Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 11

Acito et al.
related to the use of the global and the local Gaussian models. Local
model for hyperspectral data was introduced to address the multimodality that usually characterizes real scene due to the presence of
heterogeneous background classes. Thus, it is expected to tightly fit
the background by following its inherent spatial variability. In detection applications, this generally allows obtaining a high detection
rate. However, the local model has its own limits. The high data dimensionality of hyperspectral data entails a high number of degrees
of freedom in the model parameters estimation problems. For this
reason, local model may cause overfitting which, in turn, determines
the increase of the false alarm rate (FAR). It is the well-known phenomenon called the curse of dimensionality [18], [31] according to
which the number of secondary data has to be significantly higher
than the data dimensionality. Furthermore, the use of the local model
in hyperspectral ACD may cause the so-called signal contamination
problem, i.e., the neighborhood adopted to collect the secondary
data may include small size objects (targets). This may be the cause
of the corruption of model parameter estimates that may negatively
impact the performance of the algorithms [22], [24], [53].
The global model has a limited ability to adapt to the scene
heterogeneity thus potentially causing underfitting, which may result in high FAR, as well as low detection rates. Conversely, it
is more resistant than the local model to problems of overfitting
and signal contamination [31]. Furthermore, the global model is
more efficient than the local model under a computational point
of view. In fact, with reference to the presented algorithms, when
we use the global model the covariance/cross-covariance matrices computation and inversion are performed once. Instead, in the
local approach, the matrices evaluation and inversion have to be
performed for each pixel location (i, j). Thus, the computational
load for the local versions of the algorithms is approximately NL
× NS times greater than that of the corresponding global versions.
Obviously, there is not a categorical way to determine which of the
two models provides the best performance. It strongly depends on
the specific considered operating scenario.3
It is important to point out that, though in the previous section
we have also presented the local versions of the state of the art
ACD algorithms, their original derivations use the global model
[40], [41], [42]. For this reason, in the following discussion and
in the examples over real data, we refer to the global versions of
HACD, SACD, SDHACD, and SDACD.
In order to introduce the second issue we discuss, let us start
by focusing on the small size objects present in a given image. The
pixels occupied by those objects have generally spectral signatures
that are anomalous with respect to the majority of the image pixels
(background). For this reason, using a commonly adopted nomenclature, they are referred to as anomalies. The goal of ACD is to
detect small changes resulting from insertion, deletion, or displacement of small size objects (generally man-made), as well as from
small stationary objects whose spectrum changes from one image to
another, as in the case of camouflage, concealment, and deception.
3

Notice that there is a class of algorithms in which the mean vectors are estimated locally and the covariance matrix is computed
globally, This strategy provides some of the advantages of the
local approach with much less computational burden.

JULY 2017, Part II of II

The previous observations lead to the following alternative formulation of the ACD problem: within the class of all the anomalies
in the image pair, we are interested in detecting the temporal anomalies, i.e., those whose anomalousness is perceived also in the temporal domain. To be more exact, the temporal anomalies are those:
C

present in just one of the two images or that occupy different spatial location in the two images, such as the anomalies
due to insertion, deletion, or movement of small size objects
(temporal-spatial anomalies);
located at the same spatial position in the scene and having different spectral content as in the case of camouflage,
concealment, deception, and replacement (temporal-spectral
anomalies).

Consequently, in approaching the ACD problem we are not interested in detecting those anomalies corresponding to the same
small size anomalous object located in the same position in both
the images (nontemporal anomalies).
Returning to the binary decision formulation of the ACD problem, the nontemporal anomalies have to be considered as part of
the H0 hypothesis. For all the presented ACD algorithms, the model parameters in the H0 hypothesis are estimated from the pixels of
the images. Since, the small size objects are rare and occupy a very
small fraction of the scene [41], it is expected that the estimates
of the model parameters tend to characterize the background class
better than the class of the nontemporal anomalies. This might
have an impact on the behavior of each of the presented methods
in terms of capability in discriminating nontemporal and temporal
anomalies. In the following, we discuss such an issue referring to
all the presented ACD algorithms. In Section III, we provide experimental evidence for the points here discussed.
We start our discussion by considering the two algorithms derived under the assumption of the difference vector observation
model (SDACD and SDHACD). In such a case, the robustness
with respect to nontemporal anomalies is guaranteed by the fact
that when the pixels pair y(i, j) or z(i, j) are nontemporal anomalies
their values in each band are expected to be very similar. Consequently, the observation vector e(i, j) = y(i, j) − z(i, j) is likely to
be very close to zero in each band and low values for both TSDACD(i,
j) and TSDHACD(i, j) are expected. This is certainly true when the
two images are perfectly compensated for the radiometric distortion due to the different acquisition conditions. When residual
distortion remains, the observation vector e(i, j) tends to deviate
from zero and the robustness of the algorithm to the nontemporal
anomalies is not guaranteed.
The algorithms derived under the joint vector model (SACD
and HACD) need a more detailed discussion. Let us start by the
SADC algorithm. In general, the anomalous pixels (both temporal
and nontemporal) are rare with respect to the size of the analyzed
images. For this reason, the estimates μˆ 0 and Γˆ 0 in the expression
of the SADC decision statistic characterize the background of the
hyperdimensional image obtained by appending the reference image to the test image (joint image). Thus, TSACD(i, j) can be interpreted as the Mahalanobis distance between the observed vector
e(i, j) and the background of the joint image, and it is a measure

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Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2017 Tutorial XI