Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 100

4.4.2.2.1 CRLB

The Fisher information matrix of the data with respect to the parameter vector
( A , ϕ 1 , … , ϕ N ) is given by

∂

2

ln p ∂

∂∂

2

ln p ∂

∂∂

2

ln p

c c c

...

N

∂ A

2

A ϕ A ϕ 0 0

1 N

σ

2

∂

2

ln p ∂

2

ln p ∂

∂∂

2

ln p

A

2
2

c c c

...

0 0

I =− ∂∂ ϕ A ∂ ϕ

2

ϕ ϕ = (4.67)

E σ

1 1 1 N

... ... ... ..

A

2

∂
∂∂

2

ln p

A

∂
∂∂

2

ln p ∂

2

ln p 00

c c c

... σ

2

ϕ ϕ ϕ ∂ ϕ

2

N N 1 N

and the CRLB for unbiased estimation of ( A , ϕ 1 , … , ϕ N ) is given by

σ

2

0 0

N

σ

2

0 0

CRLB = (4.68)

A

2

σ

2

0 0

A

2

4.4.2.2.2 ML Estimation
The likelihood function for N statistically independent, Gaussian-distributed com-
plex observations

w

c = {( w , )} with underlying noiseless signal amplitude A

rn ,

in ,

and arbitrary phase values ϕ 1 , … , ϕ N is given by

N N

2 2

=

1

( w rn A

, −

cos ϕ ) ( w in A

, −

sin ϕ )

∏

n n

− −

LA ( ,,, | ϕϕ … {( w , w )}) e e . (4.69)

2 2

2 σ 2 σ

1 N r n , i n ,

2 πσ

2

n = 1

Taking the logarithm yields

N

1

∑

ln LN =− ln( 2 πσ

2

) − [( w A − cos ϕ )

2

− ( w

,, −

A sin ϕ ) ].

2

(4.70)

rn , n i n n

2 σ

2

n = 1

The first derivative of ln L with respect to A and ϕ n are given by

N

∂ ln L 1

2

∑

NA

= [ w cos ϕ + w sin ϕ ] − , (4.70)

rn , n in , n

∂ A σ σ

2

n = 1

∂
∂

ln L A w

=− ( sin ϕ − w s ϕ ). (4.71)

rn , n in , n

ϕ σ

2

n