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

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

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

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

Would also like: PM Restrictions Supports only meshes (orientable, 2-dimensional manifolds) Supports only meshes (orientable, 2-dimensional manifolds)

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

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 …

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

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]

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

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

Simplicial Complex VK M ^

V K M^

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

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

Simplicial Complex = {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

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

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

Generalized Vertex Split Encoding vunify

gvspl vunify aiaiaiai gvspl i = {a i },

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

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

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

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

vunify aiaiaiai gvspl i = {a i }, gvspl 1-simplices

gvspl i = {a i }, Generalized Vertex Split Encoding vunify aiaiaiai 2-simplices gvspl

gvspl i = {a i }, Generalized Vertex Split Encoding vunify aiaiaiai connectivity gvspl S

vunify gvspl i = {a i }, , vpos gvspl

Connectivity Encoding Analysis vunify example: 15 bits models (avg): 30 bits gvspl

Connectivity Encoding Constraints vunify gvspl

Connectivity Encoding Compression vunifyexample: 15 bits models (avg): 30 bits example: 10.2 bits models (avg): 14 bits gvspl

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

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

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

Simplification Results 72,346 triangles 674 triangles

Simplification Results 8,936 triangles 170 triangles

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

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