1. CS175 2003 1 Subdivision I: The Univariate Setting Peter Schröder

2. CS175 2003 2 B-Splines (Uniform) Through repeated integration 1 10 B 3 (x)

3. CS175 2003 3 B-Splines Obvious properties piecewise polynomial: unit integral: non-negative: partition of unity: support:

4. CS175 2003 4 B-Splines Repeated convolution box function x

5. CS175 2003 5 Convolution Reminder definition: translation: dilation:

6. CS175 2003 6 Refinability I B-Spline refinement equation a B-spline can be written as a linear combination of dilates and translates of itself example linear B-spline and all others… 1/2 1

7. CS175 2003 7 Refinability II Refinement equation for B- splines take advantage of box refinement

8. CS175 2003 8 Refinability

9. CS175 2003 9 B-Spline Refinement Examples

10. CS175 2003 10 Spline Curves I Sum of B-splines curve as linear combination control points

11. CS175 2003 11 Spline Curves II Refine each B-spline in sum example: linear B-spline 1/2 1

12. CS175 2003 12 Spline Curves III Refinement for curves refine each B-spline in sum refinement of control points refined bases

13. CS175 2003 13 Refinement of Curves Linear operation on control points succinctly

14. CS175 2003 14 Refinement of Curves Bases and control points

15. CS175 2003 15 Subdivision Operator Example cubic splines

16. CS175 2003 16 Subdivision Apply subdivision to control points draw successive control polygons rather than curve itself

17. CS175 2003 17 Summary so far Splines through refinement B-splines satisfy refinement eq. basis refinement corresponds to control point refinement instead of drawing curve, draw control polygon subdivision is refinement of control polygon

18. CS175 2003 18 Analysis Setup polygon mapped to polygon finite or (bi-)infinite, p i j 2 R d subdivision operator (linear for now)

19. CS175 2003 19 Subdivision Schemes Some properties affine invariance compact support index invariance (topologic symmetry) local definition

20. CS175 2003 20 Subdivision Operator Properties compact support affine invariance index invariance/symmetry

21. CS175 2003 21 Subdivision Operator local definition: weights depend only on local neighborhood Terms stationary: level independence uniform: location independence no boundaries (for now)

22. CS175 2003 22 Generating Functions Subdivision operator as convolution

23. CS175 2003 23 Examples Splines linear: quadratic: higher order…

24. CS175 2003 24 Examples Quadratic splines Chaikin’s algorithm computes new points with weights 1/4(1, 3) and 1/4(3, 1) what happens if we change the weights?

25. CS175 2003 25 Convergence How much leeway do we have? design of other subdivision rules example: 4pt scheme establish convergence establish order of continuity

26. CS175 2003 26 Analysis Simple facts affine invariance necessary condition for uniform convergence

27. CS175 2003 27 Analysis Convergence define linear interpolant over given control points and associated parametric values (knot vector) typically define pointwise

28. CS175 2003 28 Analysis Convergence in max/sup norm Theorem if then convergence of is uniform

29. CS175 2003 29 Uniform Convergence Proof linear spline subdivision operator

30. CS175 2003 30 Difference Decay Sufficient condition continuous limit if analysis by examining associated difference scheme

31. CS175 2003 31 Example Cubic B-splines stencils 44 1/8 161

32. CS175 2003 32 Example Cubic B-splines differences 31 1/8 13

33. CS175 2003 33 Difference Decay Analysis of difference scheme construction from the subdivision scheme itself

34. CS175 2003 34 Higher Orders Smoothness how to show C 1 ? divided differences must converge check difference of divided differences example 4pt scheme

35. CS175 2003 35 Smoothness Consequences 4pt scheme: decay estimate

36. CS175 2003 36 Example 4pt scheme differences of divided differences 22 1/8 6 1/8

37. CS175 2003 37 Analysis Fundamental solution gives basis functions

38. CS175 2003 38 Fundamental Solution Properties refinement relation (why?) support? non-zero coefficients:

39. CS175 2003 39 So Far, So Good I What do we know now? regular setting approximating B-splines interpolating 4pt scheme (Deslaurier-Dubuc)

