Biomedical Engineering Reference
In-Depth Information
laser intensity, and increasing location-dependent absorption with increasing
tissue depth.
A non-rigid registration algorithm that inherently supports images originating
from multiple imaging modalities was described by Rueckert
et al.
[64]. It uses
the same NMI similarity measure as the affine algorithm mentioned above. The
transformation model is a free-form deformation [68]
T
that is defined on a
data-independent, uniformly spaced control point grid (CPG)
covering the
reference image. The CPG consists of discrete control points
φ
i
,
j
,
k
, where
−
1
≤
i
<
n
x
−
1,
−
1
≤
j
<
n
y
−
1, and
−
1
≤
k
<
n
z
−
1. Points with
i
,
j
,or
k
equal to
either 0 or
n
x
−
3(
n
y
−
3 and
n
z
−
3 for
j
and
k
) are located on the edge of the
image data. The spacings between the control points in
x
,
y
, and
z
are denoted
by
δ
x
,
δ
y
, and
δ
z
, respectively. For any location (
x
,
y
,
z
) in the domain of
, the
transformation
T
is computed from the positions of the surrounding 4
×
4
×
4
control points:
3
3
3
T
(
x
,
y
,
z
)
=
B
l
(
u
)
B
m
(
v
)
B
n
(
w
)
φ
i
+
l
,
j
+
m
,
k
+
n
.
l
=
0
m
=
0
n
=
0
Here,
i
,
j
, and
k
denote the index of the control point cell containing (
x
,
y
,
z
),
and
u
,
v
, and
w
are the relative positions of (
x
,
y
,
z
) inside that cell in the three
spatial dimensions:
x
δ
x
y
δ
y
z
δ
z
i
=
−
1
,
j
=
−
1
,
k
=
−
1
,
and
x
δ
x
y
δ
y
z
δ
z
x
δ
x
−
y
δ
y
−
z
δ
z
−
u
=
,v
=
,w
=
.
The functions
B
0
through
B
3
are the approximating third-order spline polyno-
mials [31]:
B
0
(
t
)
=
−
t
3
−
3
t
+
1
/
6
,
+
3
t
2
B
1
(
t
)
=
3
t
3
+
4
/
6
,
−
6
t
2
B
2
(
t
)
=
−
3
t
3
+
3
t
+
1
/
6
,
+
3
t
2
B
3
(
t
)
=
t
3
/
6
.
The degrees of freedom of a B-spline based transformation
T
, and thus the
elements of the parameter vector
p
, are the coordinates of the control points
φ
i
,
j
,
k
.
