Biomedical Engineering Reference
In-Depth Information
holds, according to Eq. (
4.101
) in Sect.
4.10.3
,
x
k
is obtained as
H
T
H
−
1
H
T
y
k
.
ʥ
−
1
x
k
=
ʲ
+
ʲ
(4.60)
This is the update equation for
x
k
. Using the matrix inversion formula in Eq. (C.92),
this equation can be rewritten as
H
T
H
T
−
1
ʲ
−
1
I
H
T
ʣ
−
1
y
x
k
=
ʥ
+
H
ʥ
y
k
=
ʥ
y
k
.
(4.61)
Since this expression uses
ʣ
y
, it is conveniently used in the convexity-based
algorithm. Note that the auxiliary variable
x
k
is equal to the posterior mean of
the voxel source distribution because the update equation above is exactly equal to
Eq. (B.26).
ʥ
and
4.6.4 Update Equation for
ʽ
that minimizes the cost function
F(
ʽ
,
Let us derive
ʽ
z
,
x
)
. Since only the second
and the third terms in
F(
ʽ
,
z
,
x
)
contains
ʽ
(as shown in Eq. (
4.52
)), we have:
1
K
K
F(
ʽ
,
x
k
ʥ
−
1
x
k
+
z
T
ʽ
=
,
)
=
ʽ
argmin
ʽ
x
z
argmin
ʽ
k
=
1
z
j
ʽ
j
+
K
k
=
1
x
j
(
N
1
t
k
)
=
argmin
ʽ
.
(4.62)
ʽ
j
j
=
1
Thus, considering the relationship
z
j
ʽ
j
+
K
k
=
1
x
j
(
K
k
=
1
x
j
(
1
1
N
t
k
)
t
k
)
∂
∂ʽ
j
=
z
j
−
=
0
,
2
j
ʽ
j
ʽ
k
=
1
we can derive
K
k
=
1
x
j
(
1
t
k
)
ʽ
j
F(
ʽ
,
ʽ
j
=
argmin
z
,
x
)
=
.
(4.63)
z
j
4.6.5 Summary of the Convexity-Based Algorithm
The update for
z
in Eq. (
4.58
) and the update for
x
in Eq. (
4.61
) require
ʽ
to be known.
ʽ
The update equation for
in Eq. (
4.63
) requires
x
and
z
to be known. Therefore, the
convexity-based algorithm updates
z
,
x
and
ʽ
using Eqs. (
4.58
), (
4.61
) and (
4.63
), in
