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

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

3. Previous Work Progressive Meshes [Hoppe, 96] 150 M0M0M0M0 vspl 0 M1M1M1M1152 M 175 500 … vspl i … 13,546 vspl n-1 M n =M ^ M0M0M0M0 vspl 0 … vspl i … vspl n-1 Progressive Mesh (PM) representation

4. PM Features Continuous LOD sequence Smooth visual transitions (Geomorphs) Progressive transmission Space-efficient representation 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) Supports only meshes (orientable, 2-dimensional manifolds)

6. PM Restrictions Supports only meshes (orientable, 2-dimensional manifolds) Preserves topological type Supports only meshes (orientable, 2-dimensional manifolds) Preserves topological type M0M0M0M0 MnMnMnMn

7. PM Restrictions Supports only meshes (orientable, 2-dimensional manifolds) Preserves topological type Supports only meshes (orientable, 2-dimensional manifolds) Preserves topological type 167,7448,0002,522 M0M0M0M0 MnMnMnMn … M i …

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

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

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

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

12. Simplicial Complex VK M ^

13. V K M^

14. K V M^ 1 23 4 567 = {1, 2, 3, 4, 5, 6, 7} + simplices abstract simplicial complex {1}, {2}, …0-dim

15. Simplicial Complex 5 1 23 467 = {1, 2, 3, 4, 5, 6, 7} + simplices V K M ^ {1}, {2}, …0-dim {1, 2}, {2, 3}…1-dim abstract simplicial complex

16. Simplicial Complex 5 1 23 467 = {1, 2, 3, 4, 5, 6, 7} + simplices {1}, {2}, …0-dim {1, 2}, {2, 3}…1-dim VK M ^ {4, 5, 6}, {6, 7, 5}2-dim abstract simplicial complex

17. PSC representation PSC Representation M1M1M1M1 M 22 gvspl 1 M 116 … gvspl i … gvspl n-1 M n =M ^ arbitrary simplicial complexes

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

19. Generalized Vertex Split Encoding vunify

20. gvspl vunify aiaiaiai gvspl i = {a i },

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

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

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

24. gvspl i = {a i }, Generalized Vertex Split Encoding vunify aiaiaiai gvspl 0-simplices 4

25. vunify aiaiaiai gvspl i = {a i }, 4 34122 gvspl 1-simplices

26. gvspl i = {a i }, 4 34122 12 Generalized Vertex Split Encoding vunify aiaiaiai 2-simplices gvspl

27. gvspl i = {a i }, 4 34122 12 Generalized Vertex Split Encoding vunify aiaiaiai connectivity gvspl S

28. vunify gvspl i = {a i }, 4 34122 12, vpos gvspl

29. Connectivity Encoding Analysis vunify example: 15 bits models (avg): 30 bits 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 33 3 4 4 4 4 4 44 4 gvspl

30. Connectivity Encoding Constraints vunify 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 33 3 4 4 4 4 4 44 4 gvspl

31. 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 33 3 4 4 4 4 4 44 4 Connectivity Encoding Compression vunifyexample: 15 bits models (avg): 30 bits example: 10.2 bits models (avg): 14 bits gvspl

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

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

34. Unify pair with lowest cost updating costs of affected candidatesupdating costs of affected candidates Unify pair with lowest cost updating costs of affected candidatesupdating costs of affected candidates PSC Construction Form a set of candidate vertex pairs 1-simplices of K 1-simplices of K DT1-simplices of K 1-simplices of K DT Compute cost of each vertex pair E = E dist + E disc + E area + E foldE = E dist + E disc + E area + E fold Form a set of candidate vertex pairs 1-simplices of K 1-simplices of K DT1-simplices of K 1-simplices of K DT Compute cost of each vertex pair E = E dist + E disc + E area + E foldE = E dist + E disc + E area + E fold

35. Simplification Results 72,346 triangles 674 triangles

36. Simplification Results 8,936 triangles 170 triangles

37. PSC VK M^ M1M1M1M1 gvspl n progressive geometry and topology lossless n any triangulation single vertex PSC Summary arbitrary simplicial complex

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