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A Simple Class of Non-Linear Subdivision Schemes Scott Schaefer Etienne Vouga Ron Goldman Subdivision Set of rules S that recursively act on a shape p 0 1 k k p S p Converges to a smooth shape Subdivision

  1. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  2. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  3. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  4. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  5. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  6. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  7. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  8. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  9. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  10. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  11. Functional Equations  Find parametric midpoint of a function F    x x    0 1 F G ( F ( x ), F ( x )) 0 1   2  Example: F ( x ) = cos( m x+b )         ( 1 F ( x ))( 1 F ( x )) ( 1 F ( x ))( 1 F ( x )) x x    0 1 0 1 0 1 F   2 2

  12. Other Averaging Rules Function Averaging Rule   2 2   x x F ( x ) F ( x ) F ( ) 0 1 0 1 F ( x ) x 2 2      (( ( ) ( )) / 2 ( ) ( ) F x F x F x F x x x 2 F ( ) 0 1 0 1 F ( x ) x 0 1 2 2   x x F ( x ) F ( x )  F ( ) 0 1 0 1 1 F ( x )  2 ( F ( x ) F ( x )) / 2 x 0 1   x x F ( x ) F ( x )  F ( ) 0 1 0 1 1 F ( x ) 2  2 2 2 x ( F ( x ) F ( x ) ) / 2 0 1         ( ( ) 1 )( ( ) 1 ) ( ( ) 1 )( ( ) 1 ) F x F x F x F x x x F ( x ) cosh( x ) F ( ) 0 1 0 1 0 1 2 2

  13. Non-linear Maps  Given   n R  F: 1-1 function on  S: subdivision scheme ˆ   1   S F S F   Then   ˆ   1    S F S F         ˆ        0 0  S p p S F p F p

  14. Non-linear Maps  Given   n R  F: 1-1 function on  : subdivision scheme      S S S 2 S 1 d ˆ   1   S F S F   Then       ˆ     1 1 1            S F S F F S F F S F 2 1 d

  15. Non-linear Maps Example       ˆ     1 1 1           S F S F F S F F S F 2 1 d F ( x ) Lane-Reisenfeld ˆ j    S ( p ) p 1 S ( p ) F ( F ( p   ))   1 j 1 j j 2 2    ˆ  1 1   p p F ( p ) F ( p )   j j 1 j j 1 S ( p ) S ( p ) F ( )   i 1 j i 1 j 2 2

  16. Non-linear Maps Example       ˆ     1 1 1           S F S F F S F F S F 2 1 d  x Lane-Reisenfeld F ( x ) e ˆ j  j  S ( p ) p S ( p ) p     1 j 1 j 2 2  ˆ  p p   j j 1 S ( p ) S ( p ) p p    i 1 j 2 i 1 j j j 1

  17. Smoothness and Interpolation  Given   n R  F : 1-1 function on  S : subdivision scheme ˆ   1    S F S F  Then ˆ    ˆ 0 k n 0 min( k , n ) S ( p ) : C & F : C S ( p ) : C  ˆ   S :interpolatory :interpolatory S

  18. Example Four-Point [Dyn et al. 1987]

  19. Example Four-Point [Dyn et al. 1987] 9 9 16 16   1 1 16 16

  20. Example Four-Point [Dyn et al. 1987]

  21. Example Four-Point [Dyn et al. 1987]

  22. Example Four-Point [Dyn et al. 1987]

  23. Example Four-Point [Dyn et al. 1987]

  24. Example Four-Point [Dyn et al. 1987] Mobius Transform

  25. Example Four-Point [Dyn et al. 1987] Mobius Transform

  26. Example Four-Point [Dyn et al. 1987] Mobius Transform

  27. Example Four-Point [Dyn et al. 1987] Mobius Transform

  28. Example Four-Point [Dyn et al. 1987] Mobius Transform

  29. Geometric Properties  Properties: convex-hull, variation diminishing  z Linear F ( z ) e

  30. Geometric Interpretation  Modify geodesics so that the properties hold ˆ ˆ ˆ ˆ    1 1 D ( P , Q ) Dist ( F ( P ), F ( Q )) Euclidean F  1 F

  31. Geometric Interpretation  A set C is convex w.r.t. the geodesics G if the geodesic connecting any two points in C lies completely within C

  32. Geometric Interpretation  A set C is convex w.r.t. the geodesics G if the geodesic connecting any two points in C lies completely within C

  33. Geometric Interpretation  A set C is convex w.r.t. the geodesics G if the geodesic connecting any two points in C lies completely within C

  34. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  35. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  36. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  37. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  38. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  39. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  40. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  41. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  42. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  43. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  44. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  45. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  46. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  47. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  48. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  49. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  50. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  51. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  52. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  53. Intersection If convex hulls of the control points do not 1) intersect, then the curves do not intersect If each curve is approximately a straight 2) line, intersect those lines; else subdivide

  54. Computing Convex Hulls  Non-linear hulls may be curved and difficult to compute  If is monotonic, we can compute a F ' t ( ) simple piecewise linear approximation

  55. Computing Convex Hulls  Non-linear hulls may be curved and difficult to compute  If is monotonic, we can compute a F ' t ( ) simple piecewise linear approximation

  56. Computing Convex Hulls  Non-linear hulls may be curved and difficult to compute  If is monotonic, we can compute a F ' t ( ) simple piecewise linear approximation

  57. Computing Convex Hulls  Non-linear hulls may be curved and difficult to compute  If is monotonic, we can compute a F ' t ( ) simple piecewise linear approximation

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