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

338 Unsupervised Classification

but retaining the requirements in Equations (8.7) and (8.8). The para meter

q now has an e ffect and determines the “degree of fuzziness”. It is often

chosen as q = 2. The matrix U is referred to as a fuzzy class membership

matrix. The classification problem is to find that value of U which minimizes

Equation (8.30). Minimiza tion can be achieved by finding values for the u

kν

which s olve the minimization problems

E

ν

=

K

X

k=1

u

q

kν

kg(ν) −

ˆ

µ

k

k

2

→ min, ν = 1 . . . m,

under the constra ints imposed by Equa tion (8.7). Accordingly, we define a

Lagrange function which takes the c onstraints into account, i.e.,

L

ν

= E

ν

− λ

K

X

k=1

u

kν

− 1

!

,

and ...