04 MPEG-4, 1 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June D Compression Jarek Rossignac GVU Center and College of Computing Georgia Tech, Atlanta

04 MPEG-4, 2 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Definition –Compact coding of 3D representations –Efficient compression/decompression algorithms Issues –Domain –Accuracy –Ratio –Codec performance What is 3D Geometry Compression? Binary format Compress Decompress

04 MPEG-4, 3 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 The compression problem Given the boundary of a feature (triangle mesh), encode its interior –Location (coordinates) specifying where the interior vertices are –Triangle/vertex incidence, defining how the surface interpolates the points vertex location incidence Boundary? x y z x y z x y z vertex 1 vertex 2 vertex 3 Triangle 2 Triangle 1 Triangle 3 Previously decoded for a lower-resolution model

04 MPEG-4, 4 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 What to decode first? geometryincidencegeometryincidence Could estimate incidence from geometry (Delaunay, encode only edge flips) Could estimate geometry from incidence (relaxation, encode corrective vectors) Could estimate each vertex from previous vertices and from partial incidence ? ? ?

04 MPEG-4, 5 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Must decode connectivity first Cannot use geometry to estimate connectivity, because connectivity is used to predict geometry (see vertex-data compression section). Must develop connectivity compression methods independent of the vertex locations. connectivity geometry corrective

04 MPEG-4, 6 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June to-1 compression ratio x y z x y z x y z vertex 1 vertex 2 vertex 3 Triangle 2 Triangle 1 Triangle 3 Independent triangles: 576V bits (coordinates as floats 9x32T bits) Each vertex encoded ~ 6 times x y z x y z 7 x y z 10 vertex 1 vertex 2 vertex 3 Strip 2 Strip 1 L RR Triangle Strips: 214V bits (1.1x3x4B/T + 1B/S + 1b/T) Each vertex encoded ~ twice V and T tables Example: V=1000, B=10, k=0 Samples+incidence storage costs: (3B+6log 2 V)V = (30+60)V = 98V bits x y z c vertex 1 vertex 2 vertex 3 x y z c Triangle Triangle Triangle Triangle 2 Triangle 3 Triangle V(3B+k) bits V(6log 2 V) bits Can be compressed to: (14+2)V = 16V bits

04 MPEG-4, 7 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Connectivity compression: An old challenge Tutte: Theoretical lower bound (Tutte 62): can’t guarantee < 1.62 bits per triangle Itai,Rodeh: Representation of graphs, Acta Informatica, 82 Turan: On the succinct representation of graphs, Discrete Applied Math, 84: 6Tb Naor: Succinct representation of unlabeled graphs, Discrete Applied Math, 90 Keeler,Westbrook: Encoding planar graphs, Discrete Applied Math, 93: 2.3Tb Deering: Geometry Compression, Siggraph, 95 Denny,Sohler: Encode 2D triangulation as permutation of points, CCCG, 97: 0Tb Topological Surgery Taubin&Rossignac Edgebreaker Rossignac&team Valence-based Gotsman&team Cutborder Gumhold&Strasser Spirale Reversi Snoeyink&Isenburg

04 MPEG-4, 8 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 A simple and elegant solution (?) Connecting vertices into a single spiral (Hamiltonian walk) defines the left and the right boundaries (walls) of a long corridor. Given the left and right boundaries of a triangle strip (corridor), we need T (left/right) bits to encode its triangulation. ex: LRRLLRLRR L RR Store vertices in their order along the wall. (Can use former vertices to predict location of new ones.) Encode connectivity using only 1 (left/right) bit per triangle ! LLL R R R

04 MPEG-4, 9 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 But..but..wait a minute! It doesn’t work! The corridor may have warts R ? Warts are hard to avoid ? The spiral may bifurcate The corridor may bifurcate L R

04 MPEG-4, 10 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June Tb encoding of VST and TST suffices Guaranteed 3Tb connectivity (2Tb T-tree + 2Vb V-tree)

04 MPEG-4, 11 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June T bit encoding details The triangle spanning tree may be encoded using 2T bits: has-left-child has-right-child A vertex spanning tree may be encoded using 2V bits (= 1T bits): has-children has-right-sibling

