Martin Isenburg Triangle Strip Compression University of North Carolina at Chapel Hill

Introduction A new edge-based encoding scheme for mesh connectivity.

Introduction A new edge-based encoding scheme for mesh connectivity.  compact mesh representations

Introduction A new edge-based encoding scheme for mesh connectivity.  compact mesh representations  simple implementation

Introduction A new edge-based encoding scheme for mesh connectivity.  compact mesh representations  simple implementation  fast decoding

Introduction A new edge-based encoding scheme for mesh connectivity.  compact mesh representations  simple implementation  fast decoding

Introduction A new edge-based encoding scheme for mesh connectivity.  compact mesh representations  simple implementation  fast decoding  include triangle strip information at little additional costs

A Triangle Mesh

A Stripified Triangle Mesh

Triangle Strips

Technique for efficient rendering of triangle meshes. Reduce data transfer between main memory and render engine. Requires built-in buffer for two (previous) vertices. Supported in today’s graphic boards. Triangle Strips

Rendering with Triangles

Rendering with Triangle Strips

v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v0v0 7 triangles 9 vertices Sequential Triangle Strip

Sequential Generalized

v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v0v0 v 0 v 1 v 2 v 3 v 2 v 5 v 4 v 7 v 6 v 8 7 triangles 10 vertices Generalized Triangle Strip

v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v0v0 v 0 v 1 v 2 v 3 v 2 v 5 v 4 v 7 v 6 v 8 7 triangles 10 vertices Generalized Triangle Strip

v2v2 v1v1 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v0v0 v 0 v 1 v 2 v 3 v 2 v 5 v 4 v 7 v 6 v 8 7 triangles 10 vertices Generalized Triangle Strip v2v2 swap

finding good set of triangle strips (e.g. few swaps, restarts) isn’t easy computing optimal solution is NP hard proposed tools with good heuristics: –STRIPE(96, Evans et al.) –SWAPS(97, Speckmann & Snoeyink) –FGTS(99, Xiang et al.) Stripification Algorithms

Storing a Mesh

Where are the vertices located ?  mesh geometry Which vertices form a triangle?  mesh connectivity Storing a Mesh

list of vertices x 0 y 0 z 0 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of vertices list of triangles x 0 y 0 z x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of vertices list of triangles x 0 y 0 z x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of vertices list of triangles x 0 y 0 z x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of vertices list of triangles x 0 y 0 z x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of vertices list of triangles x 0 y 0 z x 1 y 1 z x 2 y 2 z 2 u 2 v 2 w x 3 y 3 z 3 u 3 v 3 w x 4 y 4 z 4 u 4 v 4 w x n y n z n..... Standard Representation

list of vertices list of triangles x 0 y 0 z x 1 y 1 z x 2 y 2 z 2 u 2 v 2 w x 3 y 3 z 3 u 3 v 3 w x 4 y 4 z 4 u 4 v 4 w x n y n z n..... Connectivity Storage costs: 6 log(n) bpv

Storing a Stripified Mesh

Where are the vertices located ?  mesh geometry Which vertices form a triangle?  mesh connectivity Storing a Stripified Mesh

Where are the vertices located ?  mesh geometry Which vertices form a triangle?  mesh connectivity Which triangles form a strip?  mesh stripification Storing a Stripified Mesh

Which vertices form a triangle strip?  mesh connectivity  mesh stripification Storing a Stripified Mesh Where are the vertices located ?  mesh geometry

list of vertices x 0 y 0 z 0 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of verticeslist of triangle strips x 0 y 0 z x 1 y 1 z x 2 y 2 z x 3 y 3 z 3 4x 4 y 4 z x n y n z n Standard Representation

list of verticeslist of triangle strips x 0 y 0 z x 1 y 1 z x 2 y 2 z x 3 y 3 z 3 4x 4 y 4 z x n y n z n Storage costs: 2~3 log(n) bpv Connectivity and Stripification

Connectivity Compression for Triangle Meshes

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] Compression Techniques

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] Short Encodings of Planar Graphs and Maps 4.6 bpv (4.6) Compression Techniques

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] Topological Surgery 2.4 ~ 7.0 bpv (--) Compression Techniques

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] Triangle Mesh Compression 0.2 ~ 2.9 bpv (--) Compression Techniques

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] Edgebreaker 3.2 ~ 4.0 bpv (4.0) Compression Techniques

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] A Simple and Efficient Encoding for Triangle Meshes 4.2 ~ 5.4 bpv (6.0) Compression Techniques

Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 99 ] Mesh Collapse Compression 1.1 ~ 3.4 bpv (--) Compression Techniques

strip-internal edges

bpv strip-internal edges

Triangle Fixer

Encoding scheme Define initial active boundary around arbitrary edge of mesh.

Define initial active gate as one of the two boundary edges. Encoding scheme

Label active gate with T, R, L, S, or E. Encoding scheme

Label active gate with T, R, L, S, or E. Which label ? Depends on its adacency relation with the boundary.

