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

Neural networks 251

Figure 6.8. This modification of the dis criminant has the advantag e that the

output signal saturates at values zer o or one for large ne gative or positive in-

puts, respectively. However, Bishop (1995) suggests that there is also a good

statistical justification for using it. Suppose the two classes are normally

distributed with Σ

1

= Σ

2

= I. Then

p(g | k) =

1

2π

exp(

−kg − µ

k

k

2

2

),

for k = 1, 2, and we have with Bayes’ Theorem

Pr(1 | g) =

p(g | 1)Pr(1)

p(g | 1)Pr(1) + p(g | 2)Pr(2)

=

1

1 + p(g | 2)Pr(2)/(p(g | 1)Pr(1))

=

1

1 + exp(−

1

2

[kg −µ

2

k

2

− kg − µ

1

k

2

])(Pr(2)/Pr(1))

.

With the substitution

e

−a

= Pr(2)/Pr(1)

we get

Pr(1 | g) =

1

1 + exp(−

1

2

[kg −µ

2