B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 111

P j ( s )= N 1 ,k ( s ) V j + N 2 ,k ( s ) V j +1 + ··· + N m,m ( s ) V j + m− 1 . (5.15)

For m = 3, the re-parameterized blending functions on the unit interval 0 ≤

s< 1 are as follows:

N 1 , 3 ( s )= (1 − s ) 2

2
N 2 , 3 ( s )= − 2 s 2 +2 s +1
2

(5.16)

N 3 , 3 ( s )= s 2

2 .

Substituting all these blending functions, the periodic quadratic B-spline curve
on the unit interval is then
2 P j ( s )=(1

2 s + s 2 ) V j +( 2 s 2 +2 s +1) V j +1 + s 2 V j +2

− −

= s 2 ( V j − 2 V j +1 + V j +2 )

(5.17)

+ s ( 2 V j +2 V j +1 +0 .V j +2

+1( V j + V j +1 +0 .V j +2

−

or, in the matrix form

P j ( s )=( S )( N )( V )

⎛ ⎞ ⎛ ⎞

1 − 21 V j

V j +1
V j +2

(5.18)

⎝ ⎠ ⎝ ⎠ .

1

2 ( s 2

= s 1) − 220

110

Likewise for periodic cubic B-spline, m = 4 and the reparameterized blending
functions on the unit interval are as follows :
N 1 , 4 ( s )= − s 3 +3 s 2

− 3 s +1

6
N 2 , 4 ( s )= − 3 s 3 +6 s 2 +4
6
N 3 , 4 ( s )= − s 3 +3 s 2 +3 s +1
6

(5.19)

N 4 , 4 ( s )= s 3

6 .

The curve in the matrix form is, therefore,

P j ( s )=( S )( N )( V )

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

− 13 − 31 V j

V j +1
V j +2
V j +3

3 − 630 (5.20)

1

6 ( s 3

s 2 = s 1) .

− 30 30

1410

Note that for any m ,

( S )=( s m− 1 s m− 2 ··· 1) 0 ≤ s< 1 . (5.21)

Cohen and Risenfeld [42] have shown the generalized form of N for periodic
B-spline curves, as given by

( m m− 1

1) r−j m
r

1 − 1)

( r + 1)) i ( N i +1 ,j +1 = ( m − − (5.22)

( m − 1)! i − j

r = j

≤ i, j ≤ m −

where 0 1.