Image Processing Reference
In-Depth Information
) denotes the minimum singular value of the matrix. It can be shown
for the MV solution that [14]
where
σ
min
(
⋅
/
2
∆
ρ
ρ
∆
S
S
∆
d
d
MV
Ψ
Ψ
≤
κ
()
S
+
4
,
(2.29)
1
Ψ
0
3
Ψ
Ψ
MV
where the subscript
Ψ
denotes a matrix or vector premultiplied by
Ψ
−1/2
, with
1
2
−
Ψ
−1/2
being defined as
ΨΛ
=−
V
1
H
assuming that
Ψ
=
V
Λ
I
H
. The earlier
result can be easily extended to the LS solution by setting
Ψ
=
I
in Equation 2.29.
2.4.2
R
EGULARIZED
R
ECONSTRUCTION
M
ETHODS
d
ρ
LS
ρ
MV
,
especially when
S
is ill-conditioned. To desensitize the solution to data noise and
model errors, regularization methods are often used. Tikhonov regularization is
perhaps the most common regularization scheme, in which we form a weighted
sum of the data misfit term
The SENSE reconstruction
(either
or
) is sensitive to
∆
,
∆
S
, and
∆Ψ
S
ρρ
||
ρ −
d
||
2
and a regularization term
||
A
(
−
)
||
2
r
ρ
reg
using a weighting factor
λ
2
, and find the solution
that minimizes this sum, i.e.,
=
arg min{||
S
ρ
−
d
||
2
+
λ
2
||
A
(
ρ
−
) || }
2
,
(2.30)
ρ
ρ
reg
r
ρ
r
where
λ
is often referred to as the regularization parameter and
is a regular-
ρ
reg
ization image. A closed-form solution for
exists for
L
2
-norm and is given by
ρρ
=+
(
SS
H
+
λ
2
AA S
H
)
−
1
H
(
d
−
S
ρ
).
(2.31)
reg
r
r
ρ
r
and is essential for this regularized recon-
struction scheme. Although this is still a research problem, several algorithms have
been proposed, which find useful practical applications. We will briefly review
some of them to illustrate the concept.
Selecting “good” values for
λ
ρ
r
2.4.2.1
Construction of
ρ
r
=
ρ
r
There are basically three schemes to construct
: (a) setting
0,
(b) recycling
ρ
r
,
an initial SENSE reconstruction to create
and (c) collecting additional data to
ρ
r
.
generate
Scheme (a) corresponds to, perhaps, the simplest version of the Tikhonov
regularization scheme. It was used in Reference 15 with some success. In
scheme (b), the conventional SENSE algorithm is used to obtain an initial
reconstruction, which is then filtered by a median filter to suppress any residual
aliasing artifacts [16]. However, if the matrix
S
is highly ill-conditioned within
a large region, the filtering step may not be effective in suppressing the aliasing
artifacts. Scheme (c) acquires additional
k
-space center lines at the Nyquist
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