Beta-Splines: A Flexible Model - Bezier and Splines in Image Processing and Machine Vision - page 132

w

− 1 ( β 1 ,β 2 ,t )=(6 β 1 [ t 2

2 t +1]+ β 1 t [ t

− − 1]

(6.29)

+ β 1 [ t 2

− 1] + β 2 t [ t − 1]) /δ

w 0 ( β 1 ,β 2 ,t )=6( β 1 t ( t 2 +1]

− t +2)+ β 1 [ −

(6.30)

t 2 ) /δ

+ β 2 t [ − t +1] −

w 1 ( β 1 ,β 2 ,t )=6 t 2 /δ. (6.31)

Therefore,

P i ( t )=( − 6 β 1 (1 − t 2 )) V i− 2 /δ

+(6 β 1 [ t 2 2 t +1]+ β 1 t [ t − − 1]

+ β 1 [ t 2 − 1] + β 2 t [ t − 1]) V i− 1 /δ

(6.32)

+6( β 1 t ( t 2 +1]

− t +2)+ β 1 [ −

t 2 ) V 0 /δ

+ β 2 t [ − t +1] −

+6 t 2 V i +1 /δ.

The second derivative of the curve P i ( t ) can be computed through

w
−

2 ( β 1 ,β 2 ,t )=12 β 1 (1

− t ) /δ (6.33)

w
−

1 ( β 1 ,β 2 ,t ) = 6(2 β 1 [ t

1] + 2 β 1 [ t

− − 1]

(6.34)

+2 β 1 t + β 2 [2 t − 1]) /δ

w 0 ( β 1 ,β 2 ,t ) = 6(2 β 1 ( − t +1) − 2 β 1 t

(6.35)

+ β 2 [ − 2 t +1] − 2 t ) /δ

w 1 ( β 1 ,β 2 ,t )=12 t/δ. (6.36)

This yields,

P i ( t ) = (12 β 1 (1 − t )) V i− 2 /δ

+ 6(2 β 1 [ t 1] + 2 β 1 [ t

− − 1]

+2 β 1 t + β 2 [2 t − 1]) V i− 1 /δ

(6.37)

+ 6(2 β 1 (

− t +1) − 2 β 1 t

+ β 2 [ − 2 t +1] − 2 t ) V 0 /δ

+12 tV i +1 /δ.