Efficient Billboard Cloud Representation for Mesh Simplification

2. Billboard Clouds Xavier Décoret† Frédo Durand† Francois Sillion Julie Dorsey‡ †MIT-CSAIL Artis (INRIA/CNRS/UJF/INPG) ‡ Yale university

3. What this is not about!

4. What this is about • New representation: • Rectangles  global shape • Textures with a  finer details (silhouette) + appearance

5. Mesh Simplification • Clustering [RB93,LT97] • Hierarchical Dynamic Simplification [LE97] • Decimation of Triangle Meshes [SZL92] • Re-tiling [Tur92] • Progressive Meshes [Hop96,PH97] • Quadric Error Metrics [GH97] • Out of Core Simplification [Lin00] • Voxel based reconstruction [HHK+95] • Multiresolution analysis [EDD+95] • Superfaces [KT96], face cluster [WGH00]

6. Mesh Simplification • Constraints on models • Error control • Simplification envelopes [CVM96] • Permission Grids [ZG02] • Image driven [LT00] • Handling of attributes (textures and colors) • Integration to the metric[GH98][Hop99] • Re-generation [CMRS98,COM98] • Extreme Simplification • Silhouette Clipping [SGG+00]

7. Alternative Rendering • Image-based rendering • Lightfield,Lumigraph [LH96,GGRC96] • Impostors [Maciel95,Aliaga96,DSSD99] • Relief Textures [OB00] • Point-based rendering • Surfels [PZBG00] • Pointshop 3D [ZPKG02]

8. Classic billboards • A modelling “trick” [RH94] • Generalization to many planes / formalism • Automaticconstruction

9. Classic Billboards

10. Classic Billboards

11. Classic Billboards

12. Principle • Illustrated in 2D polygonal model

13. Principle Simplification by planes polygonal model

14. Maximum displacement Principle (1) • Allow vertex displacement P

15. Face Valid approximation by a plane Principle (2) • Project faces onto planes

16. Principle (2) • Project faces onto planes Valid approximation by a plane

17. Problem • How many planes? Which planes?

18. Overview • Express as an optimization problem • Represent the space of planes • Measurea plane’srelevance • Find a set of planes

19. Optimization problem Define over the set of Billboard clouds: • An error function • Vertex displacement • A cost function • Number of planes Error Based: bound max error  minimize cost

20. Overview • Express as an optimization problem • Represent the space of planes • Dual representation • Discretization • Measurea plane’srelevance • Find a set of planes

21.      0  2p 0 Dual space Dual representation • Illustrated in 2D • Hough transform [Hough62] Dual of line = point Line Origin Primal space

22. Dual of a point • Set of lines going through the point  (xP,yP)  0 Origin 2p 0 Primal space Dual space

23. Dual of a point • Set of lines going through the point  (xP,yP)  0 Origin 2p 0 Primal space Dual space

24. Dual of a point • Set of lines going through the point  (xP,yP)  0 Origin 2p 0 Primal space Dual space

25. Dual of a point • Set of lines going through the point  (xP,yP)  0 Origin 2p 0 Primal space Dual space

26. Dual of a point • Set of lines going through the point  r = xPcosq +yP sinq (xP,yP)  0 Origin 2p 0 Primal space Dual space

27. Dual of a point • Set of lines going through the point  r = xPcosq +yP sinqr 0 (xP,yP)  0 Origin 2p 0 Primal space Dual space

28. Dual of a sphere • Set of lines intersecting the sphere  R P  0 Origin 2p 0 Primal space Dual space

29. Dual of a sphere • Set of lines intersecting the sphere  R Dual of center P P  0 Origin 2p 0 Primal space Dual space

30. 2R Dual of a sphere • Set of lines intersecting the sphere  R Dual of center P P  0 Origin 2p 0 Primal space Dual space

31. Dual of a sphere • Set of lines intersecting the sphere  Dual of sphere=2R-thick slice R P  0 Origin 2p 0 Primal space Dual space

32. R P’ R P Dual of a face • Planes intersecting all vertices’ spheres   0 Origin 2p 0 Primal space Dual space

33. R P’ R P Dual of a face • Planes intersecting all vertices’ spheres   0 Origin 2p 0 Primal space Dual space

34. R P’ R P Dual of a face   0 2p 0 • How to work with this complex set of planes?

35. R P’ R P Discretization  Bins  0 2p 0 • How to work with this complex set of planes?

36. R P’ R P Discretization   0 2p 0 • How to work with this complex set of planes?

37. Overview • Express as an optimization problem • Represent the space of planes • Dual representation • Discretization • Measurea plane’srelevance • Density function • Find a set of planes

38. R P’ R P Discretization   0 2p 0 • How to work with this complex set of planes?

39. R R R R R R R P’ P’ P’ P’ P’ P’ P’ R R R R R R R P P P P P P P Discretization   0 2p 0

40. A tagged bin indicates that: Discretization  There is (at least)one plane valid for the face  0 2p 0 • Relevance of this plane?

41. Use projected area of face (on central plane) Density function Relevance  Grey plane is a better approximation of face !  0 2p 0

42. Density function • Compute in plane space (a float per bin) • Represent the relevance of each plane • Accumulate face contributions into the bins

43. Planes valid for face Density + - Density function Faces  

44. Density + - Density function Faces Planes valid for face  

45. Density + - Density function Faces Planes valid for face  

46. Density + - Density function Faces Planes valid for face  

47. Density + - Density function Faces Planes valid for face  

48. Density + - Density function Faces Planes valid for face  

49. Density + - Density function Faces Planes valid for face  

50. Density function Faces Planes valid for face Density +  - 