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

Exercises 281

(d) Show that, for the one-dimensional distributions of Equation (6.79),

d

12

=

1

2



1

σ

2

1

−

1

σ

2

2



(σ

2

1

− σ

2

2

) +

1

2



1

σ

2

1

+

1

σ

2

2



(µ

1

− µ

2

)

2

.

2. (Ripley, 1996) Assuming that all K land cover classes have identical

cova riance matrices Σ

k

= Σ:

(a) Show that the discriminant in Equation (6.17) ca n be replaced by

the linear discriminant

d

k

(g) = log(Pr(k)) − µ

⊤

k

Σ

−1

g +

1

2

µ

⊤

k

Σ

−1

µ

k

.

(b) Suppose that there ar e just two classes k = 1 and k = 2. Show that

the maximum likelihood classifier w ill choose k = 1 if

h = (µ

1

− µ

2

)

⊤

Σ

−1



g −

µ

1

+ µ

2

2



> log



Pr(2)

Pr(1)



.

(c) The quantity d =

q

(µ

1

− µ

2

)

⊤

Σ

−1

(µ

1

− µ

2

) is the Mahalanobis

distance between the class means. Demonstrate that, ...