Image Processing Reference
In-Depth Information
If the mean-square error is de
ned as
h
i
Þ
T
o*
E
¼
Eo*
ð
o
ð
o
Þ
(
9
:
133
)
where o* is the estimation of o and E[
] stands for the expectation operator, then the
problem of gray-level patch selection for measurement purpose can be formulated
as a method to
1, 2, . . . , S)
that yields the least error E. There are three factors that contribute to the error E: (1)
the noise n, (2) the low rank approximation if not all of the eigenvectors are used,
and (3) the estimation error of a. The
find the matrix H (or equivalently to
find g(i) for i
¼
first two of these factors are not related to
the gray-level sample selection and, therefore, only the estimation error of a is
important.
The set of gray levels used for measurement that possess the maximum disper-
sion provides the minimum error E which is equivalent to minimizing the trace of a
matrix expressed as
n
o
1
tr
H
T
H
Optimal gray levels ¼
S
¼ min
S
(
9
:
134
)
For example, if the G
1 vector e is de
ned as the error introduced by estimation
inaccuracy, it can be evaluated as
e
¼
V
D
a
(
9
:
135
)
where
1 estimation error vector. Note that V is orthonormal,
estimation a* is unbiased, and the expected error energy can be evaluated as
D
a
¼
a*
a is a K
X
Ee
T
e
¼
E
D
a
T
V
T
V
D
a
¼
E
D
a
T
D
a
¼
Var
a*
½
(
k
)
(
9
:
136
)
k
The covariance matrix of a*, as described in Ref. [54] can be evaluated as
1
ðÞ¼
cH
T
H
Cov
a*
(
9
:
137
)
where c is a constant, yielding
h
i
X
1
¼ tr
cH
T
H
Var
a*
½
(
k
)
¼tr Cov
a*
½
ðÞ
(
9
:
138
)
k
To reduce the computation time, the potential candidates are
first pruned. In matrix
V, if two columns are proportional, speci
cally, if v
kj
¼a
v
ki
for k
¼
1, 2, . . . , K, and
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