2. Defining Point Set Surfaces Nina Amenta and Yong Joo Kil University of California, Davis IDAV Institute for Data Analysis and Visualization Visualization and Graphics Research Group

3. IntroductionIntroduction

4. MLS Surface [Levin] Mesh-independent surface interpolation

5. MLS Surface [Alexa et al.] Computing and Rendering Point set surfaces, TVCG 2001. MLS Surface

6. [Pauly et al.] Shape Modeling with Point-Sampled Geometry, SIGGRAPH 2003. MLS Surface

7. [Pauly, Gross, Kobbelt] Efficient simplification of point-sampled surfaces, IEEE Vis. 2002. [Fleishman, Cohen-Or, Alexa and Silva] Progressive point-set surfaces, TOG 2003. [Adamson and Alexa] Ray tracing point-set surfaces, Shape Modeling International 2003. [Guo and Quin] Dynamic sculpting and deformation of point-set surfaces, PG 2003. [Mederos, Velho, and de Figueiredo] Moving least squares multiresolution surface approximation, SIBIGRAPI, 2003. [Xie, Wang, Hua, Quin, and Kaufman] Piecewise C1 continuous surface reconstruction of noisy point clouds via local implicit quadric regression, IEEE Vis. 2003. [Adamson and Alexa] On normals and projection operators for surfaces defined by point sets, S. Point-Based Graphics, 2004. [Mueller, Keiser, Nealan, Pauly, Gross, and Alexa] Point based animation of elastic, plastic and melting objects, S. Computer Animation, 2004.

8. ContributionContribution MLS is a kind of Extremal Surface –Equation! Analyze properties. Framework for generalization –Points with normals. Modeling with fewer primitives

9. Extremal Surface [Medioni and Guy] Inference of surfaces, curves and junctions from sparse, noisy 3D data, IEEE PAMI, 1997. [Tang, Medioni] Extremal feature extraction from 3D vector and noisy scalar fields, IEEE Visualization, 1998. [Medioni, Lee, and Tang] A Computational Framework for Segmentation and Grouping, Elsevier, 2000.

10. Extremal Surface Definition Vector Field: n Energy Field: e

11. Extremal Surface Definition e on n(x)n(x)

12. Extremal Surface in 2D e : energy field n : vector field

13. Implicit Definition Oriented vector field. Maxima and Minima of energy field.

14. MLS Projection function x

15. x q Least squares error

16. MLS Projection function x Minimal least squares error

17. MLS Projection function x (x)(x) (x)(x) f (x) = f f (x)

18. Stationary Points x (x)(x) n(x) & e(x) ?

19. Stationary Point of MLS x

20. Vector Field of MLS n(x)n(x) x

21. x

22. Energy Field of MLS x e(x)e(x)

23. Extremal Surface MLS  an extremal surface

24. Explicit Equation Normals from derivative Surface normal  n

25. DomainDomain

26. GeneralizationGeneralization MLS surface is an example of Extremal Surface Extremal Surface provides framework for generalization of MLS Example using Surfels

27. Extremal Surface for Surfels x

28. Vector Field

29. Energy Field

30. Extremal Surface

31. General Projection Scheme

32. Other approaches [Levin], Mesh-independent surface interpolation, (on his web site) [Adamson and Alexa], On Normals and Projection Operators for Surfaces Defined by Point Sets, Eurographics Symposium on Point-based Graphics

33. Varying Energy Fields

34. Varying Weight

35. Projection Method Surfel count: 77428 Our method: 16 secs. PointShop3D (ScanTools): 9 secs. Thanks to IBM TJ Watson Research Center

36. Sparse Set and MLS surface

37. Extremal Surface for Surfels

38. Future work Sampling theory Projection methods More vector and energy fields Sharp features

39. Thank you National Science Foundation (NSF) University of California, Davis PointShop3D team David Levin Our plugin: Defining Point Set Surfaces, available on pointshop3d.com