Bivariate Bspline

1
Bivariate B-spline

• Outline
• Multivariate B-spline Neamtu 04
• Computation of high order Voronoi diagram
• Interpolation with B-spline

2
Generalizing B-spline
B-spline basis
degree k 2

• Basis function
• - a piecewise poly. defined over (dk1) knots
• compactly supported
• smooth

• Knot sets
• poly. reproduction
• local

3
Generalizing B-spline

• Basis function

Simplex spline basis de Boor 76
Geometric definition
Evaluation ( Micchelli recurrence )

• a piecewise poly. defined over (dk1) knots
• compactly supported
• smooth

4
Generalizing B-spline

• Basis function

Simplex spline basis de Boor 76
2d examples
k 1
2
3

• a piecewise poly. defined over (dk1) knots
• compactly supported
• smooth

5
Generalizing B-spline

• Knot sets
• Given a universe of knots in Rd, define family of
  knot sets of size dk1.
• multivariate B-spline Neamtu 04
• - DMS spline ( triangular B-spline ) Dahmen,
  Micchelli Seidel 92

• poly. reproduction
• local

6
Bivariate B-spline

• a knot set XXB U XI is chosen whenever there
  is a circle through XB that has only XI inside.

XB
XI
7
Bivariate B-spline

• High order Voronoi diagram

Definition A Voronoi diagram of degree i in 2d
partitions the plane into cells such that points
in each cell have the same closest i neighbors
i 1
2
3
8
Bivariate B-spline

• High order Voronoi diagram

Definition A Voronoi diagram of degree i in 2d
partitions the plane into cells such that points
in each cell have the same closest i neighbors
Property a degree k bivariate B-spline knot
set corresponds to a vertex of (k1)-Voronoi
diagram.
i 1
2
3
k 0
1
2
9
Voronoi Computation

• theory O(n log(n)) time , O(n) space
• practice O(n) time for evenly distributed points
  
• Engineering challenges
• speed ( exploit even distribution )
• robustness ( degeneracy, round-off errors )
• memory (streaming ) (demo)

10
Computation Pipeline

• A set of knots S in the plane
• A family of (k3) subsets of S (
  vertices in (k1)-Voronoi diagram )
• A set of degree-k simplex spline basis

A set of terrain samples P in 2d
terrain surface
wavelet transform
11
Surface reconstruction

• Given a set of terrain samples as input,
  construct a bivariate B-spline terrain surface.
• choosing knot positions
• What knots to use when given samples?

12
Surface reconstruction
knot positions good
bad
13
Surface reconstruction

• Given a set of terrain samples as input,
  construct a bivariate B-spline terrain surface.
• choosing knot positions
• What knots to use when given samples?
• coefficient computation
• Interpolation or approximation?

14
Computation Pipeline

• A set of knots S in the plane
• A family of (k3) subsets of S (
  vertices in (k1)-Voronoi diagram )
• A set of degree-k simplex spline basis

A set of terrain samples P in 2d
terrain surface
wavelet transform

• point ordering for wavelet transform

15
(No Transcript)