Spline Wavelets: Construction, Implication, and Uses - Bezier and Splines in Image Processing and Machine Vision - page 173

∞

|φ ( ω +2 πr )

2 =1 .

| (8.36)

r = −∞

φ ( t )is

|φ ( ω )

2 and sampling a function periodizes its Fourier
transform. Equation (8.36) is true if we choose

This is because φ ∗ |

1

a ( ω )= .

∞

| N n ( ω +2 πr )

2 ) 1 / 2

( |

r = −∞

Using the above result we can write,

2 n sin n ω/ 2 e −iβω/ 2

φ ( ω )=

,

∞

| N n ( ω +2 πr )

2 ) 1 / 2

| ω n (

r = −∞

2 n sin n ω/ 2 e −iβω/ 2

(8.37)

= ,

∞

1
( ω +2 πr ) 2 n ) 1 / 2

ω n 2 n sin n ω/ 2(

r = −∞

e −iβω/ 2

ω n √ S 2 n ,

=

where

∞

1
( ω +2 πr ) 2 n ,

S 2 n =

(8.38)

r = −∞

d 2 n− 1

1
2 2 n

1

= −

dx 2 n− 1 cot x. (2 n − 1)!

Example 1: Linear splines
For linear splines, the order of the polynomial is n =2.Alsowhen n is even,
β =0and β = 1 when n is odd. Hence, from equation (8.38),

1+2cos 2 x
sin 4 x

S 4 ( ω )= 1
48

.

Therefore,

√ 1+2cos 2 x
sin 2 x

S 4 ( ω )=

1
4 √ 3

,

and so,

4 √ 3sin 2 x
√ 1+2cos 2 x .

1
ω 2

φ ( ω )=

Example 2: Cubic splines
For cubic spline, n = 4. Hence, from equation (8.38),