Mesh Collapse Compression Martin Isenburg Jack Snoeyink University

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

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 011101101011. . .

Topology geometry x 0 y 0 z 0 x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 x 4 y 4 z 4. . . x n y n zn versus topology 0 1 2 2 3 0 2 3 1 2 4 3 5 4 3. . . .

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] • De. Floriani et al [99] • Isenburg Snoeyink [00]

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

Previous work • Deering [95] • Keeler Westbrook [95] • Taubin Rossignac [96] • Tauma Gotsman [98] • Rossignac [98] • De. Floriani et al [99] • Isenburg Snoeyink [00] Short Encodings of Planar Graphs and Maps 4. 6 bpv (4. 6)

Previous work • Deering [95] Topological Surgery • Keeler Westbrook [95] 2. 2 ~ 4. 8 bpv (--) • Taubin Rossignac [96] • Tauma Gotsman [98] • Rossignac [98] • De. Floriani et al [99] • Isenburg Snoeyink [00]

Previous work • Deering [95] • Keeler Westbrook [95] • Taubin Rossignac [96] • Tauma Gotsman [98] • Rossignac [98] • De. Floriani 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] • De. Floriani et al [99] • Isenburg Snoeyink [00] Edgebreaker 3. 4 ~ 4. 0 bpv (4. 0)

Previous work A Simple and Efficient Encoding for Triangle Meshes 4. 2 ~ 5. 4 bpv (6. 0) • Keeler Westbrook [95] • Deering [95] • Taubin Rossignac [96] • Tauma Gotsman [98] • Rossignac [98] • De. Floriani et al [99] • Isenburg Snoeyink [00]

Previous work • Deering [95] • Keeler Westbrook [95] • Taubin Rossignac [96] • Tauma Gotsman [98] • Rossignac [98] • De. Floriani 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.

From Mesh to Digon mc-edge mc-vertex new mc-edge cut and open rearrange

Various Digons dividing edge simple trivial complex

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. push. Code( “S” ); } else { degree = mc-contract( digon ); codec. push. Code( degree ); } } codec. push( “E” ); } }

mc-contract mc-edge contract loop contract new mc-edge remove select

mc-divide dividing edge divide mc-edge select mc-edge

Video

Example

Example mc-vertex

Example mc-edge

Example mc-edge digonify

Example

Example removed vertex degree of removed vertex mc-contract

Example

Example removed vertex degree of removed vertex mc-contract

Example

Example removed vertex mc-contract degree of removed vertex

Example

Example removed vertex mc-contract degree of removed vertex

Example

Example divided vertex mc-divide

Example push on stack

Example

Example mc-contract degree of removed vertex

Example

Example trivial digon

Example pop from stack

Example

Example mc-contract degree of removed vertex

Example

Example removed vertex mc-contract degree of removed vertex

Example

Example trivial digon

Example

Example mc-tree

Example mc-tree embedded mc-tree

Example mc-tree

Example mc-encoding Code word sequence Vertex permutation mc-tree

Decode algorithm decode( Mesh mesh, Codec codec ) { while ( not codec. empty( ) ) { code = codec. pop. Code( ); 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-edge added edge

Encoding a Hole hole added 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 bpv bishop 250/496 1. 4 ~ 1. 7 bunny 1524/3044 2. 7 ~ 2. 8 dragon 1502/3000 3. 1 ~ 3. 4 triceratops 2832/5660 2. 4 ~ 2. 7 beethoven 2655/5030 2. 5 ~ 2. 7 shape 2562/5120 1. 1 ~ 1. 2 mesh

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