04 MPEG-4, 12 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June T bits (Turan 84) Turan: On the succinct representation of graphs, Discrete Applied Math, 84 “Succinct representation of graphs” –Discrete Applied Mathematics Incidence = planar graph Decompose graph into VST and TST Total cost 6T bits (12v-24) guaranteed –Compare to my 3T bit code for both trees and to 1.83T bits of Edgebreaker

04 MPEG-4, 13 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Topological Surgery (TS) Taubin-Rossignac IBM’95, VRML, MPEG-4 “Geometric compression through Topological Surgery,” G. Taubin and J. Rossignac, ACM Transactions on Graphics, vol. 17, no. 2, pp. 84–115, “Geometry coding and VRML,” G. Taubin, W. Horn, F. Lazarus, and J. Rossignac, Proceeding of the IEEE, vol. 96, no. 6, pp. 1228–1243, June 1998.

04 MPEG-4, 14 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Run-length encoding of TST and VST Most nodes have a single child. - Group them into runs - Store structure of each tree (2 bits per run) - Store left/right bit per triangle

04 MPEG-4, 15 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Run-length codes detail bit per run (leaf?) + 1 bit per triangle (Left/Right) bit per run (leaf?) + 1 bit per run (last child?) +

04 MPEG-4, 16 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 TS example

04 MPEG-4, 17 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Spiral is better than other traversal orders

04 MPEG-4, 18 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Topological Surgery Compression Cut mesh by spiraling vertex-tree –Few runs in practice Encode vertex tree –1 Integer/run: Length –2 bits/run: structure Encode triangle tree –1 Integer/run: Length –1 bit/run (leaf?) –1 bit/triangle (left?) Label vertices as visited in vertex tree –Need not send vertex id (saves log(v) bits) –Encode corrections to predicted locations (discussed later)

04 MPEG-4, 19 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Topological Surgery decompression

04 MPEG-4, 20 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Results TS (MPEG-4) vs VRML

04 MPEG-4, 21 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Topological Surgery: Results and Impact Results: –Compresses to less than 10 bits per triangle Connectivity: about 2.0 T bits (increases for smaller meshes) Geometry: 6 bits per coordinate (decreases with tessellation&quantization) Publications and impact of Topological Surgery: –Interfaced to VRML 2.0 offered by IBM (1995) –Core of the MPEG-4 standard for 3D Compression –Geometric compression through topological surgery, Taubin&Rossignac ACM Transactions on Graphics, 17(2):84-116, April 1998 (Best Paper Award) –Geometry coding and VRML, Taubin, Horn, Lazarus, & Rossignac Proceedings of the IEEE, 96(6): , June 1998 –Inspired several approaches Touma,Gotsman: Triangle Mesh Compression, GI, 98 Gumbold,Straßer: Realtime Compression of Triangle Mesh Connectivity, Siggraph, 98 Taubin,Gueziec,Horn,Lazarus: Progressive forest split compression, Siggraph, 98

04 MPEG-4, 22 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 TS spiral joins topological layers

04 MPEG-4, 23 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Inserting alternate layers (Bajaj et al.) C. L. Bajaj, V. Pascucci, and G. Zhuang, “Single resolution compression of arbitrary triangular meshes with properties,” Computational Geometry: Theory and Applications, vol. 14, pp. 167–186, 1999.

04 MPEG-4, 24 Jarek Rossignac, CoC & GVU Center, Georgia Tech SM, June 2002 Summary Topological Surgery compression cuts the mesh along a spiraling VTS and encodes the VTS and its dual TST using RLE It encodes vertices in the VST order using entropy codes for residues over predictors that are linear combinations of ancestors in the VST It has been improved in various ways and forms the core of the MPEG-4 3D compression standard It encodes most meshes that have more than 300 vertices with less than 20 bits per vertex. It does not provide a guaranteed linear cost per vertex and requires between 20 and 100 bits per vertex for meshes of less than 100 vertices The implementation is somewhat complex but efficient