Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 135

then E [ y A =

2

. Furthermore, from Equation 4.18 and Equation 4.20, we know

that Var( ) y = 4 σ

22

( A + σ

2

) / N . Hence, from Equation 4.180 we then have for high

SNR

1
2

− 1

4

4 σ

2

+

[] A A

− 3

( A

2

+ σ

2

) (4.181)

E

A

N

c

σ

2

.

A 1 − (4.182)

2 NA

2

ABBREVIATIONS

CRLB Cramér-Rao lower bound

DC direct current

FT Fourier transform

ML maximum likelihood

MRI magnetic resonance imaging

MSE mean squared error

PDF probability density function

RF radio frequent

SNR signal-to-noise ratio

SYMBOLS

A signal amplitude

 estimate of the signal amplitude

 estimator of the signal amplitude

b bias

c set of complex observations (stochastic variables)

C covariance matrix

δ (.) delta Dirac function

ε (.) unit step Heaviside function

E [.] expectation operator

1 F 1 (.)

confluent hypergeometric function of the first kind

Γ (.) gamma function

I 0 (.) zeroth-order modified Bessel function of the first kind

I 1 (.) first-order modified Bessel function of the first kind

I Fisher information matrix

K number of parameters

L (.) likelihood function

m magnitude observation (number)

m magnitude observation (stochastic variable)

… magnitude variable corresponding to the observation m

µ mean value of the Gaussian distribution

N number of observations