Biomedical Engineering Reference
In-Depth Information
where
D
is a data term according to Sect.
2.1.1
and
M
is a transformation model
according to Eq. (
2.2
)or(
2.46
).
n
1
n
2
n
3
Definition 2.8 (Spline transformation).
Given a number
n
=(
,
,
)
∈
3
N
of spline coefficients and a spline basis function
b
:
R
→
R
,the
spline
y
p
(
y
p
(
y
p
(
transformation
or
free-form transformation y
p
(
x
x
x
x
)=(
)
,
)
,
))
for
T
3
a point
x
=(
x
,
y
,
z
)
∈
Ω
⊂
R
is defined as
n
1
i
=
1
n
2
j
=
1
n
3
k
=
1
p
i
,
j
,
k
b
i
(
x
)
b
j
(
y
)
b
k
(
z
)
,
y
p
(
x
)=
x
+
(2.23)
n
1
i
=
1
n
2
j
=
1
n
3
k
=
1
p
i
,
j
,
k
b
i
(
x
)
b
j
(
y
)
b
k
(
z
)
,
y
p
(
x
)=
y
+
(2.24)
n
1
i
=
1
n
2
j
=
1
n
3
k
=
1
p
i
,
j
,
k
b
i
(
x
)
b
j
(
y
)
b
k
(
z
)
,
y
p
(
x
)=
z
+
(2.25)
where
p
i
,
j
,
k
,
p
i
,
j
,
k
,
p
i
,
j
,
k
)
T
3
=
(
∈
R
|
p
}
n
1
n
2
n
3
i
∈{
1
,...,
},
j
∈{
1
,...,
},
k
∈{
1
,...,
(2.26)
n
1
are the spline coefficients and
b
i
(
x
)=
b
(
x
−
i
)
for
i
∈{
1
,...,
}
,
b
j
(
y
)=
n
2
n
3
b
(
y
−
j
)
for
j
∈{
1
,...,
}
,
b
k
(
z
)=
b
(
z
−
k
)
for
k
∈{
1
,...,
}
. Note that
T
3
n
1
this notation is only valid for
Ω
=
{
(
x
,
y
,
z
)
∈
R
|
0
≤
x
<
+
1
,
0
≤
y
<
n
2
n
3
+
1
,
0
≤
z
<
+
1
}
.
A possible choice for the basis function
b
:
R
→
R
, often called
mother spline
(cf. Fig.
2.2
), is
⎧
⎨
3
(
+
)
,
−
≤
< −
,
x
2
2
x
1
x
3
3
−
−
2
(
x
+
1
)
+
6
(
x
+
1
)
,
−
1
≤
x
<
0
,
x
3
3
b
(
x
)=
(2.27)
+
2
(
x
−
1
)
−
6
(
x
−
1
)
,
0
≤
x
<
1
,
⎩
3
(
−
x
+
2
)
,
1
≤
x
<
2
,
0
,
else
.
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