Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 143

n

f ( x )= B i,k ( x ) P i

i =1

n m

= P i α i,k ( j ) N j,k ( x )

i =1 j =1

m n

= [ P i α i,k ( j )] N j,k ( x )

j =1 i =1

where α i,k ( j ) is given by equation (7.15). Comparing this with equation (7.7),
we get the following statement:
(3) For k = 1, from equation (7.15),

a j ) 0 + − a j ) 0 + .

α i, 1 ( j )=( τ i +1 − ( τ i − (7.17)

It agrees with α i, 1 ( j ) given by case 1, for any a j ∈ [ t j ,t j +1 ). Similarly, for

k = 2 we get,

α i, 2 ( j )=[ τ i +1 ,τ i +2 ] φ j, 2 − [ τ i,τ i +1 ] φ j, 2

with

a j ) 0 + ( y

t j +1 ) .
This agrees with α i, 2 ( j ) for the case 2, described above, for any a j ν [ t j .t j +2 ).
Now to prove Theorem 1, we present two lemmas. The first lemma is due
to Marsden [120].
Lemma 1 :
For any y

φ j, 2 ( y )=( y − −

∈

and any x

∈

[ t k ,t m +1 ), we have

m

x ) k− 1 =

( y − Ψ j,k ( y ) N j,k ( x ) , (7.18)

j =1

where Ψ j,k is given by equation (7.5).
Proof (deBoor [53]):

m

For k = 1 we get from the lemma 1, 1 = N j, 1 ( x ), which follows from

j =1

equation (7.10). For k ≥ 2, one can use the recurrence relation of deBoor [53],

Cox [47]

N j,k ( x )=( x − t j ) Q j,k− 1 ( x )+( t j + k − x ) Q j +1 ,k− 1 ( x ) , (7.19)

where

Q j,k ( x )= N j,k ( x ) / ( t j + k − t j ) , t j + k >t j ,

(7.20)

=0 ,

otherwise.

Letting ζ k =( y

− x ) k− 1 in equation (7.18), we get