Microscope Image Processing

Figure 7.3 illustrates the nesting space of L

2

. While the set of scaling

functions f

j,k

(x) is the basis for V

j

, the set of wavelet functions c

j,k

(x) discussed

earlier is the basis for W

j

. The subspaces V

j

and W

j

are orthogonal to each other,

and the intersection of the subspaces V

j

is the null space {0}. Thus the basis

functions in both spaces should be orthogonal,

hf

j,k

(x),c

k,l

(x)i¼

ð

f

j,k

(x)c

k,l

(x)dx ¼ 0, j, k, l 2 Z (7:22)

The entire space can be written as

L

2

¼ V

0

 W

0

 W

1

 W

2

 (7:23)

or it can be written without the scaling space at j ¼1

L

2

¼W

2

 W

1

 W

0

 W

1

 W

2

 (7:24)

which is the expansion using the wavelet basis as in Eq. 7.17.

7.3.2 Scaling Functions and Wavelets

Using the multiresolution structure, a scaling function f(x)inV

0

, which is