Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 26
Introductory View of Anomalous Change Detection
class of EC distributions whose PDF is a monotonically decreasing
function of the Mahalanobis distance. In this case the PDFs of the
observation e(i, j) conditioned on the two hypotheses are [3]
(
)
(
)
f e ( i, j ) H 0 =
f e ( i, j ) H1 =
1
( 2π )
L
2
Γ0
1
( 2π )
L
2
)
(
)
hL d1 ( i, j )
1
2
Γ1
(
hL d 0 ( i, j )
1
2
(37)
APPENDIX II
In this Appendix we prove that the test statistics of the HACD and
the SACD are invariant to a linear transformation applied to one of
both the vectors y(i, j) and z(i, j) in e(i, j). Particularly, denoting as
HACD(y, z; i, j) the test statistic THACD(i, j) and as SACD(y, z; i, j)
the test statistic TSACD(i, j), we prove that
HACD ( y , z; i, j ) = HACD ( R ⋅ y + b, K ⋅ z + m; i, j )
SACD ( y , z; i, j ) = SACD ( R ⋅ y + b, K ⋅ z + m; i, j )
where Γ0 and Γ1 are the covariance matrices under the two hypotheses, hL(x) is a monotonically decreasing function of x and depends also on the data dimension (L), and d0(i, j) and d1(i, j) are the
squared Mahalanobis distances:
T
T
1
−1
0
0
−1
1
1
(38)
z T ( i , j ) = K ⋅ z ( i, j ) + m
1
In (38) μ0 and μ1 are the mean vectors of e(i, j)|H0 and e(i, j)|H1,
respectively.
Following the same reasoning as in Section II, we can express
the GLRT with respect to the parameter μ1 as
Λ G ( i, j ) =
Γ0
Γ1
1
2
1
2
))
( (
h ( d ( i, j ) )
max hL d1 ( i, j )
μ1
H1
>
Λ G ( i, j ) =
Γ1
1
2
1
2
hL ( 0 )
1
(
hL d 0 ( i, j )
Considering that the ratio hL ( 0 )
)
H1
>
<
λG
(40)
H0
Γ0
1/ 2
Γ1
1/ 2
is greater than zero and that
hL(d0(i, j)) is a monotonically decreasing function of d0(i, j), the
GLRT in (40) can be reformulated as
(
d 0 ( i, j ) = e ( i , j ) − μ 0
)
T
(
Γ 0−1 e ( i, j ) − μ 0
)
H1
>
<
λ*
(41)
H0
As in the Gaussian model case considered in Section II, being μ0
and Γ0 not known a priori, they are replaced by their ML estimate
Γˆ 0 and μˆ 0 from the secondary data. The resulting detector becomes
(
T ( i, j ) = e ( i, j ) − μˆ 0
)
T
(
Γˆ 0−1 e ( i, j ) − μˆ 0
)
H1
>
<
λ
(42)
H0
which is exactly the same expression as that reported in (9).
26
e T ( i, j ) = eT ( i, j ) − μˆ 0,T
μˆ 0,T = μˆ yT T , μˆ zT T
Being hL(d1(i, j)) a monotonically decreasing function of d1(i, j) it
has its maximum value in d1(i, j) = 0, thus:
Γ0
T
T
T
eT ( i, j ) = y T ( i, j ) , zT ( i, j )
(39)
λG
<
H0
0
L
where R and K are two generic L × L full rank matrices and b and
m are two generic L × 1 vectors.
We start by letting
y T ( i, j ) = R ⋅ y ( i, j ) + b
(
) Γ ( e ( i, j ) − μ )
d ( i , j ) = ( e ( i, j ) − μ ) Γ ( e ( i , j ) − μ )
d 0 ( i, j ) = e ( i, j ) − μ 0
(43)
T
(44)
with μˆ yT and μˆ zT obtained according to (14) from all the pixels yT(i,
j) and zT (i, j).
It can be easily proved that μˆ yT = R ⋅ μˆ y + b and μˆ zT = K ⋅ μˆ z + m.
So e T ( i, j ) can be rewritten as
R 0
e T ( i, j ) =
⋅ e ( i, j ) = D ⋅ e ( i, j )
0 K
R 0
D=
0 K
(45)
where 0 represents the L × L matrix with all-zero entries and
e ( i, j ) = e ( i, j ) − μˆ 0. According to the expressions in (14) and the
notation in (44) and (45), it is straightforward to obtain the following relationships:
NS NL
Γˆ yT =
1
NS ⋅ NL
y ( i, j ) ⋅ y ( i, j )
Γˆ zT =
1
NS ⋅ NL
z ( i, j ) ⋅ z ( i, j )
Γˆ yT zT =
IEEE A&E SYSTEMS MAGAZINE
T
NS NL
i =1 j =1
1
NS ⋅ NL
− μˆ yT ⋅ μˆ TyT = R ⋅ Γˆ y ⋅ R T
T
T
i =1 j =1
T
T
T
NS NL
− μˆ zT ⋅ μˆ TzT = K ⋅ Γˆ z ⋅ K T (46)
y ( i, j ) ⋅ z ( i, j )
i =1 j =1
T
T
T
− μˆ yT ⋅ μˆ TzT = R ⋅ Γˆ yz ⋅ K T
JULY 2017, Part II of II
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