Image Processing Reference
In-Depth Information
distributed [43-48]. As magnitude MR data are Rician distributed, these images
require noise estimation methods that exploit this knowledge. In this subsection,
several methods will be discussed, which can be classified as follows:
Single-acquisition methods:
In MRI, the image noise variance is commonly
estimated from a large uniform-signal region or nonsignal regions within
a single-magnitude MR image [19,39]. In this section, we consider ML
estimation of the noise variance from complex (Subsection 4.5.2) and
magnitude MR data (Subsection 4.5.3) from a region in which the
underlying signal amplitude is nonzero but constant, as well as from a
background region, i.e., a region in which the underlying signal ampli-
tude is zero[49].
Double-acquisition methods:
Furthermore, methods were developed based
on two acquisitions of the same image: the so-called double-acquisition
methods [40,50,51]. At the end of Subsection 4.5.3.3, we will briefly
describe a robust method based on a double-acquisition scheme[1,24].
4.5.2
N
OISE
V
ARIANCE
E
STIMATION
FROM
C
OMPLEX
D
ATA
2
needs to be estimated from
N
complex-valued
observations . We will consider the case of identical underlying
phase values as well as the case of different underlying phase values.
Suppose the noise variance
σ
c
=
{(
ww
,
)}
rn
,
in
,
4.5.2.1
Region of Constant Amplitude and Phase
Let us first consider a region with a constant, nonzero underlying signal amplitude
and identical underlying phase values.
4.5.2.1.1 CRLB
The Fisher information matrix of (
A
,
ϕ
,
σ
2
) is given by
∂
2
ln
p
∂
∂∂
2
ln
p
∂
∂∂
2
ln
p
c
c
c
N
∂
A
A
ϕ
A
σ
2
2
0
0
σ
2
∂
∂∂
ln
p
∂
ln
p
∂
∂∂
ln
p
2
2
2
NA
2
c
c
c
I
=−
=
0
0
,
(4.134)
E
ϕ
A
∂
ϕ
ϕ σ
σ
2
2
2
N
∂
∂∂
ln
p
A
∂
∂∂
ln
p
∂
∂
ln
()
p
c
0
0
2
2
2
c
c
σ
4
σ
2
σ ϕϕσ
2
22
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