Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 111

4.4.4.1 CRLB

We will first discuss the expressions for the CRLB for unbiased estimation of
the signal amplitude from complex and magnitude data separately.

Complex data: Analytical expressions for the CRLBs for unbiased esti-
mation of from complex data with identical and different phase values
were derived in Subsection 4.4.2. They are given by Equation 4.55 and
Equation 4.68, respectively. Note that both CRLBs do not depend on
the phase values.
Magnitude data: In contrast to the CRLBs for complex data, no analytical
expressions for the CRLB for unbiased estimation of the signal ampli-
tude from magnitude MR data with known and unknown noise variance
can be derived. However, the lower bounds given by Equation 4.80
and Equation 4.116 in Subsection 4.4.3 can be evaluated numerically.
Thereby, the expectation values can be evaluated from Monte Carlo
simulations.

A

Note also that all lower bounds are inversely proportional to the number of
data points. Figure 4.5 shows the CRLBs as a function of the SNR, for unbiased
estimation of the signal amplitude from complex as well as from magnitude data

CRLB for unbiased estimation of A ( σ = 1, N = 25)

1

complex data
magnitude data, unknown σ

0.9

2

2

magnitude data, known σ

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.5 1 1.5 2 2.5 3

SNR (A/ )

σ

FIGURE 4.5 CRLB for unbiased estimation of A from complex and magnitude data, with
known and unknown noise variance. (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.)