Biomedical Engineering Reference
In-Depth Information
to the intensity values share coefficients
C
(0)
ijk
of order higher than zero, i.e.,
all of the functions
V
(
d
N
,
d
=
1
,
2
,...,
D
have the
same edges
. The fact that the
zero-order term in Eq. (8.16) is input volume dependent means we allow for local
constant offsets between the input volumes
V
d
. This effectively provides built-in
gain correction in the scheme, since it can handle small intensity attenuation
artifacts between the multiple scans.
Moving least-squares: To solve for the expansion coefficients
C
in Eq. (8.16) we
define the moving least-squares functional
w
d
(
x
d
−
x
0
)
V
(
d
N
(
x
d
−
x
0
)
−
V
d
(
x
d
)
2
D
E
(
x
0
)
=
,
(8.17)
x
d
d
=
1
where
x
0
is the expansion point from where we are seeking edge information,
V
d
(
x
d
)
≡
V
d
(
x
d
) and where
⎧
⎨
1
−
2(
x
/
)
2
for 0
≤
x
≤
/
2
2(
x
/
−
1)
2
w
d
(
x
)
≡
for
/
2
<
x
<
(8.18)
⎩
0
for
x
≥
is a “moving filter” that weights the contribution of different sampling points,
x
d
, according to their Euclidean distance,
x
d
−
x
0
, to the expansion point,
x
0
. Other expressions for this weighting function could, of course, be used, but
Eq. (8.18) is fast to compute, has finite support (by the window parameter
),
and its tangent is zero at the endpoints. After substitution of Eq. (8.16) into
Eq. (8.35) we obtain the functional
w
d
(
x
d
−
x
0
)
C
(
d
)
D
000
−
V
d
(
x
d
)
E
(
x
0
)
=
(8.19)
x
d
d
=
1
ijk
(
x
d
−
x
0
)
i
(
y
d
−
y
0
)
j
(
z
d
−
z
0
)
k
2
N
C
(0)
+
.
i
+
j
+
k
=
1
The minimization of this moving least-squares functional with respect to the
expansion coefficients
C
requires the partial derivatives to vanish, i.e.,
w
d
(
x
d
−
x
0
)
C
(
d
)
∂
E
(
x
0
)
∂
C
(
d
)
000
=
0
=
2
000
−
V
d
(
x
d
)
(8.20a)
x
d
ijk
(
x
d
−
x
0
)
i
(
y
d
−
y
0
)
j
(
z
d
−
z
0
)
k
N
C
(0)
+
,
i
+
j
+
k
=
1
