Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 4
Introductory View of Anomalous Change Detection
The second phase is the detection step, in which the similarity
between corresponding pixels in the two images is evaluated on the
basis of the observed pixel vectors. Two possible approaches can
be adopted for the detection step: the deterministic approach and
the statistical approach. In the deterministic approach, the anomalous changes are detected by means of a measure of similarity
between corresponding pixel vectors that is derived without any
statistical distribution model assumption. The similarity measure
is defined by exploiting geometrical properties. Within this class,
different detectors were proposed in the literature, based on simple
differencing and angular metrics, such as the change vector analysis (CVA, [10]) the spectral angle mapper (SAM, [7]), the Pearsonian correlation coefficient (PCC, [14]) and the spectral correlation
mapper (SCM, [46]). The method proposed in [61] based on an
orthogonal subspace projection approach also falls in the class of
the deterministic algorithms.
The class of statistical ACD algorithms, which is the focus of
this paper, employs a detection scheme derived in the framework
of statistical decision theory. The two hypotheses H0 and H1 are
statistically characterized by means of multivariate distribution
models and the detection step is accomplished by resorting to a
specific decision rule consisting in a threshold based test [48],
[54] [55].
In this paper, we propose a general framework to approach the
ACD binary decision problem according to the statistical decision
theory. Particularly, we clearly define the observation space, the
data statistical distribution conditioned to the two competing hypotheses and the procedure followed to arrive at the solution. We
show that the most popular statistical based ACD algorithms can
be included in the proposed unified framework. Furthermore, an
original ACD algorithm is also presented within the same statistical
framework. It is worth noting that a unified framework based on the
statistical decision theory that includes different ACD techniques is
a useful aid for those researchers that are familiar with the statistical
detection theory and face the ACD problem for the first time.
In the paper, we also discuss the strategies that are generally
adopted to cope with the problems generated by different acquisition conditions and RMRE. We describe the whole ACD data processing chain that includes the above-mentioned strategies. Specifically, we show how the recalled or introduced ACD algorithms
can be applied accounting for radiometric equalization (RE) and
RMRE compensation.
We also analyze the critical issues and the strengths of the algorithms by enriching the discussion with examples on real data.
Specifically, the behavior of the presented ACD strategies is analyzed with reference to real airborne hyperspectral images, which
are part of a freely available ground-truthed data set recently released for testing and comparing detection algorithms.
The paper is organized as follows. In Section II the general
framework for ACD algorithms derivation is presented. The specific form of the algorithm decision rules is summarized in Subsection II-A, whereas Subsections II-B and II-C are dedicated to the
detailed description of the strategies for radiometric distortion and
RMRE compensation and to their integration within the ACD processing chain. Useful observations about the presented algorithms
are reported in Subsection II-D. Finally, examples over experimen4
tal data are presented and discussed in Section III. Conclusions are
presented in Section IV.
ANOMALOUS CHANGE DETECTION
Let us denote as Y ∈ R L× N S × N L and Z ∈ R L× N S × N L the two HSI taken
at different times, L being the number of spectral channels1 and NS,
NL the number of samples and lines, respectively. Using a commonly accepted nomenclature, such images are referred to as the
test image and the reference image, respectively. Let us also denote
as Y(i, j) = [Y1(i, j), Y2(i, j),..., YL(i, j)]T and Z(i, j) = [Z1(i, j), Z2(i,
j),..., ZL(i, j)]T the generic pixel vector of the at-sensor radiance of
the test and reference image, respectively. Each pixel is viewed as
an L × 1 random vector (RV) in the spatial position (i, j) (i and j
being the sample and the line indexes respectively, i = [1,...,NS], j
= [1,...,NL]).
Y(i, j) and Z(i, j) are modeled as Y(i, j) = Sy(i, j) + Ny(i, j)
and Z(i, j) = Sz(i, j) + Nz(i, j), where Sy/z(i, j) = μy/z(i, j) + S y / z(i, j)
represents the "useful signal" and Ny,z(i, j) is the noise term in the
test image and the reference image, respectively. The term "useful
signal" is here introduced to denote the contributions to the sensor's measured signal accounting for the radiation reflected and/or
emitted by the materials in the spatial resolution cell corresponding
to the coordinates (i, j). μy(i, j) and μz(i, j) are deterministic vectors,
whereas S y ( i, j ), S z ( i, j ), Ny(i, j), and Nz(i, j) are assumed to be
zero mean jointly Gaussian distributed RVs. The useful signal term
and the noise term in each image are assumed to be statistically
uncorrelated. Furthermore, the noise terms Ny(i, j), and Nz(i, j) are
also assumed to be uncorrelated. Denoting with Γ S y, Γ S z, Γ N y and
Γ N z the covariance matrices of S y ( i, j ), S z ( i, j ), Ny(i, j), and Nz(i, j),
we have that Y(i, j) and Z(i, j) are multivariate Gaussian RVs with
mean vectors μy and μz and covariance matrices Γ y = Γ S y + Γ N y and
Γ z = Γ S z + Γ N z, respectively.
It is important to recall that, in ACD, two problems should be
taken into account. First, the test and the reference images are usually collected under different atmospheric/illumination conditions.
This induces pervasive radiometric changes in all the image pixels.
Such pervasive highly correlated changes must be estimated and
suppressed to highlight and detect the subtler changes due to small
object insertion/deletion. Second, it is difficult to obtain a perfect
alignment of the two images especially when the sensor is mounted
on airborne platforms. This results in the RMRE that is detrimental
for ACD algorithms [7], [35], [58] aimed at detecting small changes having size on the order of the image pixel.
For better clarity, in this section we first (Subsection II-A)
formalize the ACD as a binary decision problem and derive the
detection algorithms assuming that the two images are perfectly
coregistered and radiometrically comparable. Then, in Subsection II-B, we extend the proposed algorithms taking into account
coregistration errors and changes in atmospheric and illumination
conditions.
1
Notice that the assumption of equal number of spectral channels
is not mandatory for all the algorithms proposed in this paper. As
clarified later, such assumption is not necessary when the joint
vector observation model is adopted for the two images.
IEEE A&E SYSTEMS MAGAZINE
JULY 2017, Part II of II
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