Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 8
Introductory View of Anomalous Change Detection
Notice that when the images are perfectly coregistered, the algorithms derived by assuming both the joint vector and the difference
vector observation models do not make any distinction between the
test and the reference image. In fact, it can be easily proved that the
test statistics of the HACD, SACD, SDHACD, and SDACD do not
vary if the roles of test and reference images are swapped.
Single Vector based Anomalous Change Detector
The SVACD is derived from (9) by assuming the local multivariate
Gaussian model under the H0 hypothesis and the SV observation
model.
To motivate the SVACD let us consider that the goal of an ACD
algorithm is the detection of temporal anomalies in two hyperspectral images of the same scene collected at different times. In general, anomalies can be detected measuring the similarity between the
pixel under test and the surrounding background. The problem of
anomaly detection in a single hyperspectral image has been deeply
investigated in the past. In this framework, one of the most popular anomaly detection algorithms is the Reed-Xiaoli (RX) detector
[43] that measures the similarity between the pixel under test and
the surrounding background by means of the squared Mahalanobis
distance. To compute such a distance, the RX detector estimates the
background mean spectrum and covariance matrix from pixels in a
local neighborhood of the pixel under test. Specifically, such parameters are estimated locally using the pixels taken in a neighborhood
of the pixel under test. The SVACD is inspired by the RX algorithm.
Specifically, it computes the similarity between each pixel of the test
image and the surrounding background. To detect temporal anomalies, rather than spatial and spectral anomalies in the individual image, the background parameters are now estimated by using the pixels of the reference image instead of those taken from the test image.
Thus, to derive the SVACD the SV observation model is adopted.
It assumes, as observed vector in the position (i, j), the pixel vector of
the test image at the same position and the H0 hypothesis is statistically characterized based on the reference image. This means that we
consider as secondary data only the pixels of the reference image.
The assumption of local stationarity for the observations is taken into account by modeling e(i, j)|H0 as a multivariate Gaussian
random process whose mean vector μ0(i, j) = μz(i, j) and covariance
matrix Γ0(i, j) = Γz(i, j) are spatially varying but locally stationary,
i.e., they do not change in a local neighborhood of the spatial position (i, j). Particularly, according to the hyperspectral data model in
[27] the covariance matrix is assumed to be more slowly spatially
varying than the mean vector. According to (9), the decision rule
associated to such data model is given by
(
)
(
TSVACD ( i, j ) = y ( i, j ) − μˆ z ( i, j ) ⋅ Γˆ −z 1 ( i, j ) ⋅ y ( i, j ) − μˆ z ( i, j )
T
)
H1
>
<
H0
λ
(24)
In (24), μˆ z ( i, j ) is the sample estimate of μz(i, j), evaluated on Nμ
secondary data, whereas Γˆ z ( i, j ) is the sample estimate of Γz(i, j),
evaluated on NΓ secondary data. They are obtained according to the
formulas in (17), with Ωμ(i, j) and ΩΓ(i, j) defined by two distinct
sliding windows centered at the position (i, j) [17], [43].
Since the mean vector is supposed to vary spatially faster than
the covariance matrix the two sliding windows are designed to
8
have . The choice of the dimension of the estimation windows is
crucial. As to ΩΓ(i, j), on one hand it has to be large enough to
accurately estimate the covariance matrix Γz(i, j) (in particular, at
least L + 1 samples are needed to guarantee the inversion of the
covariance matrix), on the other hand it should also be as small as
possible to capture a homogeneous background. This is particularly important in complex scenarios, such as urban areas, where
the background is highly structured. Moreover, if a target pixel relocates in a different position within ΩΓ(i, j), it could bias the estimates of Γz(i, j) and impair the algorithm performance. Finally, it is
important to remark that the computational burden of the SVACD
increases with the size of the estimation windows. In practice, NΓ =
B · L, B ∈[5, 10] represents [47] a good compromise between the
two conflicting requirements mentioned above.
As to Ωμ(i, j) the same tradeoff between estimation accuracy
and contamination due to nonhomogeneous background applies. In
this case, we do not have the matrix inversion constraint thus the
number of secondary data can be lower than L. However, to guarantee good performance in detecting not only those targets that are
inserted into the scene but also the ones deleted or moved within
the scene, Nμ should be on the order of their spatial extent (in pixel
units). The value of Nμ basically affects the detection of anomalous
targets that are present in the reference image and not in the test
image. To better clarify this point let us consider the behavior of the
SVACD in the spatial position corresponding to a background pixel
in the test image (y(i, j)) and a target pixel in the reference image.
With Nμ on the order of the target spatial extent, the annulus Ωμ(i, j)
mostly encloses spectral pixels of the anomalous target in the reference image and μˆ z ( i, j ) is strongly biased by the spectral content of
such a target, which is anomalous with respect to y(i, j). Thus, the
values of the squared Mahalanobis distance in (24) are expected to
be high. Conversely, for increasing values of Nμ the annulus Ωμ(i, j)
tends to include more and more background pixels making the estimate μˆ z ( i, j ) closer to y(i, j) and reducing the value of TSVACD(i, j).
To conclude this subsection, we recall that the ACD algorithms
have been derived by assuming the multivariate Gaussian model
(global and/or local). In some applications [3], [32] it has been observed that the data distributions could have heavier tails than the
Gaussian model. Thus, for those applications, the use of heavy tails
statistical models has been proposed. For example, in [57] the use of
the elliptically contoured (EC) distributions has been investigated also
for ACD applications. Specifically, the authors considered the EC-t
distribution model. We would like to point out that for the specific
class of the elliptical detectors, the analysis carried out in this subsection can be generalized to the case of the EC model whose PDF is
a monotonically decreasing function of the Mahalanobis distance. In
such a class of EC model the EC-t distribution is included. It is worth
noting that (see Appendix I), the decision rule in (9) and, consequently,
the algorithm based on that rule, can be derived by assuming the class
of EC distributions instead of the multivariate Gaussian one.
DEALING WITH RESIDUAL MIS-REGISTRATION ERRORS
In Subsection II-A, the detectors were derived by assuming that the
two images: a) are perfectly coregistered, and b) are acquired under the same atmospheric conditions and radiometrically compa-
IEEE A&E SYSTEMS MAGAZINE
JULY 2017, Part II of II
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