Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 9
Acito et al.
rable. In the following, we discuss the validity of the two assumptions in real applications and we propose processing techniques to
effectively apply ACD algorithms also when such assumptions are
not fully satisfied.
Perfect coregistration between the two images is very difficult
to obtain, especially when the images are collected by airborne
platforms. Generally, an RMRE is unavoidable and it should be
taken into account in the ACD algorithms. RMRE can be viewed
as a random shift in both the along-track and the cross-track directions, whose magnitude is typically on the order of a few pixels. It
has been shown [7], [35], [58], that RMRE is detrimental for pixelbased ACD algorithms because it leads to a detection performance
loss increasing with its magnitude.
Recently, an effective approach to cope with the performance degradation caused by RMRE has been proposed [58]. Such an approach,
known as local coregistration adjustment (LCRA), includes the a priori
knowledge on the magnitude of the RMRE into the ACD processing
chain. In LCRA the pixel vector in position (i, j) of the test image is
compared with those belonging to a given neighborhood of the coordinates (i, j) in the reference image. Such a neighborhood is selected by
means of an uncertainty window W whose size is chosen according to
the magnitude of the RMRE itself [8], [58]. In the literature, a technique
named Sequential Residual Mis Registration error Estimator (SRMRE)
[8], [45] has been proposed to estimate the mean values and the covariance matrix of the RMRE and automatically define the window W in
LCRA. Generally, rectangular windows are used centered on the mean
values μx, μy of the RMRE and having size proportional to the standard
deviations σx, σy of the RMRE in the two spatial directions, i.e.,
W=
{( r, p ) :
r − μ y < α ⋅ σ y and p − μ x < α ⋅ σ x
}
(25)
where we let C = Γˆ 0−1 − Γˆ 1−1 for the HACD and C = Γˆ 0−1 for the
SACD. μˆ 0, Γˆ 0, and Γˆ 1are obtained by properly combining μˆ y, μˆ z, Γˆ y,
Γˆ z, and Γˆ yz according to (16).
As to the SDHACD and SDACD, based on the difference vector model, T ( i, j , r , p ) has the following expression:
T
T ( i, j , r , p ) = e ( i, j , r , p ) ⋅ C ⋅ e ( i, j , r , p )
(28)
e ( i, j , r , p ) = y ( i, j ) − z ( i + r , j + p )
with C = Γˆ 0−1 − Γˆ 1−1 in the case of the SDHACD and C = Γˆ 0−1 for the
SDACD. Γˆ 0and Γˆ 1are obtained by combining Γˆ y, , and Γˆ yz according to (21) and (22), respectively.
It is important to point out that when the global model is adopted, Γˆ y, Γˆ z are not affected by the RMRE. Conversely, μˆ 0 (in HACD
and SACD) and Γˆ yz depend on the magnitude of the RMRE. However, the inaccuracy due to the RMRE in the estimation of the joint
mean vector and of the cross-covariance matrix of the two images
might not be so crucial. In fact, for RMRE on the order of a few
pixels, it is expected that the estimation error is partly compensated
by the high spatial correlation of the background.
Under the assumption of the local Gaussian model, the expressions of the sample estimates μˆ z ( i, j , r , p ), Γˆ z ( i, j , r , p ), and
Γˆ yz ( i, j , r , p ) have to be modified in addition to that of the observation vector:
1
z ( k + r, l + p )
N μ ( k ,l )∈Ωμ ( i , j )
1
T
Γˆ z ( i, j , r , p ) =
z ( k + r, l + p ) ⋅ z ( k + r, l + p )
N Γ ( k ,l )∈ΩΓ (i , j )
μˆ z ( i, j , r , p ) =
− μˆ z ( i, j , r , p ) ⋅ μˆ z ( i, j , r , p )
T
According to the LCRA approach, the statistic adopted by a given
ACD detector assumes the following general form:
(
TLCRA ( i, j ) = min T ( i, j , r , p )
( r , p )∈W
)
(
T ( i, j , r , p ) = e ( i, j , r , p ) − μˆ 0
y ( i, j )
e ( i, j , r , p ) =
z ( i + r , j + p )
JULY 2017, Part II of II
)
T
1
T
Γˆ yz ( i, j , r , p ) =
y ( k , l ) ⋅ z ( k + r, l + p )
N Γ ( k ,l )∈ΩΓ (i , j )
− μˆ y ( i, j ) ⋅ μˆ z ( i, j , r , p )
(26)
where T ( i, j , r , p ) denotes the statistic adopted by the given detector, r and p are the row and the column indexes addressing a specific position within the uncertainty window W. In the following,
for all the presented ACD algorithms, we detail the modifications
to be made in order to handle the RMRE.
As to the algorithms based on the global approach, the use of
the LCRA does not require changes in the expressions of the sample estimates μˆ y, μˆ z, Γˆ y, Γˆ z, and Γˆ yz . They continue to be obtained
from all the pixels of the image pair by supposing their perfect
coregistration. Conversely, the expression of the observation vector has to be modified in each algorithm. Specifically, for the algorithms based on the joint vector model (i.e. HACD and SACD)
T ( i, j , r , p ) has the form:
(
⋅ C ⋅ e ( i, j , r , p ) − μˆ 0
)
(27)
(29)
T
μˆ y ( i, j ) and Γˆ y ( i, j ) are the same as (17).
Notice that in the previous expressions we have reintroduced
the pixel location indexes to make explicit the spatial dependence
of the estimates.
For the joint vector model based algorithms (HACD and
SACD) T ( i, j , r , p ) has the form:
T
T ( i, j , r , p ) = e ( i, j , r , p ) − μˆ 0 ( i, j , r , p )
⋅ C ( i, j , r , p ) ⋅ e ( i, j , r , p ) − μˆ 0 ( i, j , r , p )
y ( i, j )
e ( i, j , r , p ) =
z
i
+
r
j
+
p
,
(
)
(30)
where C(i, j, r, p) is equal to Γˆ 0−1 ( i, j , r , p ) − Γˆ 1−1 ( i, j , r , p ) for the
HACD and Γˆ 0−1 ( i, j , r , p ) for the SACD. Such matrices and the
mean vector estimate μˆ 0 ( i, j , r , p ) are obtained as follows:
IEEE A&E SYSTEMS MAGAZINE
9
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