Image Processing Reference
In-Depth Information
4.5.3
N
OISE
V
ARIANCE
E
STIMATION
FROM
M
AGNITUDE
D
ATA
We will now describe the ML estimation of noise variance (and standard deviation)
from magnitude MR data. First, we will consider ML estimation of the noise
variance from a so-called background region, that is, a region in which the under-
lying signal is zero. The CRLB for unbiased estimation of both noise variance
and standard deviation will be computed, and the ML estimators will be derived.
Next, we will consider the ML estimation of noise parameters from a so-called
constant region, that is, a region in which the (nonzero) signal amplitude is assumed
to be constant. In this case, the noise parameters have to be estimated simulta-
neously with the signal amplitude.
It will be assumed that the available data is governed by a generalized Rician
distribution. The methods described in the following subsections can also be
applied to conventional Rician-distributed magnitude MR data, which would be
the case when
K
=
2.
4.5.3.1
Background Region
Suppose that a set of
N
statistically independent magnitude data points
is available from a region where the true signal value
A
is zero for each data
point (background region). Hence, these data points are governed by a Rayleigh
distribution and their joint PDF
m
=
{
}
m
n
p
m
is given by (cf. Equation 4.29)
N
2
m
m
,
K
−
1
2
∏
1
p
m
({
m
})
=
n
)
exp
−
n
(4.149)
n
(
2
σ
)
Γ
(
K
/
2
2
σ
22
K
/
2
n
=
where
{}
m
n
{
n
are the magnitude variables corresponding with the magnitude obser-
vations
m
.
4.5.3.1.1 CRLB (Variance)
The Fisher information matrix
I
with respect to
σ
2
is simply given by
∂
ln
p
2
NK
m
I
=−
=,
(4.150)
E
(
∂
σ
)
2
σ
22
4
from which the CRLB for unbiased estimation of
σ
2
is easily found [6] as
2
NK
4
CRLB
=
.
(4.151)
4.5.3.1.2 CRLB (Standard Deviation)
From the knowledge of the CRLB for unbiased estimation of
σ
2
, the CRLB for
() ()
∂
∂
τ
θ
∂
∂
τ
θ
unbiased estimation of
σ
can be derived using Equation 4.48, i.e.,
I
−
1
,
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