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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. Basis function - a piecewise poly. defined over ( d+k+1 ) knots compactly supported smooth. B-spline basis.

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Bivariate B-spline

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  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 B-spline basis degree k = 2 • 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 (d+k+1) 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 (d+k+1) knots • compactly supported • smooth

  5. k = 2 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 & Seidel92] • poly. reproduction • “local”

  6. Bivariate B-spline a knot set X=XBUXIis 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 iin 2d partitions the plane into cells such that points in each cell have the same closest ineighbors i = 1 2 3

  8. Bivariate B-spline High order Voronoi diagram Definition: A Voronoi diagram of degree iin 2d partitions the plane into cells such that points in each cell have the same closest ineighbors Property: a degreekbivariate B-spline knot set corresponds to a vertex of (k+1)-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 (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

  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 surface wavelet transform • point ordering for wavelet transform

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