More Interpolation, B-splines

CS216

Chris Pollett

Feb 16, 2010

Outline

Last day, we talked about using rational Bezier curves to represent conic sections.
We then discussed rotating a Bezier curve to sweep out a surface of revolution and then how to get the corresponding Bezier patch.
Finally, we talked about how to get a piece-wise degree three Bezier curve that interpolates (goes through) a set of points.
Our first example, of how to do this was Catmull-Rom Splines. We next look at some other techniques for interpolation...

Recall the `\vec L_i` of last day for Catmull-Rom splines where defined in terms of the `i-1`st and `i+1`st control points. Because of this we can only define `\vec p_i^+` and `\vec p_i^-` when `0 < i < m`, so for Catmull-Rom Splines the first and last points are there to make the equations of the intermediate piecewise Bezier curves work out, but will not be interpolated.
With Catmull-Rom splines, when knot points are not evenly spaced, you can get bad "overshoot" effects as seen above when two close control are next to widely separated control points.
To solve this problem, one can use a chord-length parametrization. That is, we choose the knots so that `u_(i+1) - u_i = ||\vec p_(i+1) - \vec p_i||`.
This will roughly make the arclength of the curve between `\vec p_i` and `\vec p_(i+1)` proportional to `u_(i+1) - u_i`.

To effect this parametrization, we need to modify our definitions of `\vec p_i^-` and `\vec p_i^+` using the Bessel tangent method (aka the Overhauser method).
Suppose we have a set of knots `u_i`. We can define the average velocity of the curve between `\vec p_i` and `\vec p_(i+1)` as
`\vec v_(i+1/2) = \frac{\vec p_(i+1) - \vec p_(i)}{u_(i+1) - u_(i)}`.
If the chord length approximation was good, then these will be unit vectors, and we can approximate the velocity at `\vec p_i` as weighted average:
`\vec v_i = \frac{(u_(i+1) - u_(i))\vec v_(i+1/2) + (u_(i) - u_(i-1))\vec v_(i-1/2)}{u_(i+1) + u_(i-1)}`
Finally, we can define the Bessel-Overhauser Splines, in the same way as Catmull-Rom splines but using `\vec p_i^+` and `\vec p_i^-` defined by:
`\vec p_i^(-) = vec p_i - \frac{1}{3}(u_i - u_(i-1))\vec v_i`
`\vec p_i^(+) = vec p_i + \frac{1}{3}(u_(i+1) - u_(i))\vec v_i`

Which of the following statements is true?

The glMap1* class of OpenGL functions can be used to set up a Bezier curve for drawing.
Rational Bezier curves do not make use of homogeneous coordinates.
All Catmull-Rom splines are parametrized over the unit interval.

Sometimes a curve designer would like to have their curves be "tighter" to some control points and "looser" for others.
TCB splines use three parameters: tension, continuity, and bias to allow a designer to have this kind of flexibility.
Tension controls the tightness of the curve, continuity controls the discontinuity of the first derivatives, and bias controls the degree to which the curve undershoots or overshoots an interpolation point.
The basic set-up begins again with Catmull-Rom Splines where we modify the definition of `\vec p_i^+` and `\vec p_i^-`.
We define:
`\vec p_i^+ = \vec p_i +\frac{1}{3}D\vec q_i^+ ` and `\vec p_i^(-) = \vec p_i -\frac{1}{3}D\vec q_i^-`.
`D\vec q_i^\pm` are supposed to represent the left and right derivatives of `\vec q(u)` at `u_i`. i.e., defined in terms of the limit as `u` approaches `u_i` from above for + and from below for -.

For a basic Catmull-Rom spline `D\vec q_i^\pm` is equal to
`D\vec q_i^(-) = D\vec q_i^(+) = \frac{1}{2}\vec v_(i-1/2) + \frac{1}{2}\vec v_(i+1/2)`
where `\vec v_(i-1/2) = p_i - p_(i-1)`.
To add the tension effect we define:
`D\vec q_i^(-) = D\vec q_i^(+) = (1-t)(\frac{1}{2}\vec v_(i-1/2) + \frac{1}{2}\vec v_(i+1/2))`
The continuity parameter is supposed to measure how close to `C^1`-continuous the curve is at `\vec p_i`.
This can be modeled by defining:
`D\vec q_i^(-) = \frac{1-c}{2}\vec v_(i-1/2) + \frac{1 + c}{2}\vec v_(i+1/2)`
`D\vec q_i^(+) = \frac{1+c}{2}\vec v_(i-1/2) + \frac{1 - c}{2}\vec v_(i+1/2)`
Here `c` is usually set between -1 and 0.
Finally, to model overshoot (what bias does) we can weight the two average velocities `\vec v_(i-1/2)` and `\vec v_(i+1/2)` differently by setting:
`D\vec q_i^(-) = D\vec q_i^(+) = \frac{1+b}{2}\vec v_(i-1/2) + \frac{1-b}{2}\vec v_(i+1/2)`
Here `b` is usually set between `-1/2` and `1/2` where negative values yield undershoots, and positive ones yield overshoots.

Combining our three different models yields the following equations for `D\vec q_i^\pm`:
`D\vec q_i^(-) = \frac{(1-t)(1-c)(1+b)}{2}\vec v_(i-1/2) + \frac{(1-t)(1+c)(1-b)}{2}\vec v_(i+1/2)`
`D\vec q_i^(+) = \frac{(1-t)(1+c)(1+b)}{2}\vec v_(i-1/2) + \frac{(1-t)(1-c)(1-b)}{2}\vec v_(i+1/2)`
The book shows how to generalize the interpolation problem from Bezier curves to Bezier patches.
In the surface setting there are analogs to Catmull-Rom and Bessel-Overhauser curves.
In the surface setting, one has to make use of the second derivatives for patches we defined earlier.
We are going to skip this sections; however, and move on to B-splines.

We are now going to consider another way to define curves using control points: B-splines.
B-splines were first defined in the 1940s, but gained popularity in the 1970's after good methods for defining their blending functions were discovered.
B-spline curves have several advantages over Bezier curves in terms of modeling complicated curves with a single equations, allowing for sharp bends in a curve, and sensitivity to added control points.
There are methods to translate back and forth between B-spline and piecewise Bezier curves.
We will first look at the uniform B-splines and then consider nonuniform ones.

A uniform B-spline of degree three is defined by a sequence of control points `\vec p_0, \vec p_1, \ldots, \vec p_n` together with a set of blending functions `N_0(u), N_1(u), \ldots, N_n(u)` which parametrically define a curve via the equation:
`\vec q(u) = \sum_(i=0)^n N_i(u)\cdot \vec p_i` where `3 \leq u \leq n+1`.
We will define the blending functions in a moment, but one important property that have is that `N_i(u)` equals `0` if `u \leq i` or `i+4 \leq u`.
This means if `u \in [j, j+1]`, `3 \leq j \leq n`, then the defining equation for `\vec q(u)` reduces to:
`\vec q(u) = \sum_(i=j-3)^j N_i(u)\cdot \vec p_i`.
So a B-spline has local control, if a single control point `\vec p_i` is moved, then only the portion of the curve with `i < u < i+4` is changed.
Next day, we will discuss our requirements on what the `N_i(u)`'s can be.