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

26 Images, Arrays, and Matrices

A more c onvenient and generally applicable way to find an extremum of

f(x) subject to h(x) = 0 is to determine an unconstrained minimum or max-

imum of the expression

L(x) = f(x) + λh(x). (1.61)

This is called a Lagrange function and λ is a Lagrange multiplier. The La-

grange multiplier is treated a s though it were an additio nal variable. To find

the extremum, we solve the set of equations

∂

∂x

(f(x) + λh(x)) = 0,

∂

∂λ

(f(x) + λh(x)) = 0

(1.62)

for x and λ. To see this, note that a minimum or maximum need not generally

occur at a point x

∗

for which

∂

∂x

f(x

∗

) = 0, (1.63)

because of the presence of the constraint. This poss ibility ...