Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 142

If ν and j are such that

τ ν ≤ t j +1 <τ ν +1 ,

then

f ( t j +1 )= d j
=

(7.13)

{ ( τ ν +1 − t j +1 ) P ν− 1 +( t j +1 − τ ν ) P ν } / ( τ ν +1 − τ ν ) .

Now, equation (7.7) is valid with,

α i, 2 ( j )=( t j +1 − τ i ) / ( τ i +1 − τ i ) , τ i ≤ t j +1 <τ i +1 ,

=( t i +2 − t j +1 ) / ( τ i +2 − τ i +1 ) , i +1 ≤ t j +1 <τ i +2 ,

=0 , otherwise.

Hence, we note that α i, 2 ( j )= B i, 2 ( t j +1 ). Here, we observe that the numbers
α i, 2 ( j ) are related to the B-spline B i,k for k =1 , 2. α i,k ( j ) is a discrete spline.

7.2.1 Relation Between α i,k and B i,k , k> 2

We have assumed N i,k as B-splines on a partition { t j } and B i,k as B-splines

on a coarser subpartition { τ i } . Let us now consider the following theorem.

Theorem 1 :
For all x ,wehave,

m

B i,k ( x )= α i,k ( j ) N j,k ( x ) i =1 , 2 , ··· m, (7.14)

j =1

where

α i,j ( j )=( τ i + k − τ i )[ τ i , ··· τ i + k ] φ j,k , (7.15)

a j ) 0 + Ψ j,k ( y ) ,

φ j,k ( y )=( y − (7.16)

with Ψ j,k ( y ) given by equation (7.5). Here,

( y − a j ) 0 + =1 y>a j

=0 otherwise,

a j can be chosen anywhere in [ t j ,t j + k ), and [ τ i , ··· τ i + k ] φ j,k denotes a divided

difference. We have the following remarks:
(1) α i,k ( j ) is called a discrete spline.
(2) The numbers α i,k ( j ) in equation (7.7) are the discrete B-splines given by
equation (7.15).
From equation (7.1) and equation (7.14), we have,