Beta-Splines: A Flexible Model - Bezier and Splines in Image Processing and Machine Vision - page 129

P i +1 (0) = β 1 P i (1) (6.4)

and

P

i +1 (0) = β 1 P i (1) + β 2 P i (1) .

(6.5)

This leads to

n =1 n =1

w n ( β 1 ,β 2 , 0) V i +1+ n = w n ( β 1 ,β 2 , 1) V i + n (6.6)

n = − n = −

2 2

n =1 n =1

w n ( β 1 ,β 2 , 0) V i +1+ n = β 1 w n ( β 1 ,β 2 , 1) V i + n

(6.7)

n = − 2 n = − 2

n =1 n =1

w n ( β 1 ,β 2 , 0) V i +1+ n = β 1 w n ( β 1 ,β 2 , 1) V i + n

n = − 2 n = − 2

(6.8)

n =1

w n ( β 1 ,β 2 , 1) V i + n . + β 2

n = − 2

Equating coecients of the vertices V i + n , n = − 2 , − 1 , 0 , 2 , 1, we get,

0= w 2 ( β 1 ,β 2 , 1)

(6.9)

w n− 1 ( β 1 ,β 2 , 0) = w n ( β 1 ,β 2 , 1) , r = − 1 , 0 , 1 (6.10)

w 1 ( β 1 ,β 2 , 0) = 0 (6.11)

0= β 1 w 2 ( β 1 ,β 2 , 1) (6.12)

w n− 1 ( β 1 ,β 2 , 0) = β 1 w n ( β 1 ,β 2 , 1) ,n =

− 1 , 0 , 1 (6.13)

w 1 ( β 1 ,β 2 , 0) = 0

(6.14)

0= β 1 w 2 ( β 1 ,β 2 , 1) + β 2 w 2 ( β 1 ,β 2 , 1)

(6.15)

w n− 1 ( β 1 ,β 2 , 0) = β 1 w n ( β 1 ,β 2 , 1) + β 2 W n ( β 1 ,β 1 , 1) ,n =

− 1 , 0 , 1 (6.16)

w 1 ( β 1 ,β 2 , 0) = 0 .

(6.17)