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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

$\displaystyle 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.

pdficonsmall.gif 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

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

and

$\displaystyle 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{displaymath}\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}\end{displaymath}


The First Order Blending Function

The first order blending function is just the Haar scaling function

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

and is shown by the graph

\includegraphics {figures/uniform-blending-order-1}

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

$\displaystyle 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$.

\includegraphics {figures/convolution-integration-cases-1}

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

So we have that

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

which is illustrated by

\includegraphics {figures/uniform-blending-order-2}

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

$\displaystyle 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

\includegraphics {figures/convolution-integration-cases-2a}

\includegraphics {figures/convolution-integration-cases-2b}

\includegraphics {figures/convolution-integration-cases-2c}

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

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

\includegraphics {figures/uniform-blending-order-3}

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.


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Ken Joy
2000-11-28