Progressive Simplicial Complexes

1. Progressive Simplicial Complexes Jovan Popovic Carnegie Mellon University Hugues Hoppe Microsoft Research

2. Complex Models • Rendering • Storage • Transmission 232, 974 faces

3. 150 152 500 13,546 ^ Mn=M M1 M175 M0 M0 vspl0 … vspli … vspln-1 … vspli … vspl0 vspln-1 Progressive Mesh (PM) representation Previous Work • Progressive Meshes [Hoppe, ‘96]

4. PM Features • Continuous LOD sequence • Smooth visual transitions (Geomorphs) • Progressive transmission • Space-efficient representation

5. Would also like: PM Restrictions • Supports only “meshes” (orientable, 2-dimensional manifolds)

6. M0 Mn PM Restrictions • Supports only “meshes” (orientable, 2-dimensional manifolds) • Preserves topological type

7. 2,522 8,000 167,744 PM Restrictions • Supports only “meshes” (orientable, 2-dimensional manifolds) • Preserves topological type M0 Mn … Mi …

8. PM edge collapse(ecol) vertex split(vspl) Progressive Simplicial Complexes (PSC)

9. Previous Work • Vertex unification schemes [Rossignac-Borrel ‘93] [Schaufler-Stürzlinger ‘95]

10. PM PSC edge collapse(ecol) vertex unification(vunify) vertex split(vspl) Progressive Simplicial Complexes (PSC)

11. PM edge collapse(ecol) vertex split(vspl) Progressive Simplicial Complexes (PSC) PSC vertex unification(vunify) generalized vertex split(gvspl)

12. ^ M V K Simplicial Complex

13. ^ M Simplicial Complex V K

14. ^ M 6 4 2 3 1 7 5 abstract simplicial complex = {1, 2, 3, 4, 5, 6, 7} + simplices {1}, {2}, … 0-dim Simplicial Complex V K

15. ^ M V K 6 4 2 3 1 7 5 Simplicial Complex abstract simplicial complex = {1, 2, 3, 4, 5, 6, 7} + simplices {1}, {2}, … 0-dim {1, 2}, {2, 3}… 1-dim

16. ^ M V K 6 4 2 3 1 7 5 Simplicial Complex abstract simplicial complex = {1, 2, 3, 4, 5, 6, 7} + simplices {1}, {2}, … 0-dim {1, 2}, {2, 3}… 1-dim {4, 5, 6}, {6, 7, 5} 2-dim

17. arbitrary simplicial complexes ^ Mn=M PSC Representation M1 M22 M116 gvspl1 … gvspli … gvspln-1 PSC representation

18. PSC Features Video • Destroyer PSC sequence • PM, PSC comparison • PSC Geomorphs • Line Drawing

19. vunify Generalized Vertex Split Encoding

20. ai gvspli = {ai}, Generalized Vertex Split Encoding vunify gvspl

21. Connectivity Encoding case (1) case (2) case (3) case (4) 0-dim undefined undefined 1-dim 2-dim

22. Connectivity Encoding case (1) case (2) case (3) case (4) 0-dim undefined undefined 1-dim 2-dim

23. Connectivity Encoding case (1) case (2) case (3) case (4) 0-dim undefined undefined 1-dim 2-dim S

24. 4 0-simplices Generalized Vertex Split Encoding vunify ai gvspl gvspli = {ai},

25. Generalized Vertex Split Encoding vunify ai gvspl gvspli = {ai}, 4 34122 1-simplices

26. Generalized Vertex Split Encoding vunify ai gvspl gvspli = {ai}, 4 34122 12 2-simplices

27. Generalized Vertex Split Encoding vunify ai gvspl gvspli = {ai}, 4 34122 12 connectivity S

28. vpos Generalized Vertex Split Encoding vunify gvspl gvspli = {ai}, 4 34122 12,

29. 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 Connectivity Encoding Analysis vunify gvspl example: 15 bits models (avg): 30 bits

30. Connectivity Encoding Constraints vunify gvspl 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4

31. example: 15 bits models (avg): 30 bits example: 10.2 bits models (avg): 14 bits Connectivity Encoding Compression vunify gvspl 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4

32. Space Analysis • Average 2D manifold mesh n vertices, 3n edges, 2n triangles • PM representation n ( log2n + 4 ) bits • PSC representation n ( log2n + 7 ) bits

33. Form a set of candidate vertex pairs 1-simplices of K 1-simplices of KDT candidate vertex pairs PSC Construction

34. PSC Construction • Form a set of candidate vertex pairs • 1-simplices of K 1-simplices of KDT • Compute cost of each vertex pair • ∆E = ∆Edist + ∆Edisc + E∆area + Efold • Unify pair with lowest cost • updating costs of affected candidates

35. Simplification Results 72,346 triangles 674 triangles

36. Simplification Results 8,936 triangles 170 triangles

37. ^ M PSC Summary PSC V K lossless M1 gvspl arbitrary simplicial complex single vertex • progressive geometry and topology • any triangulation

38. PSC Summary • Continuous LOD sequence • Smooth transitions (Geomorphs) • Progressive transmission • Space-efficient representation • Supports topological changes • Models of arbitrary dimension e.g. LOD in volume rendering