B-splines: Blending Functions, Non-Uniform Variants, de Boor Formula

CS216

Chris Pollett

Feb 18, 2010

Outline

Last day, we finished up talking about interpolation with Bezier curves.
We began talking about uniform B-spline curves of degree three. We gave the definition of these curves in terms of blending functions which we didn't completely specifiy.
We did say `N_i(u)` was only non-zero for `u\in (i, i+4)`.
This meant that B-spline curves have a nice "local control" property: At most four control points influence the location of any point on the curve.
Let's now look at some properties that we want our blending functions to satisfy...

We want our blending functions to have these properties:

The blending functions are translations of each other:
`N_i(u) = N_0(u -i )`.
The functions `N_i(u)` are piecewise degree three, and the breaks between pieces occur at integer values.
The functions `N_i(u)` are `C^2`-continuous.
The blending functions are a partition of unity, that is,
`\sum_iN_i(u) = 1` for `3\leq u \leq 7`.
`N_i(u) \geq 0` for all `u`.
`N_i(u) = 0` for all `u \leq i` and for `i+4 \leq u`.

By (1) and (6) on the previous slide, all of our blending functions will be fully specified once we have defined `N_0(u)` on `[0,4]`.
To do this will define `N_0(u)` in terms of four functions `R_0(u), R_1(u), R_2(u), R_3(u)` each with domain `[0,1]`.
For each integer, unit subinterval of `[0,4]`, we define `N_0(u + j) = R_j(u)`.
Finally, one choice we can make for the `R_j`'s which satisfies the properties of the previous slide is:
`R_0(u) = \frac{1}{6}u^3`
`R_1(u) = \frac{1}{6}(-3u^3 +3u^2 +3u + 1)`
`R_2(u) = \frac{1}{6}(3u^3 -6u^2 + 4)`
`R_3(u) = \frac{1}{6}(1-u)^3`

For uniform B-splines, the curve `\vec q` at `i` tends to be most strongly influence by the control point `\vec p_(i-2)`.
These splines are called uniform as the knots `u_i` corresponding to control points `\vec p_i` are evenly spaced. They are at the integers: `u_i = i`.
A nonuniform B-spline is just one where the knots are not evenly spaced.
Instead, to define such a spline one uses a sequence called a knot vector
`[u_0, u_1, \ldots, u_l]` of real values `u_0 \leq u_1 \leq u_2 \ldots\leq u_l` and corresponding control points `\vec p_0, \ldots, \vec p_n`.
For the order `m` nonuniform B-spline, these (once we're done the definition) will define a degree `k=m-1` curve and `n = l - m`.
In terms of imagining how the spline will look, one should roughly think that the point `\vec p_i` is most greatly influencing `\vec q` at the knot `u_(i+m/2)`. (Or between knots `u_(i+(m-1)/2)` and `u_(i+(m+1)/2)` for `m` odd.)
It is completely legal for `u_i = u_(i+1)`. If this happens the curve is more strongly influenced by the corresponding control point. If a knot occurs more than four times the curve can become discontinuous.

Cox-de Boor Formula is used to define non-uniform B-spline blending functions.
Assume a knot vector `u_0 \leq u_1 \leq u_2 \ldots\leq u_l` has been fixed.
The blending functions `N_(i,m)(u)` for order `m` splines are defined by induction on `m\geq 1`.
First, we define
`N_(i,1)(u) = {(1, text{if} u_i \leq u < u_(i+1)),(0, text{otherwise}):}
For `m=k+1`, the blending functions are defined by the Cox-de Boor Formula:
`N_(i,k+1)(u) = \frac{u - u_i}{u_(i+k) - u_i}N_(i,k)(u) + \frac{u_(i+k+1) - u}{u_(i+k+1) - u_(i+1)}N_(i+1,k)(u)`.

When there are repeated knots, some of the denominators may become zero.
To get around this, we use the conventions that `0/0 = 0` and `(a/0)0 =0`.
From the formula one immediately gets that the `N_(i,m)(u)` are piecewise degree `m-1` polynomials and that breaks between pieces occur at the knots `u_i`.
By induction, you can show that `N_(i,m)(u)` has support in `[u_i, u_(i+m)]`.

Let's look at a few different examples of B-splines defined using the Cox-de Boor formula...
To begin let's consider the case where the knots are uniformly spaced and `u_i = i`.
Then:
`N_(i,1)(u) = {(1, text{if} i \leq u < i+1),(0, text{otherwise}):}` and these functions are piecewise degree `0` and are discontinous at `u_i=i`.
For the order two blending functions we get:
`N_(i,2)(u) = \frac{u - i}{1}N_(i,1)(u) + \frac{i + 2 - u}{1}N_(i+1,1)(u)`
which look like a sawtooth.
These are `C^0`-continuous but not `C^1`-continuous, are a partition of unity, and are nonnegative.
If we define, `\vec q(u) = \sum_(i=0)^(l - 2)N_(i,2)(u)\vec p_(i)`, then the resulting curve consists of straight-line segments connecting the control points `\vec p_0 , \vec p_1, \ldots, \vec p_(l-2)`.
If we then go on to look at the order three and order four blending functions in the uniformly spaces setting the sawtooth we had above become progressively less pointy and the curves become `C^1` and `C^2` continuous, the order four case matching the uniform B-splines we have already considered.

Suppose we let the knot vector be `[0, 0, 0, 0, 1, 1, 1, 1]`.
If we compute the order one blending functions, we find the only nonzero one is:
`N_(3,1)(u) = {(1, text{if} 0 \leq u < 1),(0, text{otherwise}):}`
The order two blending functions will be zero except for `i = 2, 3`, and for all `m \geq 1` the order `m` blending function will be zero except on the unit interval.
The order two blending functions work out to be:
`N_(2,2)(u) = \frac{u - u_2}{u_3 - u_2}N_(2,1)(u) + \frac{u_4 - u}{u_4-u_3}N_(3,1)(u)`
`= \frac{u - 0}{0 - 0}\cdot 0 + \frac{1 - u}{1-0}\cdot 1 = 1 - u`.
`N_(3,2)(u) = u`.
Notice we are using the `(a/0)0 =0` convention we mentioned earlier.
If one keeps going, one notices that the blending functions one gets for the order four case, `N_(i,4)(u)` are exactly the blending function `N_i(u)` for a degree three Bezier curve!