B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 108

Sometimes, knot values are normalized in the range between 0 and 1. An
example for this is

[0 . 0 , 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 . 0] .

In many applications, we need a knot vector with a separation of unity and
starting value of 0. We take an example for this as

[0 , 1 , 2 , 3 , 4 , 5 , 6 , 7] .

In general, the total number of knots in this case is one less than that for the
open curves, i,e., n + m , since the initial and starting knots are identical. The
knot values t 0 ,t 1 ,

··· ,t n + m are cyclic, i.e., 0 , 1 , ··· ,n, 0 , 1 , ··· . Hence,

t m =0 ,t m +1 =1 ,t m + i = t i .

This means we choose the knot values t i = i , and reduce all basis functions
to one.

Nonuniform Knot Structure

Nonuniform knot vectors may be unequally spaced together with or without
multiple internal knots. Some of the knot vectors are [0, 0, 0, 1, 1, 1, 2, 2, 2,
3, 3], [0, 1, 2, 2, 3, 3, 4], and [0, 0.22, 0.48, 0.75, 1].

5.3 Computation of B-Spline Basis Functions

Given the knot structure, one can easily compute the B-spline basis functions
recursively using the equation (5.2) to design a curve. All these basis functions
in the recursive computation defines a triangular structure as shown below.

B i,m
B i,m− 1 B i +1 ,m− 1
B i,m− 2 B i +1 ,m− 2 B i +2 ,m− 3
.

(5.6)

. .

B i, 1 B i +1 , 1 B i +2 , 1 B i +3 , 1 ··· B i + m− 1 , 1 .

The inverse structure shows how the first order basis function B i, 1 depends
on higher order basis functions.

B i−m +1 ,m ··· B i− 1 ,m B i,m B i +1 ,m ··· B i + m− 1 ,m

. . . . . .

(5.7)

B i− 1 , 2 B i, 2 B i +1 , 2
B i, 1 .

We shall now consider a few examples so that readers get a complete under-
standing of the computation of the basis functions.