Aerospace and Electronic Systems - June 2019 - 33

Addabbo et al.
Now, exploiting [30, Th. 1], it can be shown that
U ; 1 ; 23 ; X Þ ¼ U S e
arg max LðU

ju

b 4; XÞ ¼ c À K
LðC

where u 2 ½0 2pŠ and U S 2 C3Â3 is the unitary
matrix containing the eigenvectors of S . As a consequence, the partially compressed log-likelihood
becomes
c À K log 1 À 2K log 23
À trfDiag ð½g 1 =1 ; g 2 =23 ; g 3 =23 ŠÞg
¼ Lð1 ; 23 ; X Þ:

(14)

Finally, the estimates of 1 and 23 can be found
setting to zero gradient of Lð1 ; 23 ; X Þ to obtain
g
b1 ¼ 1
K

(15)

3
1 X
b23 ¼
g
2K i¼2 i

(16)

log

i¼1

(13)

U

3
X

ng o
i

K

À 3K

(22)

and the proposition is complete.
Finally, the number unknown parameters under each
hypothesis for the homogeneous environment is given by
8
1;
>
>
<
6;
kp ðiÞ ¼
6;
>
>
:
9;

if
if
if
if

i¼1
i¼2
i¼3
i¼4

(23)

and the resulting decision statistics of the MOS rules are
 H1 :
À2c þ 6K log

&
'
tr ½S
SŠ
þ 6K þ h;
3K

(24)

 H2 :

where g i , i ¼ 1; 2; 3, are the eigenvalues of S .
Gathering the above results, the compressed loglikelihood has the following expression:
ng o

(
)
3
ng o
1 X
1
þ 4K log
À2c þ 2K log
g þ 6K þ 6h;
2K i¼2 i
K

(25)
 H3 :
(

2
1 X
À2c þ 4K log
g
2K i¼1 i

b 2 ; X Þ ¼ c À K log 1
LðC
(K
)
3
1 X
g À 3K:
À 2K log
2K i¼2 i

)
þ 2K log

ng o
3
þ 6K þ 6h;
K

(26)
(17)

3. Under H3 : This case is analogous to the previous
hypothesis with the difference that the first two
eigenvalues of C are equal. Following the same
line of reasoning as the previous case, it is possible to show that
b ¼ U S eju
U

(18)

2
1 X
g
b12 ¼
2K i¼1 i

(19)

g
b3 ¼ 3
K

(20)

 H4 :
À2c þ 2K

3
X

log

ng o

i¼1

i

K

þ 6K þ 9h:

(27)

The processing steps for the homogeneous environment case are shown in the flow diagram in Figure 5.
Starting from the three polarimetric channels, a sliding
window of K samples is used to extract the polarimetric
returns needed to compute the sample covariance matrix
S . A MOS rule (among AIC/BIC/GIC) is then computed
for each hypothesis using (24)-(27) and finally Hi^ is
selected as
Hi^ ¼ arg maxfMOSðiÞg:
i¼1;...;4

where 12 ¼ 1 ¼ 2 , and the resulting compressed log-likelihood can be recast as
(

2
1 X
g
2K i¼1 i
ng o
À K log 3 À 3K:
K

HETEROGENEOUS ENVIRONMENT

)

b 3 ; X Þ ¼ c À 2K log
LðC

(21)

4. Under H4 : Following the lead of [31], it can be
shown that the compressed log-likelihood is
JUNE 2019

In this section, we assume that the scaling factors,
s 2k > 0, k ¼ 1; . . . ; K, change due to the different
reflectivity strength of each pixel. The dependence on
these parameters can be removed exploiting the Principle of Invariance [32], and as shown in [12] and [26],
the MIS, in this case, is z k ¼ xk =kx
xk k, k ¼ 1; . . . ; K.

IEEE A&E SYSTEMS MAGAZINE

33



Aerospace and Electronic Systems - June 2019

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