Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

Change detection with polarimetric SAR imagery 399

and under the alternative hypothesis H

1

, x

1

6= x

2

, the likelihood for x

1

and

x

2

is

L(x

1

, x

2

) = p(g

1

| x

1

)p(g

2

| x

2

)

=

1

(x

1

/m)

m

(x

2

/m)

m

Γ(m)

2

g

m−1

1

g

m−1

2

e

−(g

1

m/x

1

+g

2

m/x

2

)

.

(9.34)

By taking derivatives of the log-likelihoods, it is easy to show (Exercise 9)

that the likelihood L(x) is maximized by

ˆx =

g

1

+ g

2

2

and that L(x

1

, x

2

) is maximized by

ˆx

1

= g

1

, ˆx

2

= g

2

.

Then, ac cording to Equation (2.73), the likelihood ratio test has the critical

region

Q =

L(ˆx)

L(ˆx

1

, ˆx

2

)

=

(g

1

/m)

m

(g

2

/m)

n



g

1

+g

2

2m



2m

= 2

2m

g

m

1

g

m

2

(g

1

+ g

2

)

2m

≤ k.

Equivalently,



g

1

g

2

(g

1

+ g

2

)

2



m

≤

k

2

2m

or

g

1

g

2

(g

1

+ g

2

)

2

≤

˜

k,

where

˜

k depends on k. Inverting both sides of the ...