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

90 Transformations

FIGURE 3.1

Logarithm of the power spectrum for the 3N band o f the J¨ulich ASTER image

with the Python script in Listing 3.2.

Rather than displaying P (k, ℓ) directly, which, according to ENVI’s display

convention, would place zero frequency components k = 0, ℓ = 0 in the upper

left-hand corner, use can be made of the translation property (Exe rcise 4) of

the Fourier transform:

g(i, j)e

i2π( k

0

i/c+ℓ

0

j/r)

⇔ ˆg(k − k

0

, ℓ − ℓ

0

). (3.11)

In particular, for k

0

= c/2 and ℓ

0

= r/2, we can write

e

i2π( k

0

i/c+ℓ

0

j/r)

= e

iπ(i+j)

= (−1)

i+j

.

Therefore

g(i, j)(−1)

i+j

⇔ ˆg(k − c/2, ℓ − r/2),

so if we multiply an image by (−1)

i+j

before transforming, zero frequency ...