Biomedical Engineering Reference
In-Depth Information
where
h
=(
j
−
1) mod
n
+1with 'mod' as the modulo operator, and
φ
h
+2
πn
P
−
d<k
≤
n
−
0
.
5
d
and
j>n,
φ
j
φ
j
Q
=
P
−
2
πn
−
0
.
5
d
<j,k
≤
n,
(26)
φ
h
P
otherwise
.
Application to the method outlined in the previous section requires us to rewrite
Eq. (25) as
k
+
d
k
+
d
φ
i
(
u
i
)
N
h,d
(
u
i
)+2
πN
j,d
(
u
i
)
ψ
h
=
φ
h
,
N
h,d
(
u
i
)
ψ
h
P
(27)
j
=
k
j
=
k
where
−
N
h,d
(
u
)
n
−
d<k
≤
n
−
0
.
5
d
and
j>n,
N
j,d
(
u
)=
N
h,d
(
u
)
n
−
0
.
5
d
<j,k
≤
n,
(28)
0
otherwise
.
This is equivalent to the matrix formulation
ψ
φ
,
B
∗
)
Ψ
=
BP
(
ΦB
+2
π
(29)
B
∗
is the observation matrix composed of the evaluated basis functions
N
j,d
. This is incorporated into the methodology discussed in Section 3.1 to solve
for the control point locations and corresponding weights.
where
4.
MODEL FITTING AND NONRIGID REGISTRATION
4.1. Algorithm Overview
Having discussed the mathematical preliminaries involved in fitting an NURBS
object to data, we present in Table 1 an algorithmic overview for fitting a biventric-
ular NURBS model to 4D short-axis and 4D long-axis tagged MRI data consisting
of
N
+1synchronized volumetric frames. The principles presented in Table 1
apply regardless of whether Cartesian or non-Cartesian B-splines are employed.
Note that
R
denotes the reference frame at time
t
=0(end-diastole) and
R
t
de-
notes a deformed frame later in the cardiac cycle at
t>
0. Below, we describe the
methodological components of the various steps.
4.2. Initialization: Constructionof theBiventricularNURBSModel
To construct the initial model for subsequent registration, one must select the
knot vectors, number of control points, as well as degree of spline.
Following
