Transcript Splines over iterated Voronoi diagrams

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
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 / piecewise linear
B-splines use pw linear interpolation.
Reformulate with Sibson interpolation.
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
C1
Local support
Support
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
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
1D cubic
Surface example