Digital Signal Processing Reference
In-Depth Information
and the corresponding eigenvector is estimated by
v
3
≈
v
3
. The estimated principal
eigenvalues are given by
ˆ
ˆ
λ
1
≈
λ
1
−
λ
3
,
(2.13)
and
ˆ
ˆ
λ
2
≈
λ
2
−
λ
3
.
(2.14)
v
2
. Before the devel-
opment of our segmentation algorithm, we should further analyze the statistical
properties of the above estimators such that the thresholds of the signal and noise
subspaces are reasonably designed.
While the estimated eigenvectors are
v
1
≈
v
1
and
v
2
≈
2.2.3.1
Statistical Analysis of Eigen-Structures
The covariance matrix of the interested spatial samples is given by
g
k
as in (
2.5
). The
covariance matrix
R
g
and its estimated matrix
R
g
are respectively defined in (
2.9
)
and (
2.11
). Based on
R
g
, we can obtain the estimated signal and noise subspaces.
To explore their expectations and deviations, we should first analyze the asymptotic
statistics for the eigenvalues and eigenvectors of the sampled covariance
R
g
under
the Gaussian process assumption [
28
]. Based on the perturbation formulation, the
first- and second-order moments of
ˆ
λ
i
and
v
i
can be obtained [
20
,
29
]. The eigen-
vectors of the signal subspace
v
i
and its associated eigenvalues
ˆ
λ
i
are asymptotically
ˆ
normal with noise subspaces
v
j
and
λ
j
,for
i
,
j
=
1
,
2
,
3
,
i
=
j
. According to [
20
], the
ˆ
expectation value of
v
i
and the covariance of
λ
i
can be expressed by (
2.15
)and
(
2.16
) as follows:
3
∑
j
=
1
,
j
=
i
1
2
λ
i
λ
j
(
λ
j
−
λ
i
)
E
[
v
i
]
≈
v
i
−
2
N
v
i
,
(2.15)
2
i
λ
j
)
≈
δ
λ
ˆ
ˆ
i
,
j
cov
(
λ
i
,
,
(2.16)
N
where
N
is the number of sampled pixels and
δ
i
,
j
is the Kronecter delta. The esti-
ˆ
mated values of
v
i
and
λ
i
can be expressed by
ˆ
λ
i
=
λ
i
+
ξ
i
,
(2.17)
and
v
i
=
v
i
+
s
i
,
(2.18)
where the error terms,
s
i
and
ξ
i
have the following asymptotic properties [
20
]:
2
i
N
]
≈
λ
2
σ
λ
i
λ
j
=
E
[
ξ
ξ
δ
,
(2.19)
i
j
i
,
j
