Image Processing Reference
In-Depth Information
A significant weakness of both the conventional
L
-curve and the GCV meth-
ods lies in the fact that they choose
(
x
) independently for different spatial
locations. This problem was addressed in Reference 21 with an algorithm to select
λ
λ
(
x
) jointly. Specifically, the algorithm first sets
λ
(
x
) to be within [
λ
min
,
λ
max
], and
then forms
λ
(
x
) as a linear function of the local condition number of
S
, i.e.,
λ
()
x
=
κ
()
S
+
β
,
(2.32)
ˆ
for
WWxW
/
2
−<<
/
2
. This scheme is based on the consideration that the larger
the
κ
(
S
), the heavier the regularization is needed for Equation 2.31. To determine
α
and
β
, Equation 2.32 is rewritten as
κ κ
κκ
λλ λ
()
S
−
−
λ
()
x
=
min
(
−
)
+
,
(2.33)
max
min
min
max
min
where
κ
max
and
κ
min
are the maximum and the minimum condition numbers of
all
S
, and
λ
min
and
λ
max
are determined by
&
7
max
min (
σ
σλσ
(
(
(
9
i
i
λ
=
arg min
<
K
,
(2.34)
min
+
2
/
)
(
λ
i
i
i
and
&
7
(
(
(
(
∑
x
λ
=
arg max
||
S
(
λ
)
−
d
||
≤
ε
,
(2.35)
ρ
max
reg
λ
where
σ
i
is the
i
th singular value of
S
, and
K
and
ε
are user-specified constants.
Details of the algorithm can be found in [21].
Figure 2.3
shows a set of exemplary regularized reconstructions with different
regularization parameters from real experimental data acquired with four receiver
coils and an acceleration factor of four. The importance of regularization param-
eters can be appreciated by comparing the results in (a) to (d).
2.4.2.3
Sensitivity Analysis
An upper bound for the sensitivity of the regularized solution to data noise and
model error is given by
||
∆
ρ
ρ
||
/
1
2
4
+
ρ
σ
σ
||
∆
S
S
||
|| |
∆
d
d
| |
λ
σλ
2
||
−
ρ
||
reg
max
≤
+
r
,
(2.36)
||
||
+
λ
σ
2
||
||
+
||
ρ
|| |
||
||
2
2
q
n
reg
q
where
σ
max
denotes the largest singular value of matrix
S
and
/
1
2
4
λ
σ
2
q
=
arg min
σ
+
.
(2.37)
j
j
j
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