Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 7

Acito et al.
In (16) μˆ y, μˆ z, Γˆ y, Γˆ z, and Γˆ yz are the sample estimates of μy, μz, Γy,
Γz, and Γyz, respectively. They are obtained using secondary data
from the two images as follows:
1
 y (k,l )
N μ ( k ,l )∈Ωμ ( i , j )
1
μˆ z ( i, j ) =
 z (k,l )
N μ ( k ,l )∈Ωμ (i , j )
μˆ y ( i, j ) =

(

)

TSDHACD ( i, j ) = e ( i, j ) ⋅ Γ d ⋅ e ( i, j ) = e ( i, j ) ⋅ Γ 0−1 − Γ1−1 ⋅ e ( i, j ) <> λ
T

T

H1

(18)

H0

where, according to the adopted observation model and the assumptions in (13)

1
Γˆ y ( i, j ) =
 y ( k , l ) ⋅ yT ( k , l ) − μˆ y (i, j ) ⋅ μˆ Ty ( i, j )
N Γ ( k ,l )∈ΩΓ (i , j )
1
Γˆ z ( i, j ) =
 z ( k , l ) ⋅ zT ( k , l ) − μˆ z ( i, j ) ⋅ μˆ Tz ( i, j )
N Γ ( k ,l )∈ΩΓ (i , j )

(17)

{

T

}

(19)

}

(20)

 ( i, j ) − Z
 ( i, j )  ⋅  Y



= E  Y
  ( i, j ) − Z ( i, j )  | H 0 =

Γ0

(

Γ y + Γ z − Γ yz + Γ

=

1
Γˆ yz ( i, j ) =
 y ( i, j ) ⋅ zT ( i, j ) − μˆ y ( i, j ) ⋅ μˆ Tz ( i, j )
N Γ ( k ,l )∈ΩΓ (i , j )

T
yz

)

and

where Ωμ(i, j) and ΩΓ(i, j), represent the set of position indexes
addressing the pixels assumed as secondary data for estimating the
mean spectra and the second-order static matrices, respectively. Nμ
and NΓ are the numbers of secondary data addressed by the indexes
in Ωμ(i, j) and ΩΓ(i, j).
In the expressions of the sample estimates in (17) we explicitly
reintroduced the pixel position indexes to make the formulas suitable for both the global and the local Gaussian models. Specifically, in the case of the global model Ωμ(i, j) = ΩΓ(i, j) = Ω with Ω
= {(i, j): 1 ≤ i ≤ NS and 1 ≤ j ≤ NL} and Nμ = NΓ = NL × NS. Under
the local model assumption Ωμ(i, j) and ΩΓ(i, j) are defined to select a spatial neighborhood of (i, j). In practice, they are obtained
by means of two distinct rectangular sliding windows centered in.
Notice that the size of both Ωμ(i, j) and ΩΓ(i, j) has to be chosen
as a compromise between statistical accuracy in the estimates of
the background parameters (that requires a large set of data), and
spatial nonstationarity of the background (which favors a small set
of data, [1]). Generally, Nμ < NΓ because more pixels are required
for a reliable estimate of the covariance matrix than the mean vector [33].

Straight Anomalous Change Detector
The SACD can be derived from the elliptical decision rule in (9)
and by using the joint vector observation model in (10). The expressions of μˆ 0 and Γˆ 0 are those in (16) with μˆ y, μˆ z, Γˆ y, Γˆ z, and
Γˆ yz evaluated according to (17). Also in this case, two versions of
the algorithm can be derived depending on the assumed statistical
model (global or local).
It is worth noting that the two algorithms derived from the joint
vector observation model (i.e. HACD and SACD) can be applied,
without any modification, to the case of two hyperspectral data
having a different number of spectral channels.

Simple Difference Hyperbolic Anomalous Change Detector
In order to derive the SDHACD we follow the general procedure
that led to the class of the hyperbolic detectors and we use the
difference vector observation model in (11). Furthermore, when
JULY 2017, Part II of II

the images are radiometrically comparable, in absence of changes
(H0 hypothesis) we reasonably assume that Y(i, j)|H0 and Z(i, j)|H0
have the same mean vector (μy = μz), so we can let μˆ 0 = 0.2 Thus,
(5) can be rewritten as

{

T

 ( i, j ) − Z
 ( i, j )  ⋅  Y



Γ1 = E  Y
  ( i, j ) − Z ( i , j )  | H1 =
Γy + Γz

=

In (18), instead of Γd, which is generally unknown, we use its estimate Γˆ d = Γˆ 0−1 − Γˆ 1−1 with

(

Γˆ 0 = Γˆ y + Γˆ z − Γˆ yz + Γˆ Tyz

)

(21)

and
Γˆ 1 = Γˆ y + Γˆ z

(22)

The sample estimates Γˆ y, Γˆ z, and Γˆ yz are obtained as in (17). Based
on the choice of secondary data global or local version of the algorithm is obtained.

Simple Difference Anomalous Change Detector
The SDACD can be obtained by considering the elliptical decision rule in (9) and by adopting the difference vector observation
model. In this case again, we let μˆ 0 = 0 assuming that μy = μz. The
elliptical statistic decision rule in (9) becomes
1
T
TSDACD ( i, j ) = e ( i, j ) Γˆ 0−1e ( i, j ) >< λ
H

H0

(23)

where Γˆ 0 is derived form (21) by combining the sample estimates
Γˆ y, Γˆ z, and Γˆ yz in (17). Here, again, the global and local versions of
the algorithm are obtained depending on the choice of the secondary data.
2

When different acquisition conditions characterize the two images, a specific RE algorithm is applied before proceeding with
the change detection task. RE aims at compensating the radiometric distortions between the two images thus making reasonable the assumption of equal mean vectors in the no change
hypothesis. However, with a slight modification consisting in
letting μˆ 0 = μˆ y − μˆ z the SDHACD algorithm (and the SDADC
algorithm introduced in the next paragraph) can be generalized
to the case of different mean vectors.

IEEE A&E SYSTEMS MAGAZINE

7



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