Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 5

Acito et al.

BINARY DECISION PROBLEM AND ACD ALGORITHMS
The goal in ACD is to detect changes in material radiance/reflectance occurred at a given position in the scene. Therefore, ACD
can be formally posed as a binary decision problem, where the two
competing hypotheses for the generic pixel at position (i, j) are

T

H 0 : Change Absent

(1)

H1 : Change Present

The decision has to be taken on the basis of the observed vector e(i,
j) that depends on the realizations y(i, j) and z(i, j) of Y(i, j) and Z(i,
j). In this work, different choices for the relationship (observation
model) between the observed vector e(i, j) and the pixel vectors
y(i, j) and z(i, j) are considered. Each observation model according
to the assumptions made about the statistical distributions of Y(i, j)
and Z(i, j) leads to view the observed vector e(i, j) as drawn from a
multivariate Gaussian distribution with mean value μ0 and covariance matrix Γ0 under the hypothesis H0, and with mean value μ1
and covariance matrix Γ1, under the hypothesis H1, i.e.,
e ( i, j ) H 0 ∈ N ( μ 0 , Γ 0 )

(2)

e ( i, j ) H1 ∈ N ( μ1 , Γ1 )

Since no decision cost and no prior information about the likelihood of each hypothesis is available, the decision problem in (1)
and (2) is solved by using the Neyman-Pearson approach. Accordingly, the decision strategy is based on the likelihood ratio (LR)
Λ(i, j), i.e., the ratio between the probability density functions
(PDF) of the observed vector conditioned on the two hypotheses:
Λ ( i, j ) =

(
)
p ( e ( i, j ) H )
p e ( i, j ) H1

H1
>

0

H0

<

λG

(3)

According to the conditional PDFs model in (2), the decision test
can be rewritten as

Λ ( i, j ) =

Γ0

1/ 2

Γ1

1/ 2

T
 1

exp  − e ( i, j ) − μ1  Γ1−1 e ( i, j ) − μ1  
 2
 H>1 τ (4)
<
T
 1

exp  − e ( i, j ) − μ 0  Γ 0−1 e ( i, j ) − μ 0   H0
 2


Starting from the general form of the test in (4), several ACD algorithms can be derived depending on
a.
b.

the specific assumptions made on the parameters of the
two conditional PDFs,
the observation model adopted.

In the following, we show that some of the most popular ACD
algorithms can be derived according to the decision model summarized by (1)-(4). Specifically, two classes of ACD algorithms
can be derived from (4) both based on a quadratic decision funcJULY 2017, Part II of II

tion of the observed vector e(i, j): the hyperbolic detectors and the
elliptical detectors.
The class of the hyperbolic detectors is derived by assuming
that the mean vectors are the same in the two hypotheses (i.e. μ0 =
μ1). Under such an assumption, by taking the logarithm of both Λ(i,
j) and τ the decision rule in (4) can be rewritten as
T H ( i, j ) = e ( i, j ) − μ 0  Γ d e ( i, j ) − μ 0  >< λ
H
H1

(5)

0

 Γ 1/ 2 
1
−1
−1
with λ = ln τ ⋅
1/ 2  and Γ d = Γ 0 − Γ1 . Since Γd is not a posiΓ
0



tive-definite matrix (it has negative as well as positive eigenvalues)
the contour lines of the test statistic TH(i, j) in (5) are hyperbolas.
The class of the elliptical detectors (or Mahalanobis distance
based detectors) can be derived from (4) by assuming that μ1, which
is related to the potential change occurred, is not known a priori.
To cope with this assumption, a suboptimal decision strategy is exploited. Specifically, the generalized likelihood ratio test (GLRT)
is used, that consists in replacing the unknown μ1 with its maximum likelihood (ML) estimate μˆ 1 obtained from a set of secondary
data having the same PDF as e(i, j)|H1. Generally, such secondary
data are not available in detection applications, where the targets
are rare in the scene. So, the ML estimate μˆ 1 is obtained just by using the observed vector e(i, j). According to the assumed statistical
model, we have μˆ 1 = arg max p e ( i, j ) H1 , μ1 , Γ1 = e ( i, j ). By

{(

μ1

)}

replacing μ1 with μˆ 1, (4) is rewritten as
Γ0

1/ 2

Γ1

1/ 2

1
exp  e ( i, j ) − μ 0
2

The ratio

(

Γ0

)

T

 H1
Γ 0−1 e ( i, j ) − μ 0  >< τ
 H0

(

)

(6)

1/ 2

1/ 2 in (6) is always positive and can be included in the
Γ1
threshold τ, furthermore by taking the logarithm of both sides of
the equation, the GLRT can be reformulated as

(

T E ( i, j ) = e ( i, j ) − μ 0

)

T

(

Γ 0−1 e ( i, j ) − μ 0

)

H1
>
<
H0

λ

(7)

 Γ 1/ 2 
1
with λ = ln τ ⋅
1/ 2 .
 Γ 0 

TE(i, j) is the squared Mahalanobis distance between the observed
vector and the statistical distribution of the observed vector in the
H0 hypothesis. TE(i, j) is a quadratic nonnegative function of the observed vector e(i, j) and assumes high values when a change occurs.
It is important to point out that the multivariate Gaussian model
adopted so far can be interpreted in two distinct ways. Specifically, one can assume that, conditioned to each hypothesis, all the
pixels of a given image are realizations of Gaussian RVs having
the same mean spectrum and the same covariance matrix, and the
RVs associated to the pixels of the image pair have the same crosscovariance matrix all over the scene (global Gaussian model).
Alternatively, local Gaussianity of the hyperspectral data can be
assumed (local Gaussian model). In this case, the first- and secondorder statistics of the pixels of the image pair are assumed to be

IEEE A&E SYSTEMS MAGAZINE

5



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