Estimation of Signal and Noise Parameters from MR Data - Advance Image Processing in Magnetic Resonance Imaging - page 104

Because px

x

() = 2 x p x )(cf. Appendix ), it is now easy to derive the PDF

x

ˆ

ˆ

of

A =

2

:

A

c c

ˆ

( ˆ )

ˆ

2

p

AA p A

= 2 (4.88)

ˆ

c

A c c

ˆ

2

c

A

c

N − 1

ˆ

+ 2 σ + A

2 2 2

ˆ

2

A

+ 2 σ

N

2 2

A

ˆ

c

= A

c

exp − N

c

σ A 2 σ

2 2 2

(4.89)

NA ˆ + 2 σ

2

2

A

c

ˆ

× I ε + 2 σ .

A 2

2

N − 1

σ

c

2

A c

The expectation value of is thus defined by

N − 1

ˆ

+ 2

2

σ
σ

+ A

∫

2 2 2

ˆ

2

A

+ 2 σ

N

2 2

A

ˆ

c

=

ˆ

A

c

exp − N

E

2

A

c c

σ A

2 2 2

(4.90)

NA ˆ + 2 σ

2

2

A

ˆ

c

ˆ

× I ε + 2 σ dA .

2 2

A

c

N − 1

σ

c

2

A 2

Pr[ ˆ A 2 In Figure 4.4, the probability that is negative, , is plotted as a

function of SNR for several values of N . From the figure one can conclude that

< 0 ]

0.6

N = 4

N = 8

N = 16

N = 32

0.5 N = 64

0.4

0.3

0.2

0.1

0 0

0.5 1 1.5 2 2.5

SNR

A c 2

FIGURE 4.4 The probability Pr[ 0] as a function of the SNR for various N. (From

Sijbers, J., den Dekker, A.J., Scheunders, P., Van Dyck, D (1998). Maximum likelihood
estimation of Rician distribution parameters. IEEE Trans. Med. Imaging 17(3): 357-361.)

<