Image Processing Reference
In-Depth Information
In practice, Equation 2.23 is often solved in the least-squares (LS) sense or
minimum-variance (MV) sense. The LS solution is given by
ˆ
ρ LS
=
(
SS S
H
)
1
H d
,
(2.24)
and the MV solution is given by [12]
ρ MV
=
(
SSS
H
ψ
−−
11
)
H
ψ
1
d
,
(2.25)
where
is the data noise covariance matrix. Some basic properties of the LS
and MV solutions are summarized in the following remarks.
Ψ
Remark 4: When S and
are accurate, the variance of the reconstruction
error due to data noise is given by
Ψ
σ LS
()
=
(
SS S
H
)
1
H
Ψ
SSS
(
H
) ],
1
(2.26)
x
for the LS solution, and
σ MV
()
=
[
SS
H
Ψ
−−
11
) ],
(2.27)
x
for the MV solution, where the subscript x denotes the index of the matrix
corresponding to location x .
Remark 5: The LS and the MV solutions are the same if the acceleration
factor equals the number of coils or noise is uncorrelated between coils,
in which case there is no need to measure the noise covariance matrix.
Remark 6: The SNR of the MV solution is always greater or equal to that
of the LS solution. The MV solution minimizes the variance of the recon-
struction error vector over all possible esti-
mators when the noise is Gaussian and over all linear unbiased estimators
for non-Gaussian noise. Therefore, the mean-squared error of the MV
solution is less than that of the LS solution.
E
(
∆∆
ρρ
H
)
=
trace
E
(
∆∆
ρρ
H
)
The earlier results are based on the assumption that both S and
Ψ
are accurate.
In practice, S and
Ψ
are estimated from experimental data, and any error in S
∆ρ
(denoted as
S ) and/or in
Ψ
errors (denoted as
∆Ψ
) will contribute to
.
Suppose
that
||
∆<
S
||
σ
( )
S
,
2
min
||
∆<
Ψ
||
σ
(
Ψ
)
,
(2.28)
2
min
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