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

Scaled conjugate gradient training 439

and therefo re, replacing index k by ℓ,

d

ℓ

⊤

Hw

ℓ

= d

ℓ

⊤

Hw

1

.

The step lengths are thus

α

ℓ

= −

d

ℓ

⊤

(b + Hw

ℓ

)

d

ℓ

⊤

Hd

ℓ

, ℓ = 1 . . . n

w

.

Finally, using the notation g

ℓ

= g(w

ℓ

) = b + Hw

ℓ

and substituting ℓ → k,

α

k

= −

d

k

⊤

g

k

d

k

⊤

Hd

k

, k = 1 . . . n

w

. (B.28)

For want of a better alternative, we can choo se the first search direction

along the negative local gradient

d

1

= −g

1

= −

∂

∂w

E(w

1

).

(Note that d

1

is not a unit vector.) We move, according to Equation (B.28),

a distance

α

1

=

d

1

⊤

d

1

d

1

⊤

Hd

1

along this direc tion to the point w

2

, at which the local gradient g

2

is orthog-

onal to d

1

. We then choose the new conjugate search directio n d

2

as a linear

combination ...