Image Processing Reference
In-Depth Information
where
) are the real and imaginary
variables, respectively, corresponding to the complex observation
σ
2
denotes the noise variance, and (
ω
,
ω
r
i
(
ww
,
),
with
r
i
underlying true amplitude and phase value,
, respectively. In Equation 4.1
and in what follows, stochastic (i.e., random) variables are underlined [10]. In
general, a Gaussian PDF is described by
A
and
ϕ
2
1
(
x
−
µ
σ
)
−
px
(
|
µσ
,
)
=
e
,
(4.2)
2
2
x
2
πσ
2
with
µ
and
σ
denoting the mean and standard deviation of the PDF, respectively.
4.2.1.1
Moments of the Gaussian PDF
Analytical expressions for the moments of a Gaussian PDF are given by
2
2
d
d
ν
−
1
µ
σ
µ
σ
( )
−
ν
21
ν
−
é
x
[]
=
σ
µ
e
e
,
(4.3)
2
2
2
2
µ
ν
−
1
where
E
[]
.
is the expectation operator and
ν ∈
0
[17]. For the first four moments
we have, explicitly
[]
x
= µ
,
(4.4)
E
2
[]
=µσ
2
2
,
(4.5)
E
x
[
3
=µ σ
3
3
2
,
(4.6)
E
x
4
[]
=+
µµσσ
4
6
2
2
+ .
3
4
(4.7)
E
x
4.2.1.2
Central Moments
For the central moments, we have
0
if
ν
is odd
!
[(
x
−=
!
µ
) ]
ν
.
(4.8)
E
νσ
ν
if
ν
is even
!
(/)
ν
22
2
!
ν
/
"
4.2.2
R
PDF
ICIAN
During MR data processing, it is a common practice to work with magnitude data
instead of real and imaginary data, because magnitude data have the advantage of
being immune to the effects of incidental phase variations due to radio-frequency
(RF) angle inhomogeneity, system delay, noncentered sampling windows, etc. In
this section, the PDF of the magnitude data is discussed.
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