On-Line Geometric Modeling Notes
THE UNIFORM B-SPLINE BLENDING FUNCTION

Overview

The uniform B-splines are based upon a knot sequence that has uniform spacing. This implies that the uniform B-spline blending functions $N_{i,k}(t)$ are all translates of a single blending function $N_k(t)$ where

$$N_{i,k}(t) \: = \: N_k(t-i)$$

This single blending function can be defined by convolution of blending functions of lower degree. This is the topic of these notes.

For a pdf version of these notes look here.

Definition of the Blending Functions Utilizing Convolution

The uniform $k$th order B-spline blending function $N_k$ is defined recursively by

$$N_1 ( t ) \: = \: \begin{cases} 1 \;\; & \text{if } 0 \leq t \leq 1 \\ 0 & \text{otherwise} \end{cases}$$

and

$$N_k ( t ) \: = \: ( N_{k-1} * N_1 ) ( t )$$

That is, the $k$th order blending function is defined by convolving the $k-1$st order blending function with the first order blending function. This convolution can be seen to be the integral

$$\begin{aligned}N_k ( t ) & = ( N_{k-1} * N_1 ) ( t ) \\ & = ... ... N_1(t-x) dx \\ & = \int_{t-1}^{t} N_{k-1}(x) dx \end{aligned}$$

The First Order Blending Function

The first order blending function is just the Haar scaling function

$$N_1 ( t ) \: = \: \begin{cases} 1 \;\; & \text{if } 0 \leq t \leq 1 \\ 0 & \text{otherwise} \end{cases}$$

and is shown by the graph

The support of this function is the interval $[0,1]$.

The Second Order Blending Function

To calculate the second order blending function we must calculate

$$N_2(t) \: = \: \int_{t-1}^{t} N_{1}(x) dx$$

The function $N_{1}(x)$ is nonzero only when $0 \leq x \leq 1$. Thus, we can get nonzero values in the integrand $N_1(x)$ for any $t$ where $0 < t < 2$. The integral splits naturally into the two cases shown below - for $0 \leq t \leq 1$ and $1 \leq t \leq 2$.

where in each case we have shaded the areas between the limits of integration 0 and $1$.

So we have that

$$\begin{aligned}N_2(t) & = \int_{t-1}^{t} N_{1}(x) dx \\ & = ... ...\: \: & \text{if } \: 1 \leq t \leq 2 \end{cases} \end{aligned}$$

which is illustrated by

It is clear that the support of $N_2(t)$ is the interval $[0,2]$

The Third Order Blending Function

To calculate the third order blending function, we must calculate

$$N_3(t) \: = \: \int_{t-1}^{t} N_{2}(x) dx$$

The function $N_{2}(x)$ is nonzero only when $0 \leq x \leq 2$, so we can get nonzero values in the integrand for any $t$ where $0 < t < 3$.

This is straightforward to calculate once the reader sees that there are three cases, each depending on $t$. These three cases are illustrated below as

In each case the section of the curve $N_2(x)$ that lies between the integration bounds of 0 and $1$ has been shaded.

So now we can calculate the integral by

$$\begin{aligned}N_3(t) & = \int_{t-1}^{t} N_{2}(x) dx \\ & = ... ...: & \text{if } \: 2 \leq t \leq 3 \end{cases} \\ \end{aligned}$$

This curve is a piecewise quadratic - i.e. it has quadratic pieces that are smoothly joined together. The curve is drawn as

It is clear that the support of $N_3(t)$ is the interval $[0,3]$

Summary

The uniform B-spline is somewhat unique as all blending functions are given as a translate of only one function. We have shown here that this single blending function can be calculated in an interesting way using convolution.

Bibliography

1

BARTELS, R., BEATTY, J., AND BARSKY, B.
An Introduction to Splines for Use in Computer Graphics and Geometric Modeling.
Morgan Kaufmann Publishers, Palo Alto, CA, 1987.

2

UEDA, M., AND LODHA, S.
Wavelets: An elementary introduction and examples.
Technical Report UCSC-CRL-94-47, Jan. 1994.