Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 130

2 ( σ = 1, N = 25)
complex or background magnitude data
constant region magnitude data

CRLB for unbiased estimation of σ

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 1 2 3 4 5

SNR (A/ )

σ

FIGURE 4.12 CRLB for unbiased estimation of the noise variance for complex data and
magnitude background data (dotted line) and for magnitude data from a constant region,
that is, where A is estimated simultaneously (full line). (From Sijbers, J., den Dekker, A.J.
(2004). Maximum likelihood estimation of signal amplitude and noise variance from MR
data. Magn. Reson. Med. 51(3): 586-594.)

ˆ

The variance of for complex data from a region with constant amplitude
for identical and different phase values is given by Equation 4.138 and Equation 4.144,
respectively. Only for complex data with identical phases, the variance of
asymptotically attains the CRLB. This is because for estimation from complex
data with different phases the number of unknown parameters to be estimated
simultaneously with is proportional to N .
The MSE of for complex data from a region with constant amplitude for
identical and different phase values is given by Equation 4.139 and Equation 4.145,

σ ML

2

ˆ

σ ML

2

σ

2

ˆ

σ ML

2

respectively. Both are shown as a function of N in Figure 4.13 . Moreover, the MSE
as a function of the SNR is shown in Figure 4.14 .
for magnitude data from a region with constant amplitude

ˆ

of σ ML

2

ˆ

The MSE of
can be found numerically from Equation 4.163. The results for this estimator as
a function of the SNR are also shown in Figure 4.14.
Finally, the MSE of

σ ML

2

σ

4 / N , for
magnitude and for complex data, and is independent of the phases. From this,
it is clear that the noise variance should be estimated from background data points
whenever possible. If a background region is not available, a similar reasoning
for the estimation of

from background MR data is given by σ

2

ML

σ

2 as that for the estimation of A holds. That is, estimation

of

2 from complex data with identical phases is then preferred to estimation
from magnitude data, which, in turn, is preferred to estimation from complex
data with different phases.

σ