I finished an implementation of the simplex spline approximation presented in Neamtu's latest paper. Below is a screen shot. On the left is the target function to approximate (sin(x*y)) and the on the right is the approximation with simplex spline basis from a 10X10 grid knot sets. ( The normal of the simplex spline is computed with a formula found in an old paper by Holliag. )

There are some questions that Neamtu's paper leaves unanswered about the basis functions used. Each basis function he uses for interpolation is the sum of simplex spline basis from Delaunay configurations that have identical interior knots. What's the property of the basis? In particular,

• The support of each basis is the union of a set of convex hulls, which I checked can be non-convex. Does it even have a a unique local maximum? (seems that way by eyeballing) I want to check this. Question: where is a good mathematical treatment for B-spline. Most references are too graphicy...

• How good is the choice of Greville site? (averaging of interior points). In other words, how far is it from the basis maximum?

• How many basis are there given n knots?

• a simple spline surface is a piecewise polynomial. Can we say anything about the "pieces" that this particualr choice of basis gives us?

I should also make some comparison between other interpolation schemes. In particular,

• Hans-Peter Seidel has a paper (1991) on simplex spline basis that are constructed over triangulation.
• natural neighbor interpolation.
• Bezier-based approach

Streaming will be the next implementation step. Streaming the construction of K-voronoi can be done from left to right in the plane. The relaxation operation for solving the linear system of equations should also be easily done in a streaming manner.

-- YuanxinLiu - 10 Mar 2005

The maximal of a quadratic b-spline basis over (0, a, b, 1) is located at b/(1-a+b).

For 2D, any configuration of 5 knots happen to only have one piece that is not on the boundary. Since quadratic function only have one mode, this piece must be the one with max, since other pieces all have min. (maybe the weighted convolution version will help?)

The convolution interpretation of B-spline doesn't help. Since the first degree b-spline is gotten by convolving a box with another box and is symmetrical. There is no way the maximal can be located at non-zero location. We can actually convince our selves that this can't be easily fixed up : start with a isymmetrical hat and convulve with a box. This will never produce the degenerate shape produced by (0, a, a + epsilon, 1).

-- YuanxinLiu - 15 Mar 2005
to top