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

2. 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

3. Generalizing B-spline Basis function Geometric definitionEvaluation ( Micchelli recurrence ) a piecewise poly. defined over ( d+k+1 ) knots compactly supported smooth Simplex spline basis [de Boor 76]

4. 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 23

5. Generalizing B-spline Knot sets Given a universe of knots in R d, 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

6. Bivariate B-spline a knot set X=X B U X I is chosen whenever there is a circle through X B that has only X I inside. XIXI XBXB

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 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

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 ( 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 surfacewavelet 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 ( 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 surfacewavelet transform point ordering for wavelet transform