Martin Isenburg Jack Snoeyink University of North Carolina Chapel Hill Mesh Collapse Compression

Introduction A novel scheme for encoding triangle mesh topology.

Introduction A novel scheme for encoding triangle mesh topology. –triangle mesh

Introduction A novel scheme for encoding triangle mesh topology. –triangle mesh –encoding

Introduction A novel scheme for encoding triangle mesh topology. –triangle mesh –encoding –topology

Triangle Mesh Computer Graphics Surface description Hardware support Used everyday Used everywhere Used by everybody Increasingly complex

Encoding Compressed representation Decrease storage and transmission time

Topology geometry versus topology x 0 y 0 z x 1 y 1 z x 2 y 2 z x 3 y 3 z x 4 y 4 z x n y n z n.....

Overview previous work simple meshes video example general meshes results summary

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ]

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Geometry Compression

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Short Encodings of Planar Graphs and Maps 4.6 bpv (4.6)

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Topological Surgery 2.2 ~ 4.8 bpv (--)

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Triangle Mesh Compression 0.2 ~ 2.9 bpv (--)

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Edgebreaker 3.4 ~ 4.0 bpv (4.0)

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] A Simple and Efficient Encoding for Triangle Meshes 4.2 ~ 5.4 bpv (6.0)

Previous work Deering [ 95 ] Keeler Westbrook [ 95 ] Taubin Rossignac [ 96 ] Tauma Gotsman [ 98 ] Rossignac [ 98 ] DeFloriani et al [ 99 ] Isenburg Snoeyink [ 00 ] Face Fixer 3.9 ~ 4.1 bpv (6.0)

Simple Meshes (1) the mesh is a surface composed of triangles (2) the mesh has no boundary (3) the mesh has no holes (4) the mesh has no handles

Simple Mesh

Mesh Collapse Compression i Input:Output: a simple mesh- a sequence of code words - a permutation of vertices

Compression Scheme (1) initialize - declare arbitrary directed mesh edge as current edge

Compression Scheme (2) loop until done (and usually): - contract current edge - record the removed vertex - record the degree - select new current edge

Compression Scheme (2) loop until done (but occasionally): - divide mesh along current edge - record start symbol - continue on one mesh part - record end symbol - continue on other mesh part

Digons A triangulation with exception of the outer face that is bound by only two edges.

mc-edge From Mesh to Digon mc-vertex cut and openrearrange new mc-edge

Various Digons simple trivialcomplex dividing edge

Encode Algorithm encode( Mesh mesh, Codec codec ) { stack.push( digonify( mesh ) ); while ( not stack.empty( ) ) { digon = stack.pop( ); while ( not digon.trivial ( ) ) { if ( digon.complex() ) { subdigon = mc-divide( digon ); stack.push( subdigon ); codec.pushCode( “S” ); } else { degree = mc-contract( digon ); codec.pushCode( degree ); } codec.push( “E” ); }

mc-contract contract mc-edge removeselectcontract new mc-edgeloop

mc-divide divideselect dividing edgemc-edge

Video

Example

mc-vertex

Example mc-edge

Example digonify mc-edge

Example

mc-contract degree of removed vertex removed vertex

Example

mc-contract removed vertex degree of removed vertex

Example

mc-contract removed vertex degree of removed vertex

Example

mc-contract removed vertex degree of removed vertex

Example

mc-divide divided vertex

Example mc-divide divided vertex

Example push on stack

Example

mc-contract degree of removed vertex

Example

trivial digon trivial digon

Example trivial digon

Example pop from stack

Example

mc-contract degree of removed vertex

Example

mc-contract degree of removed vertex removed vertex

Example

trivial digon trivial digon

Example trivial digon

Example

mc-tree

Example mc-tree embedded mc-tree

Example mc-tree

Example Vertex permutation Code word sequence mc-encodingmc-tree

Decode algorithm decode( Mesh mesh, Codec codec ) { while ( not codec.empty( ) ) { code = codec.popCode( ); if ( code == “E” ) { stack.push( digon ); digon = new Digon( ); } else if ( code == “S” ) { subdigon = stack.pop( ); mc-join( digon, subdigon ); } else { mc-expand( digon, code ); } mesh = undigonify ( digon ); }

General Meshes (1) the mesh is a surface composed of triangles (2) the mesh can have a boundary (3) the mesh can have holes (4) the mesh can have handles

Simple Mesh

Boundary

Holes

Handle

Handle & Boundary

Handle & Holes

Encoding a Boundary mc-vertex mc-edgeadded edge

Encoding a Hole holeadded vertex

Encoding a Handle digon in stack After mc-contract: M + index + [0..5] After mc-divide: M + index + [0..1] Hiding in a trivial digon: M + index + [0..7]

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Results vertices/triangles bpvmesh 250/ ~ 1.7bishop 1524/ ~ 2.8bunny 1502/ ~ 3.4dragon 2832/ ~ 2.7triceratops 2655/ ~ 2.7beethoven 2562/ ~ 1.2shape

Summary We presented a novel scheme for encoding the topology of triangular meshes. The achieved compression rates of 1.1 ~ 3.4 bits/vertex compete with the best results known today. Currently we work on a progressive version of mc-compression.

Acknowledgements University of British Columbia at Vancouver, Canada Gene Lee and his RASP tools International Computer Science Institute (ICSI) at Berkeley, USA funding: NSERC, IRIS, UGF