Image Processing Reference
In-Depth Information
4.5.4
D ISCUSSION
4.5.4.1
CRLB
The CRLB for unbiased estimation of the noise variance from complex data with
identical and different phase values is given by Equation 4.135 and Equation 4.141,
respectively. In both cases, independent of the signal amplitude of the data points,
the CRLB is equal to
4 / N .
Furthermore, the CRLB for unbiased estimation of the noise variance from
magnitude data has been computed for a background region as well as for a
σ
constant region. In Figure 4.14 , the CRLB is shown as a function of the SNR.
If the noise variance is estimated from N magnitude data points of a
background region, the CRLB is equal to
4 / N (cf. Equation 4.151).
Then the CRLB is the same as that for estimation from N complex data
points of a background region. This might be surprising, as estimation
from N complex data points actually exploits 2 N real-valued ( N real
and N imaginary) observations, whereas estimation from N magnitude
data points only exploits N real-valued observations. However, this is
compensated by the fact that the Rayleigh PDF has a smaller standard
deviation.
σ
If the noise variance is estimated from N magnitude data points of a
constant region, the CRLB is given by Equation 4.116. It can be shown
numerically that for magnitude data this CRLB tends to 2
4 / N when
the SNR increases, which is a factor 2 larger compared to estimation
σ
from complex data (cf. Figure 4.12 ). This is not surprising because the
Rician PDF tends to a Gaussian PDF for high SNR, with the same
variance as the PDF of the real or imaginary data. Hence for high SNR,
the difference in CRLB between magnitude and complex data can
simply be explained by the number of observations available for the
estimation of the noise variance.
4.5.4.2
MSE
For complex data from a region with constant amplitude with identical and
different phase values, expressions for the bias, variance, and MSE of the ML
estimator
ˆ
ˆ
2 were derived. The bias of is given by Equation 4.137
and Equation 4.143 for the two cases, respectively. From these expressions, it is
clear that:
σ ML
2
of
σ
σ ML
2
ˆ
Both noise variance estimators are biased. Also, the bias of
σ ML
2
is inde-
pendent of the true signal amplitude.
ˆ
For identical phases, the bias of
σ ML
2
is inversely proportional to the
ˆ
number of observations (
N
). In contrast, the bias of
σ ML
2
N
for different
phases does not decrease with increasing N ; for large
, it converges
to
σ
2 /2.
 
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