Encoding scheme Label active gate with T, R, L, S, or E. Which label ? Depends on its adacency relation with the boundary. Update active boundary and active gate.

Encoding scheme Label active gate with T, R, L, S, or E. Which label ? Depends on its adacency relation with the boundary. Update active boundary and active gate. Boundary is expanded (T), is shrunk (R and L), is split (S), or is terminated (E).

Label T before Active gate not adjacent to other boundary edge.

Label T beforeafter

Label R before Active gate adjacent to next edge along active boundary.

Label R beforeafter

Label L before Active gate adjacent to previous edge along active boundary.

Label L beforeafter

Label S before Active gate adjacent to some other edge of active boundary.

Label S Active gate adjacent to some other edge of active boundary.

Label S before Active gate adjacent to some other edge of active boundary.

Label S beforeafter

Label E before Active gate adjacent to previous and next edge along active boundary.

Label E beforeafter

Example Run: Encoding

T

T

T

T

T

R

T

T

T

T

R

T

T

T

T

R

T

H 10

R

R

R

R

R

R

R

R

E

Example Run: Decoding

E

R

R

R

R

R

R

R

R

H 10

T

R

T

T

T

T

R

T

T

T

T

R

T

T

T

T

T

Holes and Handles

Simple Mesh

Mesh with Holes

Mesh with Handle

Mesh with Handle and Holes

Encoding a hole use new label H associate integer called size with label H that specifies number of edges/vertices around hole one label H size per hole total number of labels remains equal to number of mesh edges

Label H size before hole Active gate adjacent to hole of size edges/vertices.

Label H size beforeafter hole

Encoding a handle use new label M associate three integers called index, offset 1, and offset 2 with label M that specify current configuration one label M idx,off 1,off 2 per handle total number of labels remains equal to number of mesh edges

Label M idx,off 1,off 2 before offset 1 offset 2 index in stack Active gate adjacent to some edge of boundary in stack.

Label M idx,off 1,off 2 beforeafter offset 1 offset 2 index in stack

Triangle Strip Compression

let triangle strips guide the mesh traversal replace label T with labels T R, T L, T B, and T E traversing along strip means progress for both: –encoding connectivity –encoding stripification

Label T R Triangle strip leaves triangle on the right. before

Label T R beforeafter

Label T L Triangle strip leaves triangle on the left. before

Label T L beforeafter

Label T B Triangle strip leaves triangle on the left and on the right. before

Label T B beforeafter

Label T E (case 1) Triangle strip ends in triangle. AND It’s the last triangle of current strip. It doesn’t follow a label of type T B. before

Label T E (case 1) beforeafter

Label T E (case 2) before Triangle strip ends in triangle. AND Not the last triangle of current strip. It follows a label of type T B.

Label T E (case 2) afterbefore

Example Run: Encoding

TLTL

TRTR

TRTR

TRTR

TETE

TBTB

TLTL

TRTR

TRTR

TLTL

TRTR

TRTR

TETE

TETE

H 10

R

R

R

R

R

S

L

L

E

R

R

E

Example Run: Decoding

T L T R T L T L T E T B T L T R T R T L T R T R T E T E H 10 RRR RRSLLE RRE calculate offset * Offset = 6

E

R

R

E

L

L

S

R

R

R

R

R

H 10

TETE

TE*TE*

TE*TE*

TE*TE*

TE*TE*

TRTR

TRTR

TLTL

TRTR

TRTR

TLTL

TBTB

TETE

TRTR

TRTR

TRTR

TLTL

Compressing the label sequence

map label types to unique bit-codes example: frequent labels are mapped to short bit-codes make mapping dependent on last label (e.g. one label memory) very simple implementation Fixed-Bit Encoding T  0R  01S  0001 L  001E  0011

approaches the entropy of the label sequence adaptive version three label memory due to small number of different symbols, probability tables require less than 4 KB implemented with fast bit operations Arithmetic Encoding

Results

Fandisk

Eight

Bunny

Triceratops

Skull

aac-3 vertices mesh Results triceratops bunny skull eight femur bishop shape fandisk fixed

Video

Many polygon meshes include additional information Surface Normals Texture coordinates Colours Mesh Properties

Mesh properties can be attached to Mesh Properties

Mesh properties can be attached to vertices Mesh Properties

Mesh properties can be attached to vertices faces Mesh Properties

Mesh properties can be attached to vertices faces corners Mesh Properties

Make stripification process difficult. Corner attributes must be consistent for vertices in strip. But... mapping attributes to strip corners is cheaper than mapping to triangle corners. Usually requires 1 bit per corner. Immediate savings of 50 % and more! Corner Attributes

Number of Corners strip vertices mesh triceratops bunny skull eight femur bishop shape fandisk triangles

Compact encodings for stripified triangle meshes. Exploit existing correlation between connectivity and stripification. Edge-based encoding has other useful applications (non-triangular meshes). Summary and Current Work

Bettina Speckmann for fruitful discussions on triangle strips Xinyu Xiang for triangulating some of my test models my supervisor Jack Snoeyink for reviewing the paper Acknowledgements