Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 12

Introductory View of Anomalous Change Detection
of the degree of anomalousness of e(i, j). Notice that, according
to the joint observation model in (10), TSACD(i, j) is expected to be
high when a) one of the two pixel vectors y(i, j) or z(i, j) is anomalous with respect to the corresponding image background, or b)
both the pixels are anomalous. As a consequence, the SACD is
less effective in discriminating the temporal from the nontemporal
anomalies.
Such a problem is mitigated by the HACD algorithm, whose decision statistic in (15) can be rewritten as

(

)

(

)

T
THACD ( i, j ) = TSACD ( i, j ) − e ( i, j ) − μˆ 0 Γˆ 1−1 e ( i, j ) − μˆ 0 . By exploiting the block diagonal form of Γˆ 1 (and Γˆ 1−1) the expression of
THACD(i, j) can be rewritten as

THACD ( i, j ) = TSACD ( i, j ) − TY ( i, j ) − TZ ( i, j )

(
)
( i, j ) = ( z ( i, j ) − μˆ )

TY ( i, j ) = y ( i, j ) − μˆ y
TZ

z

(
)
( z ( i, j ) − μˆ )

T

Γˆ −y1 y ( i, j ) − μˆ y

T

Γˆ −z 1

(36)

z

where TY(i, j) and TZ(i, j) can be interpreted as measures of anomalousness of y(i, j) and z(i, j) with respect to the test and the reference image background, respectively. When y(i, j) and z(i, j) are
nontemporal anomalies, not only TSACD(i, j) assumes high values
but also TY(i, j) and TZ(i, j). Thus, according to the relationship
in (30), THACD(i, j) is expected to be low and it could also assume
negative values. Conversely, when just one of y(i, j) and z(i, j)
is anomalous (temporal anomaly), TSACD(i, j) tends to be high and
only one between TY(i, j) and TZ(i, j) is high. Consequently, THACD(i,
j) is likely to assume a high value.
It is worth noting that the same mechanism that determines the
robustness of the HACD with respect to the nontemporal anomalies, may be responsible of a reduced sensitivity of the algorithms
to the temporal-spectral anomalies. In fact, when y(i, j) and z(i, j)
are both anomalous and correspond to a different spectral reflectance content (temporal-spectral anomalies), TSACD(i, j) as well as
TY(i, j) and TZ(i, j) are expected to be high. Consequently, according to (30), THACD(i, j) might assume low values, thus hindering the
detection of temporal-spectral anomalies.
The performance of the SVACD in discriminating temporal
and nontemporal anomalies depends on the relationship existing
between Nμ and the spatial extent (in pixel units) of the anomalous targets. When Nμ is close to the targets extent, and the spatial
position under test contains a nontemporal anomaly (no change),
it is expected that Ωμ(i, j) mostly includes spectral pixels of the
anomalous target in the reference image. Consequently, the background mean estimate μˆ z ( i, j ) is very similar to the pixel under
test. Thus, the difference between the observed vector and the estimated mean spectrum tends to be small thereby reducing the value
of the squared Mahalanobis distance (TSVACD(i, j)). Conversely,
when, a change occurs in the considered spatial position, the mean
background spectrum estimated from the reference image differs
from the spectrum of the pixel in the test image, and the squared
Mahalanobis distance is expected to be higher. At this point, it is
also worth making some comments on the influence of the RMRE
on ACD algorithms. Of course, RMRE is certainly detrimental for
the detection performance of each algorithm. However, the effects
12

of the RMRE on the algorithm performance might be different depending on the specific decision statistic. Regardless of the specific
ACD algorithm, RMRE increases the number of false changes especially along the edges between two background classes with different spectral characteristics. Furthermore, an increase of the false
changes caused by the nontemporal anomalies is also expected. In
fact, due to the random shift in the spatial dimensions, small objects located at the same position during both the acquisitions, tend
to occupy pixels with different row and column indexes in the test
and the reference image. Such pixels are detected as changes even
if they are not indeed anomalous under a temporal point of view.
This essentially has a negative impact on the performance of those
algorithms that allow a better discrimination between the temporal
and the nontemporal anomalies. Specifically, RMRE is expected to
affect greatly the performance of HACD, SDHACD, and SDACD
rather than that of the SACD. As to the SVACD, the local processing should mitigate the RMRE effects both for the nontemporal
anomalies and the edge pixels. In fact, when the RMRE strength
is lower than the size of the window for the mean spectrum estimation and comparable with the spatial correlation extent of the
observed data, the bias of the spectral anomalous pixels or the edge
pixels in the reference image provides a mean spectrum that does
not differ significantly from the pixel under test. As a consequence,
the difference between the observed vector and the estimated mean
spectrum as well as the squared Mahalanobis distance in (24) tends
to remain small.
It is worth pointing out that an effective way to limit the abovementioned negative effects of the RMRE is that summarized in
Subsection II-B which exploits the LCRA approach.
To conclude this section, in Fig. 1 we show the block diagram
that summarizes the typical processing chain of an ACD algorithm.
First, the two images are made radiometrically comparable by
means of the RE block,4 then the RMRE magnitude is estimated
(if no a priori information is available) and finally the decision rule
is applied according to the LCRA approach in order to take into
account the RMRE. The ACD process can be further enforced by
means of additional preprocessing and postprocessing techniques.
Improvements in terms of detection performance are expected
from the use of preprocessing techniques aimed at reducing the
noise in the two images [4], [29]. Moreover, benefits in terms of
computational cost can be obtained by using data dimensionality
reduction algorithms [1], [28], [40].
Postprocessing includes techniques to reduce the number of
false detected changes (false alarms mitigation). For instance, given that the goal of ACD is the detection of small size objects, false
alarms mitigation can be achieved by discarding the detected set of
connected pixels whose size is greater than a predefined threshold.

EXPERIMENTAL RESULTS
To illustrate with practical examples how the ACD processing chain
in Fig. 1 can be applied to real data, we present some results obtained
on the "Viareggio 2013" data set (freely available online at http://
4

Notice that the RE task is not necessary when the images are
collected under similar conditions (e.g with a very small time
delay)

IEEE A&E SYSTEMS MAGAZINE

JULY 2017, Part II of II
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