Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 147

α ( μ, r +1)= β 1 ;
μ 2= μ 2 − 1;

end
One can also use an alternative algorithm to compute d ( j ) as given by
equation (7.29). Now, before we describe the algorithmic steps, we first con-
sider the underlying background of it. By equation (7.25),

μ

d ( j )= α i,k ( j ) P i

i = μ−k +1

μ

= [( t j + k− 1 − τ i ) β i,k− 1 ( j )+( τ i + k − t j + k− 1 ) β i +1 ,k− 1 ( j )] P i .

i = μ − k +1

Since β μ−k +1 ,k− 1( j )= β μ +1 ,k− 1 ( j ) = 0 by (1) in the corollary, we get,

μ

α i,k− 1 ( j ) P [2] d ( j )=

i,j ,

i = μ − k +2

where

P [2]
i,j

=[( t j + k− 1 − τ i ) P i +( τ i + k− 1 − t j + k− 1 ) P i− 1 ] / ( τ i + k− 1 − τ i ) .

In general, for r =1 , 2 , ··· ,k ,

μ

α i,k−r +1 ( j ) P [ r ] d ( j )=

i,j ,

i = μ − k + r

with

P [1]
i,j

= P i

and

P [ r +1]
i,j

τ i ) P [ r ] t j + k−r ) P [ r ]

=[( t j + k−r −

i,j +( τ i + k−r − 1 ,j ] / ( τ i + k−r −

τ i ) .

i −

Therefore, when r = k ,wehave

d ( j )= α μ, 1 ( j ) P [ k ]

μ,j

= P [ k ]

μ,j .

With this, we write down Algorithm 2 to compute d ( j ).
Algorithm 2:
Step 1: μ 2= μ k +1;

Step 2: for i = μ 2 ,μ 2 +1 ,

−

··· ,μ do

begin
P [1 i = P i ;
Step 3: for r =1 , 2 ,

··· ,k −

1do