Image Processing Reference
In-Depth Information
for known and for unknown noise variance. From the figure, one can see that the
following:
•
For low SNR, the CRLB for unbiased estimation of
A
•
From complex data is significantly smaller than for estimation from
magnitude data.
•
From magnitude data with known noise variance is significantly
larger than for estimation from magnitude data with unknown noise
variance (i.e., in which the noise variance is a nuisance parameter).
•
Recall that knowledge of the noise variance is not required when
estimating the signal amplitude from complex data.
•
For increasing SNR, the CRLBs for unbiased estimation from magni-
tude data tend to the CRLB for unbiased estimation from complex
data, in which the CRLB equals
σ
2
/N.
4.4.4.2
MSE
The bias, variance, and MSE of the ML estimators of
A
were computed, where the
number of data points was set to
N
1. For
complex data with identical and different phase values, the bias, variance, and MSE
of were computed from Equation 4.64 and Equation 4.65, and Equation 4.76
and Equation 4.77, respectively. On the other hand, for magnitude data with known
and unknown noise variance, the bias, variance, and MSE of were obtained from
a Monte Carlo simulation experiment with a sample size of 10
5
. Thereby, was
obtained by maximizing the log-likelihood function (Equation 4.96) with respect to
A
and {
A
,
=
25 and the true variance was set to
σ
2
=
A
ML
A
ML
A
ML
2
} using Equation 4.98 and Equation 4.120, respectively.
The bias of
σ
A
has been plotted as a function of the SNR in
Figure 4.6
. In
Figure 4.7
, the MSE of
ML
A
has been plotted as a function of the SNR. Both
figures show the results obtained for complex data with identical and different
phase values as well as for magnitude data with known and unknown noise
variance. From these figures, one can observe that in terms of the MSE:
ML
A
•
for complex data with identical phase values performs best, inde-
pendent of the SNR.
ML
A
•
for magnitude data with known noise variance is significantly
better compared with
ML
A
for magnitude data with unknown noise
ML
A
variance and
for complex data with different phase values.
ML
Also, for increasing SNR, the performance difference in terms of the MSE
between the ML estimators
of based on complex and magnitude data tend to
zero. As, in practice, the assumption of identical phases for complex data is
generally invalid, it may be concluded that the signal amplitude is preferentially
estimated from magnitude MR data for which the noise variance is known. The
latter requisite is not too restrictive as often, in practice, the noise variance can
be estimated with a much higher precision than the signal amplitude.
A
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