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

Triangle Meshes {f 1 } : { v 1, v 2, v 3 } {f 2 } : { v 3, v 2, v 4 } … connectivity geometry {v 1 } : (x,y,z) {v 2 } : (x,y,z) … face attributes {f 1 } : “skin material” {f 2 } : “brown hair” …

Triangle Meshes {v 2,f 1 } : (n x,n y,n z ) (u,v) {v 2,f 2 } : (n x,n y,n z ) (u,v) … corner attrib. {f 1 } : { v 1, v 2, v 3 } {f 2 } : { v 3, v 2, v 4 } … connectivity geometry {v 1 } : (x,y,z) {v 2 } : (x,y,z) … face attributes {f 1 } : “skin material” {f 2 } : “brown hair” …

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

Previous Work Model M=(K,V,D,S) Progressive mesh representation Model M=(K,V,D,S) Progressive mesh representation

Simplification: Edge collapse 13, faces M0M0M0M0 M1M1M1M1 M 175 ecol 0 ecol i ecol n-1 MnMnMnMn ecol(v s,v t, v s ) vlvlvlvl vrvrvrvr vtvtvtvt vsvsvsvs vsvsvsvs vlvlvlvl vrvrvrvr (optimization) ’ ’

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

Application: Progressive transmission Transmit records progressively: M0M0M0M0 Receiver displays: time M0M0M0M0 vspl 0 vspl 1 vspl i-1 MiMiMiMi (~ progressive GIF & JPEG) MnMnMnMn vspl n-1

Application: Continuous- resolution LOD From PM, extract M i of any desired complexity. M0M0M0M0 vspl 0 vspl 1 vspl i-1 vspl n-1 MiMiMiMi 3,478 faces? 3,478 M0M0M0M0 MnMnMnMn MiMiMiMi ~400K faces/sec!

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 …

PM Restrictions Supports only “meshes” (orientable, 2-dimensional manifolds) Preserves topological type minimal vertex num of closed g genus mesh Supports only “meshes” (orientable, 2-dimensional manifolds) Preserves topological type minimal vertex num of closed g genus mesh

PM Restrictions Volumetric data :

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

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

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

Abstract simplicial complexes (ASC)

Abstract simplicial complexes

Geometric vs toplogical realizations

Graph representation. First level – principle simplicesFirst level – principle simplices discrete attributes discrete attributes

Abstract simplicial complexes Manifold-adjecent Not manifold-adjecent

aiaiaiai bibibibi aiaiaiai s s’

* MN – manifold adjacent neighbors

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

Geomorps. (smooth transform.)

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

Generalized Vertex Split Encoding vunify aiaiaiai aiaiaiai bibibibi

gvspl vunify aiaiaiai gvspl i = {a i }, aiaiaiai aiaiaiai bibibibi

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 aiaiaiai aiaiaiai bibibibi

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

gvspl i = {a i }, Generalized Vertex Split Encoding vunify aiaiaiai 2-simplices gvspl aiaiaiai bibibibi 1 2

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

vunify gvspl i = {a i }, ,  vpos gvspl aiaiaiai bibibibi

Generalized Vertex Split Encoding vunify gvspl i = {a i }, ,  vpos gvspl aiaiaiai bibibibi If a simplex has split code c in {1, 2}, all of its parents have split code c.If a simplex has split code c in {1, 2}, all of its parents have split code c. If a simplex has split code 3, none of its parents have split code 4.If a simplex has split code 3, none of its parents have split code 4.

Connectivity Encoding Constraints vunify 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

PSC Construction Form a set of candidate vertex pairs 1-simplices of K some pairs from different components1-simplices of K some pairs from different components Form a set of candidate vertex pairs 1-simplices of K some pairs from different components1-simplices of K some pairs from different components For each candidate vertex pairs a, b : calculate accuracy of vunify (a,b) calculate accuracy of vunify (a,b) Sort candidate vertices according to accuracy For each candidate vertex pairs a, b : calculate accuracy of vunify (a,b) calculate accuracy of vunify (a,b) Sort candidate vertices according to accuracy

PSC Construction How to get pairs from different components Octree of model’s bbox Octree of model’s bbox How to get pairs from different components Octree of model’s bbox Octree of model’s bbox

PSC Construction How to get pairs from different components Delaunay triangulation K DT Delaunay triangulation K DT How to get pairs from different components Delaunay triangulation K DT Delaunay triangulation K DT No point in P is inside the circumcircles of any simplex in K DT Center of circumcircles gives Voronoi diagram

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 fold∆E = ∆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 fold∆E = ∆E dist + ∆E disc + ∆ E area + E fold

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

Computing ∆E

Simplification Results 72,346 triangles 674 triangles

Simplification Results 8,936 triangles 170 triangles

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

PSC Features Video

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