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

310 Supervised Classification Part 2

Ignoring the r equirement of Equation (7.33) for the time being, a Lagrange

function for minimization of Equation (7.31) under the constraint of Equation

(7.32) is

L = R

⊤

Σ

−1

R

R + 2λ(

K

X

i=1

α

i

− 1)

= (G − M α)

⊤

Σ

−1

R

(G − Mα) + 2λ(

K

X

i=1

α

i

− 1),

the last equality following from Equation (7.30). Solving the set of equations

∂L

∂α

= 0,

∂L

∂λ

= 0,

and replacing G by its realiz ation g, we obtain the estima tes for the mixing

coefficients (Exer cise 4)

ˆ

α = (M

⊤

Σ

−1

R

M)

−1

(M

⊤

Σ

−1

R

g − λ1

K

)

ˆ

α

⊤

1

K

= 1,

(7.34)

where 1

K

is a column vector of K ones. The first equation determines the

mixing coefficients in terms of known quantities and λ. The second equation ...