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

Minimum noise fraction 115

in a manner similar to the principal components transformation, Equation

(3.42). The covariance matrix of Y is (compare with Eq uation (3.43))

Σ

′

= A

⊤

ΣA = I, (3.58)

where I is the N × N identity matrix.

It follows from Equation (3.54) that the SNR for eigenvalue λ

i

is just

SNR

i

=

a

⊤

i

Σa

i

a

⊤

i

(λ

i

Σa

i

)

− 1 =

1

λ

i

− 1. (3.59)

Thus the projection Y

i

= a

⊤

i

G corresponding to the smallest eigenvalue λ

i

will have largest signal-to-noise ratio. Note that with Equation (3.55) we can

write

Σ

N

A = ΣAΛ, (3.60)

where Λ = Diag(λ

1

. . . λ

N

), a diagonal matrix with diagonal elements λ

1

. . . λ

N

.

3.4.2 Minimum noise fraction in ENVI

The MNF transformation is