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
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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
In general, the function has support over the interval and has the height of the function scaled by .
The Two-Scale Relation for Uniform B-Splines
Given the general B-Spline blending function of order , the two-scale relation is written as
The Two-Scale Relation for Uniform Linear B-Splines
The uniform nd order B-spline blending function is defined by
The two-scale relation for this function is given by
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 rd order B-spline blending function is defined by
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.