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Progressive Simplicial Complexes

This paper explores the innovations in Progressive Simplicial Complexes (PSC) proposed by Jovan Popovic at Carnegie Mellon University and Hugues Hoppe from Microsoft Research. We review the core features of PSC, including continuous Level of Detail (LOD) sequences for smooth visual transitions, and the effective encoding and transmission of complex geometrical data. The study highlights the efficiency of PSC in storing, rendering, and progressively transmitting complex models while preserving topological properties. Key techniques such as vertex unification and generalized vertex splits are discussed, demonstrating their significance in modern geometric modeling systems.

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Progressive Simplicial Complexes

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  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

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