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

Principal components 105

From the second requirement we have

0 =

Z

∞

−∞

xψ(x)dx =

3

X

k=0

(−1)

k+1

c

k

Z

∞

−∞

xφ(2x − 1 + k)dx

=

3

X

k=0

(−1)

k+1

c

k

Z

∞

−∞

u + 1 − k

4

φ(u)du

=

0

4

·

Z

∞

−∞

uφ(u)du +

−c

0

+ c

2

− 2c

3

4

Z

∞

−∞

φ(u)du,

using Equation (3.39). Thus

−c

0

+ c

2

− 2c

3

= 0. (3.40)

Equations (3.35), (3.36), (3.39), and (3.40) comprise a system of fo ur (non-

linear) equatio ns in four unknowns. A solution is given by

c

0

=

1 +

√

3

4

, c

1

=

3 +

√

3

4

, c

2

=

3 −

√

3

4

, c

3

=

1 −

√

3

4

,

which are known as the D4 refinement coefficients. Figure 3.9 shows the cor -

responding scaling function, determined with the cascade algorithm described

earlier (Listing 3.4).

The D4 scaling function and the subspaces that it generates a