Biomedical Engineering Reference
In-Depth Information
w
d
(
x
d
−
x
0
)
C
(
d
)
∂
E
(
x
0
)
∂
C
(0)
D
000
−
V
d
(
x
d
)
lnm
=
0
=
2
x
d
d
=
1
ijk
(
x
d
−
x
0
)
i
(
y
d
−
y
0
)
j
(
z
d
−
z
0
)
k
N
C
(0)
+
i
+
j
+
k
=
1
×
(
x
d
−
x
0
)
l
(
y
d
−
y
0
)
m
(
z
d
−
z
0
)
n
(8.20b)
.
This defines a system of linear equations in the expansion coefficients
C
(
r
)
ijk
that
can be solved using standard techniques from numerical analysis, see Eqs. (8.21)
and (8.23).
Equations (8.20a) and (8.20b) can then be conveniently expressed as
A
p
,
q
c
q
=
b
p
,
(8.21)
q
where
A
is a diagonal matrix, and
b
,
c
are vectors. In this equation we have
also introduced the compact index notations
p
≡
(
i
,
j
,
k
,
r
) and
q
≡
(
l
,
m
,
n
,
s
)
defined as
p
∈
i
,
j
,
k
,
r
∈
N
+
i
=
j
=
k
=
0
,
1
≤
r
≤
D
∪
i
,
j
,
k
,
r
∈
N
+
1
≤
i
+
j
+
k
≤
N
,
r
=
0
,
(8.22a)
q
∈
l
,
m
,
n
,
s
∈
N
+
l
=
m
=
n
=
0
,
1
≤
s
≤
D
∪
l
,
m
,
n
,
s
∈
N
+
1
≤
l
+
m
+
n
≤
N
,
s
=
0
.
(8.22b)
The diagonal matrix
A
and the vectors
b
,
c
in Eq. (8.21) are defined as
δ
r
,
d
+
δ
r
,
0
δ
s
,
d
+
δ
s
,
0
x
d
A
p
,
q
≡
w
d
(
x
d
−
x
0
)
d
×
(
x
d
−
x
0
)
i
(
y
d
−
y
0
)
j
(
z
d
−
z
0
)
k
(8.23a)
×
(
x
d
−
x
0
)
l
(
y
d
−
y
0
)
m
(
z
d
−
z
0
)
n
,
δ
r
,
d
+
δ
r
,
0
w
d
(
x
d
−
x
0
)
V
d
(
x
d
)
b
p
≡
d
×
(
x
d
−
x
0
)
i
(
y
d
−
y
0
)
j
(
z
d
−
z
0
)
k
,
(8.23b)
c
p
≡
C
(
r
)
(8.23c)
ijk
.
Next the matrix equation
A
c
=
b
must be solved for the vector
c
of dimen-
sion (
N
+
3
)
+
D
−
1, where
N
is the order of the expansion in Eq. (8.16) and
D
is the number of nonuniform input volumes. As is well known for many moving
least-square problems, it is possible for the condition number of the matrix
A
to become very large. Any matrix is singular if its condition number is infinite
