If this (...) leaves you a bit wondering what multivariate splines might be, I am pleased. For I...

If this (...) leaves you a bit wondering what multivariate splines might be, I am pleased. For I don’t know myself.

Carl de Boor

Splines over iterated Voronoi diagrams

Gerald Farin

Overview

• Voronoi diagrams

• Sibson’s interpolant

• quadratic B-splines

• quadratic iterated splines

• the general case

History

• B-splines: 1946 - Schoenberg• Finite elements: 1950’s - Zienkiewicz...• Simplex splines: 1976 – de Boor• Recursion: 1972 – de Boor, Mansfield, Cox• Bezier triangles: 1980’s – Sabin, Farin• Box splines: 1980’s – de Boor, de Vore• B-patches: 1982 – Dahmen, Micchelli, Seidel

Voronoi diagrams

Voronoi diagrams

Voronoi diagrams

Voronoi diagrams

Sibson’s interpolant

Sibson basis function

Support

Properties

• linear precision

• 1D: piecewise linear

• on boundary(CH): piecewise linear

• C1 except data sites, C0 there

• not idempotent

• dimension independent

Sibson / de Boor

de Boor algorithm: pw linear interpolation.

Now:

pw linear Sibson

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic B-spline functions

Quadratic surfaces

Quadratic surfaces

Quadratic surfaces

Quadratic surfaces

Quadratic surfaces

Quadratic surfaces

Reminder: Sibson’s...

Quadratic surfaces

Quadratic surfaces

P.Veerapaneni

Quadratic surfaces

Properties

• Linear precision

• 1D: quadratic B-splines

• dimension independent

• C2 (C1 at ui)

• Local support

• quadratic reproduction

Support / Smoothness

Support / Smoothness

Support / Smoothness

Support / Smoothness

Support / Smoothness

Support / Smoothness

Basis function

“Tangent planes”

P. Veerapaneni

“Tangent planes”

“Tangent planes”

The general case

• start: set of sites U0

• iterate Voronoi diagrams U1...Un-1

• assign function values Z0 at Un-1

• insert point v0

• generate (locally) refined Voronoi diagram V0

• find Voronoi diagrams V1...Vn-1

• compute Zi at Vi; i= n-1,...,1• result: Point Zn at v0

Surface example

polynomial precision

1D cubic

1D cubic

1D cubic

1D cubic

1D cubic

1D cubic

1D cubic

1D cubic