Bivariate B-spline Bivariate B-spline

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

Generalizing B-spline Generalizing B-spline

• Basis function- a piecewise poly. defined

over (d+k+1) knots – compactly supported– smooth

• Knot sets – poly. reproduction – “local”

degree k = 2

B-spline basis

Generalizing B-spline Generalizing B-spline • Basis function

Geometric definition Evaluation ( Micchelli recurrence )

• a piecewise poly. defined over (d+k+1) knots

• compactly supported• smooth

Simplex spline basis [de Boor 76]

Generalizing B-spline Generalizing B-spline • Basis function

• a piecewise poly. defined over (d+k+1) knots

• compactly supported• smooth

Simplex spline basis [de Boor 76]

2d examples

k = 1 2 3

Generalizing B-spline Generalizing B-spline • Knot sets

Given a universe of knots in Rd, define family of knot sets of size d+k+1.

– multivariate B-spline [Neamtu 04]

- DMS spline ( triangular B-spline ) [Dahmen, Micchelli & Seidel 92]

• poly. reproduction • “local”

k = 2

Bivariate B-spline Bivariate B-spline a knot set X=XB U XI is chosen

whenever there is a circle through XB that has only XI inside.

XIXB

Bivariate B-spline 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

Bivariate B-spline 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

Property: a degree k bivariate B-spline knot set corresponds to a vertex of (k+1)-Voronoi diagram.

k = 0 1 2

Voronoi ComputationVoronoi 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)

Computation PipelineComputation Pipeline A set of knots S in the plane

A family of (k+3) subsets of S ( vertices in (k+1)-Voronoi diagram )

A set of degree-k simplex spline basis

A set of terrain samples P in 2d

terrain surface wavelet transform

Surface reconstructionSurface reconstructionGiven a set of terrain samples as input, construct a

bivariate B-spline terrain surface.

• choosing knot positions– What knots to use when given samples?

Surface reconstructionSurface reconstructionknot positions:

good

bad

Surface reconstructionSurface reconstructionGiven 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?

Computation PipelineComputation Pipeline A set of knots S in the plane

A family of (k+3) subsets of S ( vertices in (k+1)-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