40. CS175 2003 40 So Far, So Good II What do we know now? differences continuity differentiability not quite general enough current setting assumes a particular parameterization

41. CS175 2003 41 More General Settings Non-uniform in spline case better curves subdivision weights will vary knot insertion interpolation

42. CS175 2003 42 4pt Scheme I Where do the weights come from? example of Deslaurier-Dubuc given set of samples use interpolating polynomial to refine ii+ 1 i+ 2 i-1 Interpolating polynomial for 4 successive samples sample here

43. CS175 2003 43 4pt Scheme II Weight computation grind out interpolating polynomial resulting weights: 1/16(-1, 9, 9, -1) Deslaurier-Dubuc generalization of same idea higher orders yield higher continuity tends to exhibit “ringing” (as is to be expected…)

44. CS175 2003 44 Deslaurier-Dubuc Local polynomial reproduction choose s k accordingly ( d =1 for 4pt) non-uniform possible, increasing smoothness, approximation power, limit for increasing d is sync fn.

45. CS175 2003 45 Irregular Analysis New tools generating functions not applicable instead: spectral analysis (why?) for irregular spacing only one parameter: ratio of spacing On to spectral analysis

46. CS175 2003 46 Analysis We need a different approach the subdivision matrix a finite submatrix representative of overall subdivision operation based on invariant neighborhoods structure of this matrix key to understanding subdivision

47. CS175 2003 47 Example Cubic B-spline 5 control points for 1 segment on either side of the origin S j j+1

48. CS175 2003 48 Neighborhoods Which points influence a region? for analysis around a point 1

49. CS175 2003 49 Subdivision Matrix Invariant neighborhood which  (i-t) overlap the origin? tells smoothness story

50. CS175 2003 50 Eigen Analysis What happens in the limit? behavior in neighborhood of point apply S infinitely many times… suppose S has complete set of EVs control points in invariant neighborhood eigen vectors

51. CS175 2003 51 Subdivision Matrix Properties eigen vectors of non-zero eigen values identical proof by extension of y if defective, use generalized eigen vectors and values

52. CS175 2003 52 Subdivision Matrix Eigen vectors and eigen functions no overbar

53. CS175 2003 53 Scaling Relation Eigen functions scale in neighborhood of the origin

54. CS175 2003 54 Smoothness (at Origin) Lemma for functions which scale I II III

55. CS175 2003 55 Necessary Conditions Continuity at eigen functions

56. CS175 2003 56 Necessary Conditions Spectrum must be 2 -i for 0 · i · k and corresponding eigen functions must be monomials generalized eigen vectors? 0 must be simple ?

57. CS175 2003 57 Sufficient Conditions Check at origin eigen functions for | | < 2 - k must be checked

58. CS175 2003 58 Subdivision Operator Spectrum necessary conditions for C k must have i =2 -i for i · k eigen functions are polynomials generally not enough 4pt scheme has 1,1/2,1/4,1/4,1/8,-1/16, -1/16 approximation properties

59. CS175 2003 59 Convergence Limit position let j go to infinity if 0 =1 and | i |<1, i=1,…,n-1 example: cubic B-spline

60. CS175 2003 60 Geometric Behavior Move limit point to origin look at higher order behavior tangent vector

61. CS175 2003 61 More General Settings Subtleties generalized eigen values more subtle smoothness analysis non-uniform subdivision completely irregular subdivision boundaries formule de commutation

62. CS175 2003 62 A Note Size of subdivision matrix for analysis need enough support to parameterize a finite neighborhood of the origin for evaluation need only enough support to zoom in on origin e.g., cubic spline needs 5 respectively 3 control points

63. CS175 2003 63 Eigen Analysis Summary invariant neighborhood to understand behavior around point Eigen decomposition of subdivision matrix helpful limit point: a 0, tangent: a 1 General setting more complicated...