Image Processing Reference
In-Depth Information
where Notice that for statistically independent observations, the
joint PDF is given by the product of the marginal PDFs of the observations:
=
(
,...,
) .
T
1
K
N
∏
p
(
x
|
)
=
p
(
x
|
),
(4.50)
x
n
x
n
n
=
1
with is the PDF of
To construct the ML estimator, we first substitute the available observations
px
n
(
|
)
x
.
x
n
x
1
,
,
x
N
for the corresponding independent variables in Equation 4.38. Because
these observations are numbers, the resulting expression depends only on the ele-
ments of the parameter vector
…
. In a second step, we regard the fixed true param-
eters
as variables. The resulting function
L
(
|
x
1
,
…
,
x
N
) is called the
likelihood
function
of the sample. The ML estimate
of the parameters
is defined as the
value of
that, within the admissible range of
θ
, maximizes the likelihood function
[27,28]:
{
}
=
arg max(ln
L
) .
(4.51)
ML
θ
The most important properties of the ML estimators are the following [25]:
Consistency:
Under very general conditions, ML estimators are consistent,
i.e.,
ˆ
∀∈ |
ε
+
{ |
Pr
− < =
|
ε
)
1
if
N
→∞
,
(4.52)
ML
with denoting the set of positive real numbers.
Asymptotic efficiency:
Under not too restrictive conditions, the covariance
matrix of an ML estimator equals the CRLB asymptotically.
Asymptotic normality:
If the number of data points increases, the proba-
bility density function of an ML estimator tends to a normal distribution.
Invariance property:
If
+
is the ML estimator of the
K
×
1 parameter
ˆ
(
ˆ
ML
vector
, then
ML
)
is the ML estimator of the
L
×
1 vector
(
)
=
(
1
(
),
…
,
L
(
))
T
of functions of
.
4.4
SIGNAL AMPLITUDE ESTIMATION
4.4.1
I
NTRODUCTION
In this section, the problem of signal amplitude estimation from MR data is
addressed. In particular, we focus on the estimation of the magnitude magneti-
zation values from data acquired during an MR imaging procedure.
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