Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 141

n

d j = α i,k ( j ) P i (7.7)

i =1

for some α i,k ( j ). Consider two different cases for understanding.
Case 1: k = 1 (step-functions)
In this case,

n

f ( x )= B i, 1 ( x ) P i , (7.8)

i =1

where

B i, 1 =1 , τ i ≤ x<τ i +1 ,

=0 , otherwise.

If

m

f ( x )= N j, 1 ( x ) d j , (7.9)

j =1

where

N j, 1 =1 , t j ≤ x<t i +1 ,

(7.10)

=0 , otherwise,

then

d j = P i , τ i ≤ t j <τ i +1 .

Therefore, in equation (7.7) we must have,

α i, 1 ( j )=1 , τ i ≤ t j <τ i +1 ,

=0 , otherwise.

Hence, one can easily note that α i, 1 ( j )= B i, 1 ( t j )
Case 2: k = 2 (piecewise linear functions)
For this case, we can consider

n

f ( x )= B i, 2 ( x ) P i ,

(7.11)

i =1

where

B i, 2 ( x )=( x − τ i ) / ( τ i +1 − τ i ) , τ i ≤ x<τ i +1 ,

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

=0 , otherwise.

Now suppose

m

f ( x )= N j, 2 ( x ) d j , (7.12)

j =1

where

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

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

=0 , otherwise.