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

Wavelets and filter banks 137

where we have introduced the quantity

(Hf )

k

= hf

n

(x), φ(2

n−1

x − k)i, k = 0 . . . c/2 − 1. (4.9)

This is the kth component of a vector Hf representing the r ow of pixels in

V

n−1

. The notation implies that H is an operator. Its effect is to aver age the

pixel vector f and to reduce its leng th by a factor of two. It is thus a kind of

low-pass filter. More specifically, we have from Eq uation (4.8),

(Hf )

k

=

c−1

X

j=0

f

j

hφ(2

n

x − j), φ(2

n−1

x − k)i. (4.10)

The dilation Equation (3.28) with normalized basis functions can b e written

in the form

φ(2

n−1

x − k) =

X

k

′

h

k

′

φ(2

n

x − 2k − k

′

). (4.11)

Substituting this into Equation (4.10), we have