On-Line Geometric Modeling Notes
Overview
The uniform B-splines are based upon a knot sequence
that has uniform spacing. This implies that the
uniform B-spline blending functions
are all translates
of a single blending function 
where
A remarkable property of this single blending function is that it can
be written as a sum of scaled and translated copies of itself. Such a
property is called a two-scale relation and is essential to
defining wavelets on spaces of functions.
For a postscript version of these notes look here
Translating and Scaling the Blending Function
The uniform B-spline blending function 
can be scaled and
translated simply by redefining the parameterization of the function.
For example the function
which is shown in the figure below (in relation to the
blending function 
),
translates the blending function so that its support begins at t=4,
and ends a t=5.5, and the height of the function has been scaled by
.75.
In general, the function
has support over the interval
and has the height scaled by c.
The Two-Scale Relation for Uniform B-Splines
Given the general B-Spline blending function of order k, the
two-scale relation is written as
where
That is, we can take translated and scaled copies of the basic
function, add them together, and get the basic function back.
The development
of the coefficients
utilizes the fact that the uniform B-spline blending
function can be defined by convolution.
The Two-Scale Relation for Uniform Linear B-Splines
The uniform 2nd order B-spline blending function 
is
defined by
which is illustrated by
The two-scale relation for this function is given by
The four components of this equation are shown in the following figure,
where the original blending function is shown with dashed lines and
the three scaled and translated functions are shown using solid lines.
The original blending function is obtained by summing the three scaled and translated functions at each point.
The Two-Scale Relation for Uniform Quadratic B-Splines
A less obvious example is given by the quadratic blending function.
This 3rd order B-spline blending function 
is
defined by
The two-scale relation for this function is given by
The five components of this equation are shown in the following
figure,
where the original blending function is shown with dashed lines and
the four scaled and translated functions are shown using solid lines.
The original blending function is obtained by summing the four scaled and translated functions at each point.
Summary
The two-scale relation is an important identity when dealing with uniform B-splines (especially in relation to the definitions of B-spline wavelets), and is not easily duplicated with non-uniform splines. The proof of the general identity is also interesting as it uses the fact that the blending function can be defined using convolution.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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