω,t
.
B
ω
WΩ
t
B
ω
Ψ
t
=(
B
ω
WB
ω
)
P
The generalization of the fitting strategy given above in using separate obser-
vation matrices reflects an additional modification to the NURBS fitting algorithm.
The Cartesian coordinate system, used for the NURBS models with a Cartesian
description, is aligned with the normals of the three sets of tag planes. Therefore,
a single sample point normal displacement is limited to influencing only one of
the three coordinate values of the control points. However, for the NURBS mod-
els with a prolate spheroidal control point description the normal displacement
of a single sample point affects all three coordinate values. Similarly, the nor-
mal displacement of a sample point from a short-axis tag plane affects both the
r
and
θ
coordinate values, whereas the longitudinal displacement affects only the
z
coordinate value. This separation in the fitting strategy is an added advantage
associated with the NURBS models with a Cartesian description.
4.4.2. Warping
R
→R
t
(Steps 4-6 in Table 1)
R
t
→
R
in the previous section, we obtain the
models
S
i
E
(
u, v, w, t
=0)and
S
E
(
u, v, w, t
=
i
). Spatial displacements between
the two frames are easily found as
After calculating the mapping
S
i
E
(
u, v, w,
0)
.
V
E
(
u, v, w
)=
S
E
(
u, v, w, i
)
−
(34)
In order to construct a comprehensive 4D myocardial deformation model that
includes temporal smoothing, one must use the spatial displacements,
V
E
, to con-
struct a Lagrangian description of deformation. This is done by using the volu-
metric NURBS model constructed from the epicardial and endocardial contours at
time
t
=0. This model was denoted above by
S
L
(
u, v, w,
0). The displacement
field,
V
E
, is densely sampled. For each point
p
j
=
S
i
E
(
u
j
,v
j
,w
j
,
0), conjugate
gradient descent is used to find the parameters (
u
j
,v
j
,w
j
) in
S
L
(
u
j
,v
j
,w
j
,
0)
that correspond to
p
j
. A least-squares fit is then performed to find the model
S
L
(
u, v, w, i
) using the parameters (
u
j
,v
j
,w
j
) and the associated displacements
V
E
(
u
j
,v
j
,w
j
). This is the Lagrangian fitting portion of the algorithm. Once this
process is completed for all time frames, temporal lofting is possible.
