Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 6

Introductory View of Anomalous Change Detection
stationary in a given neighborhood of each pixel location (i, j) and
not all over the scene. Generally, such a local model is adopted in
conjunction with the additional assumptions made in [27], where
a generic optical image is modeled as a nonstationary multivariate
random process with a quickly spatially varying mean vector and a
more slowly varying second-order statistics.
To be rigorous, under the assumption of the local model, the
previous equations should be modified by introducing the pixel
location indexes (i, j) in μ0, μ1, Γ0, and Γ1 to make clear their spatial
dependence. In the following, without loss in generality, we omit
the indexes (i, j) excepted for those cases where we need to explicitly distinguish the local from the global model.
Notice that, in deriving (5) and (7), we assumed that μ0, Γ0, and
Γ1 were known. In practice, such parameters are not available and
must be replaced by their estimates μˆ 0, Γˆ 0, and Γˆ 1 Consequently, the
detection rules in (5) and (7) assume the form:

(

T H ( i, j ) = e ( i, j ) − μˆ 0

)

T

(

Γˆ d e ( i, j ) − μˆ 0

)

H1
>
<

λ

H0

Γˆ d = Γˆ 0−1 − Γˆ 1−1

(8)

cross-covariance matrix Γyz has no zero entries. Conversely, under the hypothesis H1, when different materials occupy the spatial
position (i, j), it is reasonable to assume that Y(i, j) and Z(i, j)
are uncorrelated RVs (i.e., with zero cross-covariance matrix). In
formulas:

} {

{

(

T E ( i, j ) = e ( i, j ) − μˆ 0

)

(

Γˆ 0−1 e ( i, j ) − μˆ 0

)

<
H0

λ

(9)

The binary decision framework introduced in this section can be
used to derive different ACD algorithms drawn from the decision
rules in (8) and (9). The algorithms can be derived by considering
different observation models, different forms (global or local) of
the multivariate Gaussian model and different sets of secondary
data used to estimate the model parameters μ0, Γ0, and Γ1.
Specifically, three different observation models are analyzed:
C

the joint vector model, that considers e(i, j) as obtained by
concatenating the pixel vectors y(i, j) and z(i, j):

 y ( i, j ) 
e ( i, j ) = 

 z ( i, j ) 
C

the difference vector model, in which the difference between
the pixel vectors y(i, j) and z(i, j) is assumed to be the observed vector in the spatial position (i, j):

e ( i , j ) = y ( i , j ) − z ( i, j )
C

(10)

(11)

the single vector (SV) model, that assumes as the observed
vector in the position (i, j), the pixel vector of the test image
at the same position:

e ( i, j ) = y ( i, j )

(12)

Regardless of the specific observation model, we may reasonably
assume that when no change occurs between the two images, the
pixel vector y(i, j) is strongly related to z(i, j) because they correspond to the same material. So, it can be assumed that the RVs
Y(i, j) and Z(i, j) are statistically correlated [54], [55] and that their
6

}

(13)

The two equations represent the cross-covariance matrices between
the test and the reference image, under the two hypotheses H0 and H1.
 ( i, j ) = S ( i, j ) + N ( i, j ), Z ( i, j ) = S ( i, j ) + N ( i, j ),
In (13), Y
y
y
z
z
E{} denotes the expectation operator and 0 is the L × L matrix with
all zero entries.
In the following, we show that five different ACD algorithms
can be derived according to the proposed framework. Specifically,
two hyperbolic detectors are considered:
C

H1
>

}

 ( i, j ) ⋅ Z T ( i, j ) | H = E S ( i, j ) ⋅ S T ( i, j ) | H = 0
E Y
y
z
1
1

C
T

} {

{

 ( i, j ) ⋅ Z
 T ( i, j ) | H = E S ( i, j ) ⋅ S T ( i, j ) | H ≠ 0
Γ yz = E Y
y
z
0
0

C

C
C

the hyperbolic ACD (HACD), that was proposed in [56];
the simple difference hyperbolic ACD (SDHACD) that has
a decision rule similar to that of the CE standard proposed
in [55] and three algorithms based on the elliptical decision
rules in (9):
the straight anomalous change detector (SACD) introduced
in [54] and [55];
the simple difference ACD (SDACD) proposed in [55];
the SV based ACD (SVACD) which is a completely new
detector derived assuming a local multivariate Gaussian
model.

Hyperbolic Anomalous Change Detector
The HACD algorithm can be derived from the detection rule in (8),
using the joint vector observation model. Specifically, we can start
from the expression of the hyperbolic decision rule in (5) where, in
accordance with the hypotheses in (13):
 Γy
Γ0 =  T
 Γ yz

Γ yz 

Γz 

Γ
Γ1 =  y
0

0

Γz 

(14)

The mean vector μ0 of e(i, j), assumed to be the same under both
μ y 
the hypotheses, is μ 0 =   with μy and μz representing the mean
μ z 
vector of the test image and the reference image, respectively.
According to (9), the decision rule for the HACD is

(

THACD ( i, j ) = e ( i, j ) − μˆ 0

with
μˆ 
μˆ 0 =  y 
 μˆ z 

IEEE A&E SYSTEMS MAGAZINE

)

T

(

Γˆ d e ( i, j ) − μˆ 0

)

H1
>
<

λ

(15)

H0

Γˆ d = Γˆ 0−1 − Γˆ 1−1

 Γˆ y
Γˆ 0 =  T
 Γˆ yz

Γˆ yz 

Γˆ z 

 Γˆ
Γˆ 1 =  y
 0

0

ˆΓ 
z

(16)

JULY 2017, Part II of II



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