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

Hyperspectral image analysis 315

a different mean vector µ. In the notation of Definition 2.7, the maximized

likelihoods are given by

max

θ∈ω

0

L(θ) =

m+1

Y

ν=1

exp



−

1

2

(g(ν) −

ˆ

µ

b

)

⊤

ˆ

Σ

−1

b

(g(ν) −

ˆ

µ

b

)



and by

max

θ∈ω

L(θ) =

m

Y

ν=1

exp



−

1

2

(g(ν) −

ˆ

µ

b

)

⊤

ˆ

Σ

−1

b

(g(ν) −

ˆ

µ

b

)



· exp



−

1

2

(g(m + 1) −

ˆ

µ)

⊤

ˆ

Σ

−1

b

(g(m + 1) −

ˆ

µ)



,

where

ˆ

µ

b

and

ˆ

Σ

b

are the maximum likelihood estimates of the mean and

cova riance matrix of the background. But

ˆ

µ = g(m + 1) (there is only one

observation), so the last exponentia l factor in the above equation is unity.

Therefore the likelihood ratio test has the critical region

Q =

max

θ∈ω

0

L(θ)

max

θ∈ω

L(θ)

= exp



−

1

2

(g(m + 1) −

ˆ

µ

b

)

⊤

ˆ

Σ

−1

b

(g(m + 1) −

ˆ

µ

b

)



≤ k.

Thus we r