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

100 Transformations

The constants c

k

are called the refinement coefficients. For example, the

dilation eq uation for the Haar scaling function is

φ(x) = φ(2x) + φ(2x − 1),

so that the refinement coefficients are c

0

= c

1

= 1, c

k

= 0 otherwise. Note

that c

2

0

+ c

2

1

= 2. This is a general property of the refinement coefficients:

1 = hφ(x), φ(x)i = h

X

k

c

k

φ(2x − k),

X

k

′

c

k

′

φ(2x − k

′

)i =

1

2

X

k

c

2

k

and therefo re,

∞

X

k=−∞

c

2

k

= 2, (3.26)

which is also called Parseval’s formula. In a similar way, one can show (Ex-

ercise 7)

∞

X

k=−∞

c

k

c

k−2j

= 0 for all j 6= 0. (3.27)

3.2.3.2 The cascade algorithm

Some of the scaling functions which generate an MRA cannot be expressed

as simple, analytical