1 / 60

Curvature Interpolation at Local Maximum

This research paper explores the problem with interpolation using Catmull-Rom spline and proposes a method to interpolate at maximum curvature points in curves. The desired properties include interpolating all control points and ensuring that maximum curvature points only occur at control points. The paper discusses related works and presents an optimization algorithm to achieve curvature interpolation at local maximum points.

ahardiman
Download Presentation

Curvature Interpolation at Local Maximum

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. -Curves: Interpolation at Local Maximum Curvature Zhipei Yan1, Stephen Schiller2, Gregg Wilensky3, Nathan Carr2, Scott Schaefer1 1Texas A&M University, 2Adobe Research, 3Adobe

  2. Problems with Interpolation Catmull-Rom Spline

  3. “… human visual system is sensitive to minima and maxima of curvature …” “… all curvature extrema should coincide with the given control points …” [Levien 2009]

  4. Desired Properties • Interpolate all control points • Max curvatures (salient points) only occur at control points ---- No cusps except at control points • Curvature continuous

  5. Clothoid Linear curvature Not continuous with motion of control points

  6. Quadratic spline

  7. Quadratic Bezier Curve

  8. Quadratic Bezier Curvature

  9. Interpolation at Max Curvature

  10. Interpolation at Max Curvature

  11. Quadratic Time Parameter

  12. Quadratic Time Parameter

  13. Quadratic Time Parameter

  14. Quadratic Time Parameter

  15. Quadratic Time Parameter

  16. Quadratic Time Parameter

  17. Quadratic Time Parameter

  18. Quadratic Time Parameter Unique from , , and

  19. Computation of Max Curvature Time (1) (2) Substitute (1) into (2) a Unique as a function of , , and

  20. Related works [Schaback 1989]

  21. Related works [Schaback 1989] Use input points as splliting points

  22. Related works [Feng et al. 1995] Interpolate tangent vectors at control points No curvature control.

  23. G2 Condition

  24. G2 Condition

  25. G2 Condition

  26. G2 Condition D C B A C=(1-λ)B+λD E

  27. G2 Condition

  28. G2 Condition

  29. G2 Condition

  30. Curvature

  31. G2 Condition

  32. G2 Joint C D B A E

  33. Inflection Point E • Curvatures have different signs C D B A

  34. Inflection Point

  35. Optimization

  36. Optimization Input

  37. Optimization Initialization

  38. Optimization

  39. Optimization Iteration 0 pi ci,2 ti ci,0

  40. Optimization Iteration 0 ci+1,1 ci,1 λi ci,0 ci+1,2

  41. Optimization Iteration 0 ci,2 ci,1 ci,0 ti Only is variable.

  42. Step 2 =

  43. Optimization Iteration 1

  44. Optimization Iteration 2

  45. Optimization Iteration 3

  46. Optimization Iteration 4

  47. Optimization Iteration 5

More Related