Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 145

Proof of Theorem 1:
Suppose a j / ∈{ t j +1 , ··· ,t j + k− 1 } . We can apply the divided difference equation

(7.22) [ τ i , ,τ i + k ] on both sides of equation (7.22). Multiplying also by τ i + k −

τ i , equation (7.14) follows. Since the right-hand side of equation (7.22) is
constant as a function of a j ∈

···

[ t j ,t j + k ), α i,k ( j ) is also independent of a j . One

can then let a j ∈{ and take limits from either left or right.

We next describe a recurrence relation in Theorem 2 for α i,k ( j ). One can
see its proof in the article by Cohen et al. [41].
Theorem 2 :
Suppose τ i + k >τ i and that α i,k ( j ) is given by equation (7.15). Then

t j +1 ,···,t j + k− 1 }

α i, 1 ( j )=1 , τ i ≤ t j <τ i + j ,

(7.24)

=0 , otherwise.

Moreover for k ≥ 2 and for all i, j ,

α i,k ( j )=( t j + k− 1 − τ i ) β i,k− 1 ( j )+( τ i + k − t j + k− 1 ) β i +1 ,k− 1 ( j ) ,

(7.25)

where

β i,k ( j )= α i,k ( j ) / ( τ i + k − τ i ) , τ i + k >τ i ,

(7.26)

=0 , otherwise.

The discrete splines α i,k ( j ) is thus seen to have properties similar to those for
B i,k .

7.2.2 Some Properties of α i,k ( j )

If α i,k ( j ) are as in Theorem 2, then we can consider some of its properties in
the following corollary.
Corollary 1:
(1) α i,k ( j )=0for i/

∈{ μ − k +1 , ··· ,μ } with 1 ≤ j ≤ m and μ be such that

τ μ ≤ t j <τ μ +1 ;

(2) α i,k ( j ) ≥ 0 , ∀ ( i, j );

n

(3) α i,k ( j )=1 ,τ k ≤ t j <τ n +1 .

i =1

Property (1) says that for each j , there are at most k discrete B-splines
α μ−k +1 ,k ( j ) , ,α μ,k ( j ) with a (possible) nonzero value.

One can now compute d j in equation (7.7) when P i s are known. Equation
(7.7) can be written as

···

n

d ( j )= α i,k P i . (7.27)

i =1

α i,k ( j ) is a discrete B-spline and d ( j ) is a linear combination of α i,k ( j )and
so it is a discrete B-spline. Discrete splines have local support, as can be seen