Image Processing Reference
In-Depth Information
,
The diagonal elements of represent the variances of the elements of
whereas the nondiagonal elements represent the covariances between the elements
of the estimator.
C θ
4.3.3
A CCURACY
The accuracy of an estimator can be described in terms of its bias. The bias of
an estimator is defined as the deviation of the expectation value of the parameter
(vector) from the true value:
(4.42)
Hence, the bias represents the systematic error. If the expectation of the esti-
mator equals the true value of the parameter, the estimator is said to be unbiased.
Otherwise, it is biased.
b ()
=
[]
− .
E
4.3.4
MSE
A potential measure of the quality of an estimator, taking account of both accuracy
and precision, is given by the MSE. The MSE of the k th element of the estima-
tor
is defined as
2
MSE(
)
=
(
)
.
(4.43)
E
k
k
k
Note that the MSE can also be written as the sum of the variance of the
estimator and its bias squared:
( ˆ
( ˆ
( ˆ
MSE
)arcsin
θ =
b
2
)
+
Var
).
(4.44)
k
k
k
The MSE of the vector estimator
is given by the scalar value:
T
MSE( )
=
[(
) (
)]
(4.45)
E
K
k
MSE( ˆ
=
).
(4.46)
k
=
1
4.3.5
CRLB
The same parameter can be estimated using different estimators. Generally, dif-
ferent estimators have different precisions. Then, one might ask what precision
might be achieved, or, in other words, is there a lower bound on the attainable
variance? The answer is that such a lower bound exists. It can be computed from
the joint PDF of the observations (i.e., data points) as follows [25].
 
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